The present review is devoted to the current status of microfield notion that was so successful and profitable for experimental and theoretical studies of plasma in gas discharges and thermonuclear modeling installations for many decades. The physical aspects and ideas of the main generally used microfield models are described and analyzed in detail. The review highlights the remaining vague and unclear questions in the subject.

1. Microfield Notion

1.1. The Term of Microfield

The term “microfield” was introduced to designate the electric and magnetic fields, whose action is essential on microscales intra different media [112]. This was done to distinguish microfield from the fields of other origin essential, for example, for macroscopic description of a medium. As a rule the average microfield over macroscopic volume is equal to zero.

Plasma on microscales is characterized by noticeable deviation from quasineutrality conditions and appearance of strong electric fields due to separation of charges [312]. Namely, those electric fields, essential on microscales, usually are implied under term “microfield.” The magnitude of this field and its direction are subjected to fortuitous variations from point to point in space and in time.

Thus, from the very beginning, the microfield calculations represent itself challenging, complex, statistical, and kinetic problems. Being defined by the medium properties and composing it separate particles, the microfields action in its turn affects the medium characteristics and physical processes between these composing particles. Hence, the physical phenomena that somehow or other became involved and connected with microfields are very diverse. The voluminous literature [1198], which is not confined so far by the named list and devoted to the study of various physical processes related to microfield characteristics and its affect on medium properties and composing it particles, just confirms the variety of aspects and complexity of a problem.

The characteristics of microfields could be, in principle, determined with the help of hydrogen-like atoms placed inside plasma, which experience the Stark or Zeeman effects in electric or magnetic fields correspondingly [1, 2]. Those effects lead to the line splitting into separate sublines—Stark or Zeeman components. Thereby, the simplest quantum systems could serve as some kind of microprobes for the measurements of plasma parameters on microscales and perform the role of the so-called test particles. The measured signal from these microprobes on microscales is their emission in spectral lines or other spectral characteristics, perturbed by plasma environment.

However, the emission of spectral lines practically impossible to register locally from the volume with characteristic microsizes. That is why the radiation is registered simultaneously from different microvolumes. As far as the probability of field realization with the given magnitude and direction is different in space, this is equivalent to average of observed spectral lines profiles over the field configurations with various microfield magnitudes and directions, which leads to some extent to the smoothed-broadened contour.

Basing on pointed out dependencies spectral lines of atoms, molecules and ions with simple energetic structure are used for diagnostics of plasma parameters [1, 2]. Usually the methods of measurements correspond to the so-called passive diagnostics, when the observed quantities are the distributions of intensity and polarization in discrete spectrum, emitted by plasmas. However, as meanwhile the measurements mainly has integral character, the success of their interpretation depends on construction of adequate model notions on the interaction of radiator with plasma medium, better corresponding to observed characteristics.

The formation of spectral line contour is influenced by dynamics of interaction of radiator of the electric field with that or another frequency spectrum, and by statistics of such interactions, describing the average over probabilities of appearance of the fields in plasmas. The real problem is complicated due to the strong difference in masses of negative and positive charges in plasmas, which leads to the strongly differing characteristic time scales of corresponding electric fields alterations [312]. For example, in equilibrium plasmas with density in a range of 1017 cm-3 and temperature about 1 eV, the ions of the electric fields vary more slowly than the electrons ones. So, the conventional picture of spectra formation is composed by splitting the energy levels in slowly varying ion microfield into Stark sublevels, broadening of these sublevels due to transitions between them, induced by more swift electron flights, and further averaging of spectrum over ion microfield distribution in plasmas [1, 2].

Near the series limit, the lines strongly overlap, and their intensity starts to decrease due ionization in plasma microfields [1]. However, the contribution of continuum noticeably increases in this region, and that is why visually the lines, located in sequence of decreasing intensity to the series limit, look as if ascending up the hillside, describing the increasing intensity of continuum.

For emitters with more complex internal structure, the contribution of line satellites, induced by transitions from doubly excited states of ions with preceding ionization stage, becomes important. On the other hand, under plasma creation by femtosecond laser pulses ionization evolves from K—and L—atomic shells of the targets, and the ions of hole configurations are created. In this case, the observed spectrum acquires quasicontinuous character. The plasma microfield even in these more complex conditions noticeably modifies the discrete spectrum of radiation.

The plasma microfield is stipulated as by Coulomb electric fields of charged particles, as by self-oscillations of plasma, playing the decisive role in nonequilibrium conditions. These fields are subdivided by terms of “individual” and “collective” components of microfield, respectively [1, 3].

In the wide range of plasma parameters, the quasistatic approximation is efficient for the description of interactions with ion microfield. It is grounded on the notion of instantaneous static microfield distribution function [1, 2]. However, in these conditions, the broadening by some part of ions has impact character, and this is of principal significance for a family of simulation methods [13].

It would seem, from general considerations, that the solution of spectral line broadening problem in a medium could be found using only statistical, and even, moreover, thermodynamic methods. However, in truth, a phenomenon of spectral lines broadening has inseparably linked to each other dynamical and statistical aspects. For example, the processes of spectral line shape formation and population of quantum states are interrelated, and have to be considered self-consistently [108, 109]. The important factor in finding the solution is physically a correct choice of zero-order wave functions of a problem and its direction of quantization, adequately corresponding to physical observables [53].

Thus inadequacy of only statistical or only dynamical descriptions of a problem makes necessary the search of solutions based more or less on synthesis of these notions [104109]. To a considerable extent, the necessity of such synthesis is stipulated also by actually restricted power of recent supercomputers for numerical modeling of complex multidimensional problems [1417].

It should be noted that the whole row of phenomena exists in which microfield plays the important role but more amply its characteristics show up just in spectra of atoms and ions, immersed into plasmas. That is why in this introductory part the main attention was paid for the broadening of spectral lines.

This work presents the review of current ideas about plasma microfields, physical models, and methods for describing the quasistatic instantaneous distribution functions and temporary microfield evolution. The most ample previous reviews of this problem could be found in [1, 2, 11, 1417] and the recently published papers [195198].

1.2. Dipole Approximation as Basement of Microfield Formalism

So, a consideration of medium influence on test particles serves as a source of information on origin and character of interactions in various media and in its turn about the media state.

In plasmas this impact is due first of all to charged particles—plasma electrons and ions. If to expand the interaction potential of test particles with the medium into series over multipoles, assuming large remoteness of the medium (field) particles from the test ones in comparison with distances between the test particles, then the first term of expansion becomes zero due to condition of quasineutrality. (Here the case of charged plasmas, where this condition does not fulfill is not considered.)

The first not equal to zero term of this expansion just is due to the electric fields of plasma particles and proportional to the scalar product of the vector of dipole moment of a system of test particles and the summary electric field strength vector of plasma particles. This summary electric field of medium on microscales, becoming zero under average over macrovolume due to quasineutrality condition, was called microfield, as its action shows up at microscales, where the quasineutrality condition does not hold and the charge separation is essential.

Thereby, a possibility to describe the test particles interaction with environment (plasma) in terms of microfield is linked with conditions of predominance of long range components of potential over short range ones, when the distances between particles in a test system are less than the distances between particles of a medium. On the other hand, the possibility of such description depends on the existence of dipole moment in a test system. That is why approximate representation of potential in terms of microfield corresponds to the dipole approximation.

In the case of the electric fields of collective plasma oscillations, the implementation of dipole approximation is evidently admissible, as the sizes of test systems are typically much less than the wavelength of those oscillations.

1.3. Applicability Criteria for Quasistatic Approach

Notion of quasistatic microfield is based as a rule on a simple reasoning that summary electric plasma microfield does not alter on some effective for radiation time scales [1, 2]. Within such settings, this condition turns out depending not only on microfield statistical properties but also on quantum properties of a radiator. For example, the smallness of frequency of temporary microfield changes 𝐹 in comparison with the hydrogen atom dipole moment 𝑑𝑛 (𝑛—the principal quantum number) [1, 2] frequency precession in this field is considered as such aforementioned condition:

𝑑𝑛𝑛𝐹||||̇𝐹𝐹||||.(1) For the other condition of this kind, the smallness of life time of atom quantum state 𝜏e in comparison with characteristic life time of microfield might serve or, when the characteristic frequency of atomic decay exceeds the characteristic frequency of microfield changes

||||̇𝐹𝐹||||𝜏1e,(2) indicating that an atom could not have enough time to response to temporary microfield variation. Often both these conditions are considered in aggregate with each other.

Besides the mentioned criteria, which are called “integral,” there are other types of conditions, requesting, for example, smallness of spectra variations, calculated using quasistatic microfield distribution functions 𝑊𝑠𝑡(𝐹) with small corrections 𝐼𝑠𝑡(Δ𝜔)+𝛿𝐼(Δ𝜔), accounting to microfield evolution with time 𝑊𝑠𝑡(𝐹)+𝛿𝑊(𝐹(𝑡))

𝐼𝑠𝑡𝑊(Δ𝜔)𝛿𝐼(Δ𝜔),𝑠𝑡(𝐹)𝛿𝑊(𝐹(𝑡)),(3) where Δ𝜔=𝜔𝜔0, 𝜔 is the circular frequency of radiation, 𝜔0 is the unperturbed circular frequency of transition.

Per se this requires the complete solution within perturbation theory in assumption of small effective times [1, 2]. Such type of criteria dependent on circular frequency detuning from the line centers Δ𝜔 are used to call “spectral.”

More definitive quantitative characteristics are provided by integral and spectral criteria, derived from consideration of power law potentials of binary interaction of particles with respect to problems of spectral line broadening theory (see [2]).

1.4. Quantum and Classical Theory

The necessity of quantum microfield description mainly appears in connection with degeneracy of electron plasma component [7]. That is why from practical point of view the account of “quantumness” or the extent of degeneracy of electron gas in this concrete case touches upon mainly the character of plasma ions shielding by electrons [312]. Prescriptively, this could be reduced to the function of plasma ions shielding by electrons, which sufficiently well describes all limiting cases (see [712]).

However, for example, for plasma of metals very often, the range of parameters, where the effective charge of field ions noticeably differs from the charge of bare nuclear, is of main interest. Then, the appearance of quantum exchange and correlation effects due to ion core becomes essential. Evidently, the consequent account of quantum structure of radiator also has definite contribution. The description of these effects was suggested to perform in terms of formalism of local density functional, the application of which will be discussed in Section 2.7.

Additionally, for very low temperatures, the account of quantum description might become necessary even for the translational motion.

1.5. Significance of Models

We have to comprehend that plasma is a medium with very complicated physical characteristics [311].

Namely, due to this complexity, it was not possible to elaborate universal rigorous and self-consistent theory of plasma microfield in spite of numerous papers published on the subject up to now [1198]. However, each time, some tractable but limited picture is achieved only in the frames of more or less trustable assumptions, obvious physical ideas, some solvable mathematical formalism, and various approximations. All the aforementioned components together constitute that or another physical model for microfield description.

For example, plasma could be considered as continuous medium [175] or as medium, which constitutes from many separate discrete particles [18]. Indeed, the commonly used ion-sphere model for microfield description is the typical sample of continuous models (see, e.g., [11, 45, 110]). So, it is natural to divide models on continuous and discrete ones. There are also some mixed models, where continuous and discrete approaches are applied to the different subsystems (see, e.g., [9, 11, 110]). One can consider point particles [18] and particles with finite sizes as well [176].

The deviation of plasma main parameters temperature and density also provides a variety of physical conditions—weakly and strongly coupled plasmas [9, 11], nonrelativistic and relativistic plasmas [177], degenerate electron plasma component [43], and so forth.

There is also a lot of complications connected with the choice of interaction potential, that is different for weakly and strongly coupled plasmas, for movings particles and particles at rest [42, 125]. Its working form depends on effective characteristic time scales that are prescribed to the microfield action [84, 85, 135137], which in their turn due to Fourier transform could be determined further by detunings from the line center [83, 95].

The microfield in plasma could be due to many-body interactions with discrete charged particles, or due to plasma self-oscillations or plasma waves [18, 30, 31, 4952]. Moreover, it is important on what space and time scales it is necessary to define microfield. The space scales could be limited by formalism as well, and introduction of additional constraints such as energy conservation law [178, 179]. Indeed, for weakly coupled plasma on the microdistances less than Debye radius, the fluctuations of energy of particles is of the order of temperature. So, there is no reason to implement energy conservation law, but on macroscales, the fluctuations are much smaller and this restriction starts to hold. Interestingly, both mentioned restrictions or usage of the formalism, which from very beginning is derived for macro scales like the formalism of dielectric functions, should lead to the different from discrete models microfield distributions. In this context, we remind the old dispute around Hunger and Larenz works [178], who obtained instead of Holtsmark distribution Gauss type distributions introducing additionally energy conservation law constraints. The illuminating analysis of these results and polemics around them is presented in the work of Kogan and Selidovkin [179], who found explicit mistakes in analytical derivations of Hunger and Larenz works.

The distribution functions of microfield conventionally are obtained in the case of the so-called thermodynamical limit for 𝒩 and 𝒱, so that 𝒩/𝒱=const=𝑁 (𝒩 is total number of particles in the system, 𝒱 is the system volume, 𝑁 is the density of particles) [1416]. However, it is possible to introduce microfield distribution functions for the finite number of particles as well [135137]. There are obvious contradictions between various views on microfield definitions. For example, the different notions of instantaneous static ion microfield and ion microfield, obtained as a result of thermodynamic average, appeared from the consideration of the same physical object. However, the difference between output distributions, based on the distinct initial assumptions could result only in difference of the shielding constants that had to be used in the expression of “elementary” ion microfield.

The classification of microfield models in terms of their accounting for correlations between subsystems of plasma electrons and plasma ions was presented, for example, in the work of Ortner et al. [188]. These plasma models accounting for correlations are called “Two Component Plasma” (TCP) models in distinction from “One Component Plasma” (OCP) models [9].

The microfield models are additionally subdivided on those that attempt to describe static fields and ones depending on time. Using some assumptions on microfield statistics and some other approximations, the Method of Model Microfield (MMM) [8694], Collective Coordinates for Ion Dynamics [180], Frequency Fluctuation Model (FFM) [99, 100], and Frequency Separation Technique (FST) [181] were proposed. Also, the direct computer simulations methods were elaborated firstly for static microfield distributions like the Monte-Carlo method [2729, 3539, 110, 113] and after for modeling the evolution of electric microfields versus time: Computer Simulations (CS) with particles moving along prescribed type of trajectories [101105, 142145] and Method of Molecular Dynamics (MD) [107, 109, 135139, 182, 183].

Thus we see that notions of microfield and models that are designed for its description are complicated and diverse, reflecting the diverse and complex plasma properties.

2. Quasistatic Distribution Functions

It is used to distinguish (although it could be done only approximately) plasma microfield additive components, having essentially different frequency and spatial characteristics. Firstly, it is possible to single out the electric fields of high and low frequency plasma collective oscillations and individual component of electric microfield, being a summary field of separate plasma particles.

Furthermore, an individual component in its turn could be divided into high frequency, induced by plasma electrons, and low frequency, induced by plasma ions, parts. Evidently, such separation should happen automatically under implementation of sufficiently adequate mathematic approaches to the complete system and specifics of that or another problem. Although such attempts were done, they did not lead to formulated goal. In fact, as was underlined in the previous subsection, the microfield theory is constructed based on model and intuitive ideas as necessary solutions for a whole row of problems could not be obtained using conventional thermodynamic methods.

The interaction of the point field ions with an emitter in dipole approximation could be represented in terms of the electric ion microfield 𝐹 in assumption, that perturbing particles are situated sufficiently far from emitter, so that the radius-vector of radiating electron is much less than radius-vectors of perturbing particles with respect to the emitter nuclear. Using the condition of vector additivity of electric fields of all ions 𝐹𝑗, we have

𝐹=𝑗=0𝐹𝑗.(4) Then the statistical microfield distribution function 𝐹 could be obtained from the next thermodynamic average

𝑊𝐹=𝛿𝐹𝑗=0𝐹𝑗,(5) where symbol designates the average over plasma ensemble of ions. Moreover, as a rule, this average encircles passage to the limit, under which the number of particles (ions) in ensemble 𝒩𝑖 and the system volume 𝒱 are indefinitely increasing, while their ratio is kept constant and equal to the particle density lim𝒩𝑖,𝒱𝒩𝑖/𝒱=𝒩𝑖.

The field strength of electric microfield 𝐹 and its components 𝐹𝑗 in assumption of a point test particle is evaluated in the place of its localization, which usually is chosen as an origin of reference frame of coordinates (in the case of test particles at rest).

The average value calculation, mentioned earlier, is a complex problem due to its many-body character, vector properties of quantities under evaluation, multicomponent system of plasma particles, correlations and interactions between them, and specific peculiarities of a test system.

It is important to comprehend what a function in the sense of performed average character would more correctly correspond to the posed problem. From the arsenal of mathematical methods of statistical physics the average over canonical or microcanonical ensembles, chaotic phases, fast subsystems, and so forth [312] could be recovered.

However, for obtaining such averages as a rule, the infinite time interval is needed, while the used in many physical solutions Fourier transform itself limits the effective duration of time average. For example, under Fourier transform for line profile calculation at the circular frequency detuning from the line center Δ𝜔=𝜔𝜔0, the effective time of profile formation is of the order Δ𝜔1, where 𝜔, 𝜔0 are perturbed and unperturbed circular frequencies of radiation. The value Δ𝜔1 determines thereby the allowed characteristic scales of average over the time of stochastic variables entering expression for line contour versus the frequency detuning. As a rule, it is implicitly assumed that instantaneous distribution function of ion microfield, when the average could be performed before ions would change essentially their space configuration, is used. Here is evidently some mismatch of descriptive methods and requested from physical consideration result. However, spectra depend not only on line profiles but also on spectral lines intensities, proportional to population of excited levels. The populations in many cases are determined by the balance of thermodynamically equilibrium processes. Hence, the real situation is rather diverse. The elaborated up-to-date approaches give only approximate solutions for the aforementioned row of problems. In this paper, only those that are used more often will be enlightened.

2.1. Hotltsmark Function

Historically, the Holtsmark function [18] became the first and physically significant solution of a problem of static microfield distribution, derived for isotropic ideal gas of charged particles with the same sign of charges [1, 2, 1421]. This function describes the probability of outcome for ions configuration for the given value of microfield module 𝐹 without account of plasma ion-ion and ion-electron interactions versus the reduced dimensionless microfield value 𝛽=𝐹/𝐹0, where 𝐹0=2𝜋(4/15)2/3𝑒𝑍𝑖𝑁𝑖2/3 is the normal Holtsmark field value:

(𝛽)=2𝛽𝜋0𝑑𝑥𝑥sin(𝛽𝑥)exp𝑥3/2.(6) The important characteristic of this distribution is its asymptotic behavior at small 𝛽1 and large 𝛽1. At large 𝛽, it is proportional to 𝛽5/2 and stems to the distribution of the nearest neighbor, which corresponds to small distances between the perturbing and emitting (test) particles. At small 𝛽, this distribution is proportional to 𝛽2, which corresponds to the many-body law of summary field formation at large distances between field particles, when the field values due to separate particles are small. Those asymptotic dependencies in fact are universal features and of more realistic microfield distributions [1, 2, 1417]. The basic technical element for obtaining this and and other results is the Fourier-transform of 𝛿-function, which allows to reduce the problem in the isotropic case to calculation of characteristic function 𝐴(𝑘):

𝑊(𝛽)=2𝛽𝜋0𝑑𝑘𝑘sin(𝑘𝛽)𝐴(𝑘),0𝑑𝛽𝑊(𝛽)=1.(7) This expression is universal and based only on isotropy of distribution function, does not depend on density, which enters only in the definition of the normal field. At the same time, the functional dependence of ln𝐴(𝑘) is determined by the Coulomb law of electric field. The graph of universal Holtsmark distribution function will be given in what follows in comparison with more sophisticated distributions of Ecker-Müller [22, 23], and Hooper [2729].

2.2. Ecker-Müller Distribution

The first step to account of plasma specifics became the Ecker and Ecker-Müller microfield distribution functions [22, 23]. In its derivation, it is assumed that the potential of plasma field ion is shielded by plasma electrons and obeys Debye law. The interaction between field ions is neglected. So, the difference from the Holtsmark distribution is only using the expression for electric field for plasma ions, shielded by plasma electrons according to Debye:

𝐸𝑟=𝑍𝑝𝑒𝑟1+𝑟De𝑟exp𝑟De𝑟𝑟3,(8) where 𝑍𝑝𝑒-is the charge of field ion, 𝑒 is the electron charge, 𝑟𝑑 is the electron Debye radius [311]. In various publications, the total Debye radius is substituted in this expression, simultaneously including the shielding by electrons and ions [1417]. However, from physical point of view, it is not always justified.

The Ecker-Müller distribution became a function of two variables—the reduced electric field value 𝛽 and dimesionless parameter 𝛿, proportional to the number of field ions in the Debye sphere:

𝛿=4𝜋3𝒩𝑖𝑟3De.(9) However, later, the labeling of distribution functions with the parameter

𝑎=𝛿1/3=𝑅0𝑟De(10) became conventional, where 𝑅0 is the mean distance between field ions.

For weakly coupled plasmas, only for which there is a sense to apply this distribution, the parameter value 𝑎 is limited from above by unity. At 𝑎=0, the Ecker-Müller distribution coincides with the Holtsmark distribution, and its maximum is shifted to the lesser reduced values of microfield while parameter 𝑎 is increasing. As due to quasineutrality condition, the ion density could be expressed via electron density, and from the aforementioned, it follows that the Ecker-Müller distribution is also a function of electron temperature, of course, via dependence on parameter 𝛿 or 𝑎.

The comparison of Ecker-Müller distribution and Holtsmark function versus parameter 𝛿 and values of electric reduced field values /0 is presented in Figure 1.

2.3. Baranger-Mozer Cluster Expansion

The Baranger-Mozer papers [24, 25] appeared approximately 2 years after the works of Ecker and Müller and were significant advance as according to the physical formulation as to the development of adequate mathematical formalism.

The notions of high-frequency electronic and low-frequency ionic components of plasma microfield, ion-ion correlations were introduced in [24, 25]. It was pointed out on inadmissibility of usage the total Debye radius in expression for ion microfield and was demonstrated the different character of distributions in the neutral and charged points.

The adequate formalism in [24, 25] is based on the cluster expansion methods, developed firstly for virial coefficients [712] and giving the possibility to represent ln[𝐴(𝑘)] in power series over density ordered versus the extent of correlations weakening. Let us consider in more detail the instructive derivation of these results. The summary field of ions 𝐹 satisfies the vector additivity condition, that is,

𝐹𝐹=1+𝐹2+𝐹3𝐹++𝒩.(11) The distribution function of summary microfield 𝑊(𝐹) could be transformed to the form

𝑊𝑘=1(2𝜋)3𝑑3𝑘𝐹𝐴𝑘,𝐴𝑘=𝑑𝑘exp𝑖3𝑥1𝑑3𝑥𝒩𝑖𝑘𝐹exp1+𝐹2𝐹++𝒩𝑃𝑥1,𝑥2,,𝑥𝒩,(12) where 𝑃(𝑥1,𝑥2,,𝑥𝒩) designates the probability of given configuration from 𝒩 particles.

Furher on, the standard procedure, which is performing the identical operations with each of multiplicands 𝑘𝐹exp(𝑖𝑗) in the integrand, is applied:

𝑖𝑘𝐹exp𝑗=1+𝜑𝑗,𝒩𝑗=11+𝜑𝑗=1+𝑖𝜑𝑖+𝑖,𝑖𝜑𝑖𝜑𝑖+.(13) Then integrating over free variables, the characteristic function could be represented as a sum

𝐴𝑘=𝐴𝑘,𝐴(14)=𝑑3𝑥𝑖𝑑3𝑥𝑠𝜑𝑖𝜑𝑠𝑃𝑥𝑖,,𝑥𝑠,(15) where the summation is extended on all combinations of particles from 𝒩.

Then the idea about strong decreasing of correlations versus increasing their order is explicitly implemented:

𝒱𝑃𝑥𝑖,,𝑥𝑠=𝑖𝑔1𝑥𝑖+2𝑔2𝑥𝑗,𝑥𝑘𝑖𝑔1𝑥𝑖+22𝑔2𝑥𝑗,𝑥𝑘𝑔2𝑥𝑙,𝑥𝑚𝑖𝑔1𝑥𝑖+222++3𝑔𝑠𝑥𝑗,𝑥𝑘,𝑥𝑙𝑖𝑔1𝑥𝑖+33𝑔3𝑥𝑗,𝑥𝑘,𝑥𝑙𝑔3𝑥𝑚,𝑥𝑛,𝑥𝑝𝑖𝑔1𝑥𝑖+333++32𝑔3𝑥𝑗,𝑥𝑘,𝑥𝑙𝑔2𝑥𝑚,𝑥𝑛𝑖𝑔1𝑥𝑖+4+,(16) where 𝒱 is the system volume, and the single particle probability function is

𝑃1=1𝑥𝒱𝑔1𝑥.(17) The sum 2 designates the summation over all pairs of particles from particles, the sum 22 over two different pairs of particles from particles. In the sum 32,0𝑥0200𝑑 the summation goes over all possible combinations of different triplets and pairs of particles from particles. In each term of this series, those particles, included in the product from particles, do not constitute triplets, pairs, and so forth. The 𝑔 functions due to the extraction of factor 𝒱 are defined so that do not depend on volume 𝒱 for large values of 𝒱. Generally speaking, the cluster diagram could be confronted to each term of this expansion [7].

In the limit of large 𝒩 and large 𝒱, but for constant concentration 𝑁=𝒩/𝒱=const, 𝐴(𝑘) could be represented as

𝐴𝑘=𝐺1𝑘𝐺2𝑘𝐺3𝑘𝐺,(18)𝑃𝑘=1+𝒱𝑃𝑃𝜑𝑖𝜑𝑠𝑔𝑃𝑥𝑖,,𝑥𝑠𝑑3𝑥𝑖𝑑3𝑥𝑠+𝒱2𝑃𝑃𝑃𝜑𝑖𝜑𝑣𝑔𝑃𝑥𝑖,,𝑥𝑠𝑔𝑃𝑥𝑡,,𝑥𝑣𝑑3𝑥𝑖𝑑3𝑥𝑣+𝑃𝑃𝑃+.(19) Here, the single sum covers all possible combinations with 𝑃 particles from 𝒩 ones, the double sum over all possible combinations of different two clusters with 𝑃 particles from 𝒩 ones, while all particles in a cluster are being different, and so forth.

The difference of expression (18) from (14) is that there are no the same particles in each term from (14), represented as the expansion according to (15)–(17), whereas according to definition (18), one particle, entering in 𝐺𝑃, could coincide with one of particles, that compose 𝐺𝑄. That is why (18) contains a part of additional terms which do not appear in (14). However, as stated in [24, 25], the number of coinciding terms in both expansions, under the tendency of the total number of particles to infinity, is 𝒩 times larger the number of additional terms, whose contribution to the total sum thus occurs negligible [24, 25].

If to take into account that under 𝒩 all terms in each sum become equal, then calculating the number of those terms, one could obtain the following closed expression for 𝐺𝑃(𝑘)

𝐺𝑃𝑘𝑁=exp𝑝𝑝!𝑝𝑘,𝑝𝑘=𝜑1𝜑2𝜑𝑝𝑔𝑝𝑥1,𝑥2,,𝑥𝑝𝑑3𝑥1𝑑3𝑥2𝑑3𝑥𝑝,(20) and hence the expression for 𝐴(𝑘) takes the form

𝐴𝑘=exp𝑃=1𝑁𝑝𝑝!𝑝𝑘.(21) In contrast to the virial expansion, the convergence of integrals 𝑝(𝑘) and its sum are more rapid due to the appearance of powers of additional factors 𝜑𝑗(𝑘) in the integrands for terms of cluster expansion series, which drastically narrows the range of effective values of variables, providing the main contribution to integrals.

As the Bogolubov-Born-Green-Kirkwood-Yvon chain [712], the cluster expansion is based on two very significant semi-intuitive notions: (i) about monotonous decreasing of correlation functions versus increase of the correlation order; (ii) about a sufficiently rapid decrease of correlation functions versus increase of the distance between particles.

For the low-frequency distribution of ion microfield, the electric field produced by single-field ion at the origin of reference frame was taken in the form of Coulomb electric field statically shielded by plasma electrons according to Debye as was already mentioned in the previous paragraph. The Debye approximation was implemented in expressions for pair correlation functions, and calculations were limited by the pair correlations in neutral point and the triple correlations in charged point. In the case of the electric field distribution in charged point, the triple correlation function was disentangled with the help of the Kirkwood superposition approximation [712]. Thereby, only the two first terms of cluster expansion of ln[𝐴(𝑘)] were taken into account, where the second term describes ion-ion correlations. For the pair correlations function of field ions, the linearized Debye approximation was used for description of ion-ion correlations, which is the first not equal to zero term of expansion [312].

The high-frequency function, describing the Coulomb field of plasma electrons, practically was not used later, but the low frequency component of plasma microfield had got applications in spectroscopy.

Formally, this distribution due to ion-ion correlation additionally to the dependencies on 𝛽 and 𝑇𝑒 also is a function of ion temperature 𝑇𝑖 through dependence on additional dimensionless parameter 𝑅𝑐/𝑅0=𝑒2𝑍2𝑝/𝑇𝑖𝑅0, which practically coincides with the definition of ionic coupling parameter Γ𝑖, where 𝑅𝑐 is the ionic Coulomb radius. It should be pointed out that in the second of cited works [24, 25], the linearized Debye pair correlation function, used for description of ion-ion correlations, contains as a shielding length the total Debye radius, where the ion-ion shielding also is accounted for [24, 25].

Regretfully, in the tabulation of ion microfield distribution functions in [24, 25] the numerical mistakes were detected, which led to undeserved disavowal of developed approach. Later, Pfennig and Trefftz found and removed these inaccuracies [26] together with distrust to approach in general.

The important advantage of Baranger-Mozer formalism is the possibility of its generalization for arbitrary plasma ionization composition, that is presented, particularly, in 2.9.1. The explicit results of 2.9.1. allow to obtain more ample apprehension on practical receipts of Baranger-Mozer approach implementations.

The graphical behavior of Baranger-Mozer distribution functions after removal of numerical inaccuracies coincides with the Hooper distributions, obtained within the different model and represented in the following subsection.

The main progress of these two works is distinguishing the high-frequency electric microfiled component, whose time variation is governed by the motion of electrons, and the low-frequency electric microfiled component, whose characteristic time scale is determined by ion motion. At the same time, it is assumed that the average of high-frequency component on the ion microfield time scale contributes to the summary low-frequency microfield component via the Debye electronic shielding of ion electric field due to the electron clouds surrounding ion charges [24, 25]. Having in mind the problem of the Stark broadening of spectral lines, the authors aimed to obtain the distribution of, namely, “instant” microfield and not the average “thermodynamic” microfield. It should be underlined, thus these declarations although quite sound and reasonable from physical sense contradict with the available formalism, which is, of course, thermodynamic in its origin [24, 25].

As the properties of correlation functions with the order larger than 2 are studied still insufficiently up to now, only the two first terms of expansion were considered in [24, 25]: the first one is being linear dependent on density and the second one is being proportional to the density squared. Thus, the first term describes certain type of independent quasiparticles, characterized by some interaction potential with the test particle, while the second term is responsible for pairwise or reduced triple correlations between field particles.

2.4. Hooper Model

The Hooper model implements Bohm and Pines “collective coordinates method” (CCM) [30, 31]. This method devoted to an attempt to separate formally Hamiltonian of the system of Coulomb particles into two Hamiltonians, characterizing almost independent subsystems one of which represents itself the plasma collective characteristic oscillations, and the other one represents the subsystem of independent quasiparticles “dressed” by the screening due to separated collective degrees of freedom [30, 31]. It was shown [30, 31] that under specific assumptions, this separation is possible to accomplish by applying the specifically defined sequence of canonical transformations of variables. These results had great impact on the further development of ideas of plasma microfield and were used later as a basis in order to determine how to separate the collective microfield component due to the plasma characteristic oscillations from many-body but “individual” microfield component due to particles or quasiparticles [4851]. Firstly for constructing the static microfield distributions, this method was proposed by Broyles [32, 33]. The Broyles papers [32, 33] contain deep and very interesting original physical analysis of microfield problem, as well as several innovative suggestions for development of appropriate mathematical formalism.

Meanwhile, in the same period of time, the Monte Carlo (MC) procedure was formulated and published providing a powerful tool for consideration of thermodynamically equilibrium conditions and calculations of correlation functions and various static microfield distributions [110] (see [3539, 111113]. The overall situation at that time with the Baranger-Mozer results was not clear, and the progress in Monte Carlo and ideas of Broyles inspired Hooper to reconsider the derivation of static microfield distribution functions in some different original way [2729]. Hooper adopted the ideas of Baranger-Mozer on the separation of high- (electron) and low-frequency (ion) microfield components, but he introduced the separation of the interaction potential into the so-called central (corresponding to the interaction with the test particle) and noncentral (corresponding to the interactions between field particles) parts [2729]. He also formally included the scalar product of vectorial Fourier variable on the vector of elementary electric field strength of the single particle into the central part of the interaction potential. After this, Hooper constructed the analog of the two term Baranger-Mozer cluster expansion but for complex central potential, which had certain impact on the definition and determination of the correlations functions, for example. It was supposed that the screening of the ion field in the central part of potential is determined by the electronic Debye radius while the screening length for the noncentral part also is described by Debye potential but with a screening length equal to the electronic Debye radius multiplied by fitting parameter “𝛼” to be determined later. Using the Bohm and Pines method of collective variables as a mathematical trick according to the Broyles ideas, Hooper was able to derive the formulae for the microfield distribution function that as in the case of Baranger-Mozer is expressed through the finite number of subsidiary integrals and functions [2729]. Performing calculations along with this derivation and comparing their results with Monte Carlo method for the same values of parameters, Hooper found the rather wide ranges for the “𝛼” parameter variation, in which the results of calculations with the prescribed accuracy practically do not alter and coincide with Monte Carlo results for the same set of plasma parameters and assumptions concerning the interaction potential. The examples presented in [2729] showed that for the low-frequency ionic microfield component, 𝛼 could change from 1.3 to 1.8 at 𝑎=0.8, and from 1.8 to 4.0 at 𝑎=0.2. Hence, this strangely means that in some range of fitting parameter variation, the results in question do not depend on its values. When his article was altogether in print [2729], Hooper became aware of the article of Pfennig and Trefftz [26], and after comparison he found that the results of his calculations do not differ from the improved for digital mistake results of Baranger-Mozer [24, 25]. Thus, it was rather dramatic point because no words the Hooper's method of derivation was much more complicated and that is why, probably, lesser convincing than the Baranger-Mozer one. However, tables of microfield distributions presented in Hooper's works became widely used in practical calculations in plasma spectroscopy and thus frequently cited, although their values practically coincide with the values prescribed by Baranger-Mozer approach!

Alas, the derivation of Hooper results is substantially unclear [2729] mostly due to the very complicated formalism used in [2729], although initial settings do not differ from Baranger-Moser ones. Hooper also stated that he used nonlinear form of Debye-Hückel correlation function. However, in this case the dependence of the second term of cluster expansion starts to be more complicated and could not be reduced only to the second power of density. Regrettably, the noticeable difference of the effective shielding length from the Debye value did not get any physical treatment in [2729] and posterior works, exploiting these initial Hooper ideas.

In Hooper works, there are no details on Monte-Carlo method used in the model. The Monte-Carlo method in its essentials corresponds to the infinite interval of time average and thus includes the total ion-ion screening, which is inadmissable in the case of its implementation for the description of quasistatic ion broadening of spectral lines in plasmas. So, now it is well known that the results of Hooper approach do not differ from corrected Baranger-Mozer results [24, 25]. At the same time Hooper approach is more laborious and could not get unequivocal interpretation. The later Hooper works with coauthors showed that the developed formalism does not have simple extension on the case of arbitrary plasma ionization composition, and even the case of binary composition needs tremendous computing efforts [34]. The Hooper distributions also are limited by values of parameter 𝑎1.

In Figure 2, one can see microfield distribution found by Hooper for several values of 𝑎 in the neutral point, and in Figure 3 in the charged point. Both distributions are calculated for the singly charged field ions and singly charged test ion. The designations in the figures are as in the original Hooper papers. The distribution for 𝑎=0 in neutral point coincides with the Holtsmark distribution. The other distributions coincide with the Baranger-Mozer ones for the same conditions as mentioned earlier.

The Hooper formalism for construction of distribution functions, based on using in mathematical calculations the collective coordinates method and cluster expansion, could not be generalized directly on quantum case or plasmas with complex ionization composition. In fact, to our knowledge, his results and formalism were never reproduced independently from the author [2729]. However, the distribution functions presented in [2729] and other papers are very trusted by professional community and popular in doing practical calculations.

2.5. Monte-Carlo Method

The calculation of microfield distribution functions by Monte-Carlo method (MC) is based on computer statistical sampling of probability of fall-out of various spatial configurations of field particles [3539].

Firstly, the systematic description of Monte-Carlo method was published in [110] (see also [111113]) and formally is not limited by only weakly coupled plasmas.

Until recently, the majority of results for microfield distribution functions for strongly coupled plasmas were obtained namely by this method [39]. The notion of strongly coupled plasmas encircles also conditions, when electronic Γ𝑒=𝑒2𝑁𝑒1/3/𝑇𝑒 and ionic plasma parameters Γ𝑖=𝑍2𝑝𝑒2𝑁𝑖1/3/𝑇𝑖 of coupling exceed unity not at the same time.

In the main part of MC studies up to date, the Debye form is chosen for the initial ionic potential with the effective screening length taking into account the degeneracy of electronic component. The size of the cell 𝐿 is determined by the density of modeled conditions, namely, by the number of particles in MC simulations 𝒩 plus the test particle, and connected with the ion density 𝑁𝑖 by the relation

𝑁𝑖=(𝒩+1)𝐿3.(22) For including the influence of remote particles, the cell is reproduced by its “self-images” with step equal to 𝐿, and the total sum of potentials is evaluated by the Evald method [114, 115]. The important advantage of MC is that it easily matches any boundary conditions.

During simulations, the field ion and its location inside the cell are chosen in a random manner. If, during modeling, the ion occurs outside the cell, then it is substituted by its image. Under the usage of powerful computers like Cray, the first 104 configurations were discarded in order to avoid dependence on initial conditions.

For searching equilibrium solution, the Metropolis algorithm is used [105]: the difference of energy Δ𝑊 between two consequent configurations is calculated, and if this difference is negative, then the configuration is included with the weight factor equal to 1, and if it is positive then with the weight factor equal to exp[Δ𝑊/𝑇]. This allows to avoid the system trapping in local random minima. Evidently, during approaching the equilibrium, Δ𝑊0. All equilibrium values of microfield are calculated after reaching the equilibrium. For example, in the widely used by experts results of MC modeling [3539], the number of particles in the cell was 700–800, while the number of configurations after reaching the equilibrium 107. It should be noted that unlike the initial version of method [110], the later results [3539] are obtained after an average of total potential over the angles of radius vector of test particle, which accelerates the convergence and secure the fulfilment of conditions of isotropy.

In a recent paper [39], the rather simple approximate functions of reduced microfield and coupling parameters for various regions of plasma parameters were proposed during fitting procedure to results of MC calculations of plasma microfield distributions.

In regions of very small and very large reduced microfield values, 𝛽 MC has very large fluctuations and could not provide prescribed accuracy. For description of distribution functions in this regions, the matching with known asymptotic results is applied [21].

2.6. Adjustable Parameter Exponential Approximation

The Hooper's ideas gave rise to another approach in the theory of microfield distribution that is called Adjustable Parameter Exponential Approximation (APEX) developed in series of papers by Iglesias et al. (for current version, see [43, 184189]). This method was aimed to describe first of all the microfield distribution at highly charged test ions in strongly coupled plasmas, where other theoretical approaches as Baranger-Mozer one fail, while MC at that time was considered as inconvenient and expensive for large-scale calculations together with magneto-hydrodynamic and radiation transfer codes in the laser inertial confinement fusion (LICF) studies [4042]. Hence, the main motivation for APEX derivation [4044] was an attempt to give alternative with respect to MC description of microfield distributions at test ions in strongly coupled plasmas. However, APEX from the beginning was formulated as ad hoc approach.

APEX also singles out the high frequency-electron and low frequency-ion components of plasma microfield. The constructions of microfield distributions for those components are rather different. The APEX model for high-frequency component could be considered in our classification as a mixed one, because it uses the notion of point separate electrons and the notion of uniform continuous positive background due to ions. In this APEX derivation, the results of the so-called one component plasma model (OCP) were applied [712]. Here, the narration mostly concerns the APEX results for ionic low-frequency part [4244].

The key point in the APEX construction is the assumption of the Yukawa-type effective interaction potential between ions with the screening length, which is proportional to the adjustable parameter “𝛼” to be determined later. Also, APEX utilizes the exact relation that have to be fulfilled at the test particle with charge equal to 𝑍0 [43]. At the same time if to remember that in the strongly coupled plasma the Debye radius as a rule is less than the mean interparticle distance, the validity and applicability of the Debye potential start to be doubtful for these conditions.

According to the APEX ideology, the introduction of the APEX effective field should account for high-order correlations and thus should make it possible to consider effectively noninteracting quasiparticles [43]. Thus, the initial APEX starting formulation and idea was using transformations of cluster expansion for ln𝐴(𝑘), like used by Hooper [2729], to obtain single-term representation of cluster series with the help of more accurate methods for constructing the correlation functions than those provided by Debye approximation [712].

For transformation of cluster series to one term it was proposed to substitute in ln𝐴(𝑘) not a real, but some effective electric field and corresponding distribution of quasiparticles, equating the products of local probability density on the value of local field, namely:

𝑒𝐶𝜎𝑍𝜎𝐺𝜎(𝑥)𝑓𝜎(𝑥)=𝑒𝐶𝜎𝑍𝜎𝑔𝜎𝐺(𝑥)𝑓(𝑥),𝜎(𝑥)=𝑔𝜎(𝑥)𝑓(𝑥)𝑓𝜎,𝑓(𝑥)𝜎(𝑥)=exp𝛼𝜎𝑥𝑥21+𝛼𝜎𝑥,[]𝑓(𝑥)=exp𝑎𝑥𝑥2(1+𝑎𝑥),(23) where 𝑎=𝑅0/𝑅𝐷𝑒; 𝜎designates the field ions species; 𝐶𝜎=𝑁𝜎/𝑁, 𝑍𝜎 are partial concentration and the charge of field ions species 𝜎 correspondingly; 𝑔𝜎(𝑥) is the correlation function of the field and test ions; 𝑓𝜎(𝑥) is the APEX effective field, depending on fitting parameter 𝛼𝜎; 𝐺𝜎(𝑥) is the effective distribution function of the field particles density with the charge 𝑍𝜎 around the test ion with the charge 𝑍0; 𝑓(𝑥) is the reduced initial, screened by electrons according to Debye, this “so-called” elementary electric field is the adopted dependence for the electric field of single plasma ion.

Thus, the Hooper ideas of implementation of additional fitting parameter under optimization of distribution functions got in APEX alike, but another realization.

The set of fitting parameter {𝛼𝜎} according to [42] has to be found from the exact relation for mean square of microfield at test ion

𝐸2=4𝜋𝑁𝑇𝜎𝐶𝜎𝑍𝜎𝑍0𝜓𝜎𝜓(𝑎),𝜎(𝑎)=𝑎20𝑑𝑥𝑥𝑔𝜎[],(𝑥)exp𝑎𝑥(24) where 𝑍0 is the test ion charge.

The left-hand side of this equation in APEX takes the form [42]

𝜎𝑍2𝜎𝐶𝜎0𝑑𝑥𝑥2𝑔𝜎(𝑥)𝑓𝜎=1(𝑥)𝑓(𝑥)𝑍0Γ𝑖𝜎𝑍𝜎𝐶𝜎𝜓𝜎(𝑎).(25) It is assumed in [42] that the solution could be found for each species separately, which gives the set of equations for all 𝜎:

𝑍2𝜎0𝑑𝑥𝑥2𝑔𝜎(𝑥)𝑓𝜎𝑍(𝑥)𝑓(𝑥)=𝜎𝑍0Γ𝑖𝜓𝜎(𝑎).(26) The correlation functions could not be determined within APEX. To close APEX scheme, the correlation functions are calculated separately within the hypernetted chain approximation (HCN) [712], when the so-called “bridge function” is put to zero [43]. The HCN correlation functions are considered as very precise and remarkably differ from Debye ones for large plasma coupling parameters and reproduce rather well MC and MD correlation functions [712, 4043]. The illustration of correlation function behavior for strongly coupled plasma is shown in Figure 4 from [43]. Utilizing HCN correlation functions is one of the APEX significant advantages that gives possibility to describe microfield distributions in strongly coupled plasmas (SCP) [43]. At the same time as could be judged by laconic APEX papers, the starting potential in HCN is again Debye potential [42], which could be invalid for very large plasma coupling parameters.

The APEX results very well reproduce the MC simulations, considered by the APEX authors as more time consuming than APEX. However, recently MC programs were substantially improved and could compete with APEX speed of computations [3739]. As shown in APEX publications, the value of fitting parameter 𝛼1 can exceed unity several times [68, 69]. In Figure 5, the example of APEX distribution in the mixture of field ions with equal concentrations and its comparison with the results of other authors is presented [43]. It is seen that the APEX better reproduces MC calculations than it cold be done in the frames of TH [34] or HDG [72, 73] approaches, which were not designed to describe strongly coupled plasmas. In Figure 6, the calculated in APEX [68, 69] variation of the reduced fitting parameter 𝛼 in the reciprocal Debye length units 𝑘DH for the hydrogen-like Ar ions at temperature 𝑇𝑒=800 eV is shown versus density. These results demonstrate that the effective APEX potential has all the time the radius of shielding (2÷4) times less the Debye radius in the interval of density variation of 4 orders of magnitude. So, the effective interaction is more short-ranged in comparison with Debye potential.

The recently improved APEX version [43] can address to nonequilibrium plasma parameter-non-equality of ion 𝑇𝑖 and electron 𝑇𝑒 temperatures. The example in Figure 7 shows the case when the ion temperature by an order of magnitude less than the electron one could lead to two times difference of the quasistatic Stark line profile halfwidth [43]. The improved APEX scheme allows to consider the degeneracy of electronic component [43]. Although the significance of the degeneracy effects evidently signalize about an uprise of the quantum effects, it has almost no consequences on the derivation of the practically classical microfield distribution function beside changing the screening length of the interaction potential [43]. However, the APEX microfield distribution function itself could be changed quite considerably, which is well illustrated in [43, Figure 8]. It is worthy to discuss the 𝑊(𝛽) behavior at large Γ𝑖. In this limit, the pairwise Radial Distribution Function should acquire additional maxima on the scales, corresponding to short-range and long-range ordering. Moreover, it should be escorted by an uprise of pronounced anisotropy. From physical point of view [45] in this case, the distribution function should be alike Gaussian, describing small deviations from equilibrium particles positions in the vicinity of crystallization Γ𝑖150, which was really observed in modeling of strongly coupled plasmas. Hence, the asymptotic of distribution function could not be the nearest neighbor NN distribution [21, 39]. Although it is stated in APEX that at Γ the APEX is approaching Gaussian, this transition was not followed in detail, whereas it is doubtful how so qualitatively different asymptotic laws would replace each other. The available results do not allow to clear this question yet.

In Figure 9 from [43], the microfield distributions for 𝑍0=𝑍𝑠=25, 𝑁𝑒=1024 cm3, 𝑇𝑒=50 eV, giving Γ=50 from [43] are presented, where it is seen that at these conditions the APEX is approaching already Gaussian in some regions of 𝛽 variation. However, APEX asymptotic at Γ=50 is still more alike nearest neighbor (NN) [43]. Also, one can see the comparison of APEX distributions constructed with HCN and MD Radial Distribution Functions (RDFs) [43]. The current methods of simulations ab initio, the molecular dynamics (MD) [68, 69, 96, 101109, 135139] and MC [3539], provide rather large noise with increasing of reduced microfield values, as was illustrated recently in [43].

For description of microfield distributions in neutral point, the APEX approach was reformulated [44]. In this case, the relations (24)–(26) do not take place. For the case of neutral point, it is suggested to determine 𝛼 from the condition of equality of the second order derivative over Fourier-variable 𝑘 to zero at 𝑘=0 of any term beside the first one of specially renormalized cluster expansion [44]. If to return to the charged point, then it was stated that the new relation could be reduced to the form introduced for the charged point [44].

The APEX model was recently interestingly combined in works of Nersisyan et al. [184186] for the classical two component plasmas (TCP) with the “potential-of-mean-force” (PMF) approximation very similar to the earlier work of Yan and Ichimaru [187]. Here the basic APEX ingredients like the expressions for the elementary electric fields are changed and modeled by the Coulomb fields modified in the case of attractive interactions by diffraction corrections [184186]. The fitting parameter “𝛼” is not introduced at all since the second moment relation is satisfied exactly [184186]. The new model called PMFEX [184186] preserves the APEX way for generating the correlation functions-HCN approximation and demonstrates rather good coincidence with MD simulations and admirable stability in providing data in the region of large reduced microfield values. Astonishingly, PMFEX has more natural generalization to obtain microfield distribution functions (MDF) in neutral point than APEX itself.

At last, the APEX procedure was generalized to extend it for modeling liquid domain in [189], where additional parameter of scaling is introduced

𝑟𝜅=WS𝜆𝑒,(27) where 𝑟WS is the Wigner-Seitz radius, 𝜆𝑒 is the electron screening length. In this work, the set of analytical formulas are proposed for acceleration of computations depending beside reduced field 𝛽 on 𝜅 and the coupling plasma ion parameter Γ𝑖, defined as

Γ𝑖=𝑍𝑒2𝑘𝐵𝑇𝑟WS,(28) alike it was done in [39], where obtained by Dr. Dominique Gilles, data in MC simulations were fitted by multi-parametric approximate expressions. In [189] and the aforementioned expression, the thermodynamic equilibrium is assumed 𝑇=𝑇𝑖=𝑇𝑒, 𝑍 is the residual ion charge. However, to our opinion, an extension of microfield ideas on liquids with Yukawa type of interaction potential between particles is complicated and disputable subject.

In conclusion of this section, it should be resumed that in spite of evident success of APEX applications, the APEX itself is essentially ad hoc semiempirical method, whose reproduction is almost impossible without the help of its authors. At the same time the new APEX modifications evidently expand the range of successful implementation of these ideas.

2.7. Density Functional Theory

The most close to conventional thermodynamic notions method of construction of plasma microfield distribution functions was proposed in the work of Dharma-Wardana and Perrot [46, 47]. This method is based on generalization of local density functional theory of Kohn and Sham (LDFT) [116, 117] to finite temperatures. The outstanding research of these coauthors, who performed a row of fundamental studies, made possible the regular application of LDFT methods in plasmas (see, e.g., [118123]).

The physical idea of this approach is the implementation of the self-consistent description of dense plasmas, which can reflect the influence of its properties on the quantum characteristics of free electrons with the arbitrary extent of degeneracy and partially ionized core of field ions and actually the states of emitter, determined simultaneously and self-consistently with correlation functions.

The request for self-consistency to some extent corresponds to the solution of kinetic problem, giving the answer on a question what partial concentration, temperature, and effective charge would have that or another plasma component at given temperature and density of free plasma electrons. Namely, relying on this initial information, the distribution functions are constructed in the other nonself-consistent approaches. It is evident that self-consistent approach would be by far more complicated due to necessity to find simultaneously with a distribution function the distribution of electron density, effective charges of ions cores, and various correlation functions.

The range of plasma parameters on which such a description is pretended corresponds to large values of electron plasma-coupling parameter Γ𝑒1 and strong ion-electron correlation due to influence of bounded electron states of emitter and field ions, but at the same time to values of ion plasma coupling parameter mainly less than unity Γ𝑖<1.

The proper variational methods of local density functional is finding self-consistent distribution of electron density with simultaneous solution of the Schödinger equation for determining the wave functions and energy levels in potential, which in its turn is a functional of the electron density distribution [46, 116, 117].

The computational realization of approach is accomplished in the so-called correlation sphere of finite radius. This recalls the principles of mean ion plasma model (MIP), assuming finite size of ion sphere, in which the quasineutrality condition is fulfilled. It is common to refer on this procedure as on solution of DFT-Schrödinger equation [116].

In [46], this distribution of electron density is used further on in solution of Ornstein-Zernike equation in HCN approximation [712] for calculations of ion-ion correlation functions 𝑔𝑖𝑖(𝑟).

These functions are substituted then in two terms cluster expansion of Baranger-Mozer type [24, 25] for calculations of the logarithm of characteristic function with some amending modifications, connected with possibilities of partial summation of chain terms of higher orders in the so-called “weighted-chain-sum” (WCS) approximation [46]. This amendment functionally is expressed in appearance of majorizing factor for the second-order density term in the Baranger-Mozer expression for the characteristic function logarithm [46].

The principal moments of this approach [46] are (i) the choice of Baranger-Mozer scheme of cluster expansion, that allows generalization on quantum case in distinction from, for example, limited by classical approximation Hooper model; (ii) the criticism of a choice of the Yukawa type potentials for describing pairwise interactions in plasma; (iii) the determination of 𝑔ii(𝑟) with the help of special self-consistent procedure within HCN approximation; (iv) the way of calculation of the electric field value at the origin of reference frame due to field ion with the effective charge 𝑍𝐵 according to the exact result of pseudopotential theory of the second order:

𝑍𝐸(𝑟)=𝐵𝑟2+1𝑟2𝑟0𝑛𝑔𝑒𝑖(𝑥)14𝜋𝑥2𝑑𝑥.(29) The expression for 𝐸(𝑟) is convenient to represent in the form

𝐸(𝑟)=𝑍𝐵𝑞(𝑟)𝑟2,1𝑞(𝑟)=1𝑍𝐵𝑟0𝑑𝑥4𝜋𝑥2Δ𝑛(𝑥),Δ𝑛(𝑥)=𝑛𝑔𝑒𝑖.(𝑥)1(30) The last equation determines the total nonlinear excess of electron density around ion “B,” including exchange and correlations effects, which is found from the solution of DFT-Schrödinger equation in [46, 116, 117]. This pileup of electron density around ion is defined with respect to the level of uniform neutralizing background of free plasma electrons.

It should be noted that ionization equilibrium in DFT [46, 47, 116, 117] does not obey Saha equation because the correct condition of thermodynamic equilibrium is the free energy minimization. To the same resume, Hammer and Michalas arrived at one year later during analysis of the microfields influence on the equation of state [154156].

At the same time, one drawback of this approach could be hidden here. Indeed, the emitter or field ion of “finite” size is inserted in the uniform electron background, but the plasma effects like lowering of ionization potential and kinetics of establishing of equilibrium with continuous spectra are not included in the description of levels population and realization of the bound electron states of the upper levels, as tried to formulate Hammer and Michalas [154156].

As could be judged by original formulation [46, 47, 116, 117], it seems that DFT capability to describe nonequilibrium plasma conditions with complex chemical and ionization composition and different temperatures of electron and ion subsystems appears to be doubtful.

In the equation for electric field, the shielding of only one ion (field ion or emitter) is included. Due to the authors statement [46], the accounting for analogous terms for the second ion is beyond the accuracy of the used second-order pseudopotential theory.

It is important to underline that the electric field defined by the aforementioned equations in the quantum case could not be equalled as in the classical limit to the gradient (with opposite sign) of pairwise potential of ion-ion interaction. This is because in these conditions, this gradient will include nonelectrostatic terms connected with exchange and other purely quantum effects. Thereupon in [46] it is demonstrated that the usage of the effective pairwise potential of Yukawa type provides inadequate results.

The discussed approach operates with the following quantities: the effective charge of field ions 𝑍 and their mean density 𝜌, and the mean density of free electrons 𝑛, associated with quasineutrality condition:

𝑛=𝑍4𝜌,3𝜋𝑟3𝑠𝑛=1,(31) where 𝑟𝑠 has the sense of the electron sphere radius. The reduced and normal fields are determined by expressions

𝐸𝐸=𝐸0,𝐸0=𝑍𝑟20,415(2𝜋)3/2𝑟30𝑟𝑛=1,0=0.9991178𝑟𝑠,(32) where 𝑍 represents itself the charge of field ions. (This choice of the normal field, although admissible in principle, could be misleading. More adequate to our opinion would be 𝐸0𝑖=𝑍/𝑟20𝑖, (4/15)(2𝜋)3/2𝑟30𝑖𝜌=1.) The distribution functions depend on parameter

𝑎𝑟mf=𝑟0𝑟De=0.999123Γ𝑒1/2,Γ𝑒=𝑒2𝑇𝑟𝑠𝑟𝑐𝑟𝑠,(33) where the electron Debye radius 𝑟De is determined also on the basis of the mean free electrons density 𝑛, the electron plasma coupling parameter Γ𝑒 is defined in the same way as Hooper did, and 𝑟𝑐 is the Coulomb radius.

The classical ion plasma coupling parameter is determined from the expression

𝑍Γ=2𝑟𝑐𝑟WS,𝑟WS=34𝜋𝜌1/3,(34) where 𝑟WS is the radius of Wigner-Seitz cell. In this model, 𝑍 and 𝑍 are related to each other by

𝑍=𝑍𝑛𝑏,(35) where 𝑛𝑏 is the number of bounded electrons per ion, calculated on the basis of DFT approach.

The DFT microfield distribution functions in distinction from the distribution functions in classical plasmas depending not only on the parameter 𝑎 but also on the parameter 𝑇=𝑇/𝑇𝐹 at least, where 𝑇𝐹 is the electron Fermi temperature, defined by the equation

𝑘𝑇𝐹=𝐸𝐹=123𝜋2𝑛2/3,a.u.(36) Moreover, according to authors [46, 47] opinion, the extent of plasma coupling in quantum case is characterized more correctly by parameter Γ, which is determined by the basis of notion about the mean ion radius 𝑅 assigned to each ion, so that the mean number of free electrons per ion 𝑛𝑖𝑓 is equal to

𝑛𝑖𝑓=43𝜋𝑅3𝑛𝑓,(37) where 𝑛𝑓(=𝑛) is the density of plasma free electrons. On the other hand,

𝑍=𝑛𝑏+𝑛𝑖𝑓,𝑛𝑏=2𝑅04𝜋𝑟2𝑛𝑏(𝑟)𝑓𝑠(𝑟)𝑑𝑟,Γ=𝑍2𝑟𝑐𝑅,(38) where Γ could be considered as the effective ionic plasma coupling parameter, corresponding to “equivalent” classical plasma, 𝑓𝑠(𝑟) is the Fermi factor, describing the character of electron states filling and depending on temperature and chemical potential. Thereby, the value 𝑍 or 𝑛𝑏 are also the results of self-consistent solution of DFT-Schrödinger equations.

In the first article [46] of the authors of this approach, devoted to constructing microfield distribution functions, the Kirkwood approximation [712] was applied for disentanglement of the three particle correlations. This procedure was supplemented by separation of the central and noncentral parts of interactions in [47] on the basis of methods, elaborated in papers for description of quantum Hall effect. In particular, this improvement was connected with the APEX authors criticism of the DFT results for strongly coupled plasmas, where noticeable discrepancy was observed between the predictions of APEX and the first version of DFT-approach [46].

To illustrate this, the results of DFT-approach in comparison with APEX [40, 41] for Al plasma are presented in Figure 10 according to [46]. It is necessary to note that data of [40, 41] correspond to the so-called the high-frequency component of microfield, which describes the distribution of Coulomb electric fields of particles with the charge 𝑍𝑒 [40] or with arbitrary composition of ions of different species [41], inserted in the uniform neutralizing electron background. However, the authors of [46] did not point out for which values of 𝑍 or 𝑍 they took the data from [40, 41]. However, curiously enough, the most open for criticism moment of this approach is its complete self-consistency, which leads to the loss of possibility of the conventional identification of observed spectral transitions. In other words, in this DFT version, the self-consistent wave functions and the energy structure of emitters in plasma do not remember the corresponding characteristics of free emitters, which, generally speaking, are tools for decoding of observations.

Probably the cause of that is the insufficient accuracy in the description of the bounded excited states within DFT [46, 47]. The DFT version under discussion, however, could be successfully applied for calculations of the thermodynamic characteristics, when the affixment of results to observed properties of radiation in spectral lines is not important.

Moreover, it is not quite clear how to track the time scales of microfield variations in this approach, and the character of averages, applied in its derivation, more corresponds to purely thermodynamic notions. That is why the doubt arises in possibility to construct with the help of this approach instantaneous ion microfield distribution functions. Partially, it is connected with the very orientation of this method on description of quantum effects in the microfield distributions and necessity of possible reexamination of the “instantaneous distribution functions” notion in this case. However, this criticism concerns equally and Monte-Carlo method, and so forth.

In comparison with the other theoretical approaches to the microfield distribution functions construction, touched here, this method is perhaps the most laborious and complicated for realization, as the procedure of finding solution is very complicated and cumbersome, and requires preliminary complex calculations of additional auxiliary functions.

Beside, this method essentially does not give universal results for actually the microfield distribution functions, as the distribution from the very beginning depends on the specific quantum properties of field ions and the emitter. At the same time, the doubt arises on how adequate and ample the developed notions about the character and the speed of establishing equilibrium between the emitter (test particle) and plasmas. Apparently DFT approach, of course, could not pretend on all completeness of plasma kinetic description, which is necessary for determination of plasma parameters. The latter are necessary as initial (input) data for the calculations of microfield distribution functions. Thus, the self-consistency of DFT approach in certain sense is limited.

Up to now, the DFT calculations of microfield distribution functions are performed only for several concrete cases and did not get wide-spread implementation. It is also unknown if there are any accessible for applied usage DFT codes for calculation of microfield distribution functions that are similar, for example, to APEX, although during the past years after the paper [47], the DFT approach as a method for description of atomic properties was considerably improved [120123].

At the same time, no essential progress was achieved in the extent of adequateness of description with the help of DFT the excited atomic states, and consequently the radiative processes with their participation [122]. However, the DFT application to calculations of atomic, molecular, and chemical properties are all over considered currently as quite effective from point of view of universality, simplicity, and also due to the high speed of performing corresponding calculations on contemporary computers [123].

However, in spite of pointed out drawbacks the studies, performed within DFT approach, provided very interesting and instructive physical results, which without any doubt are very valuable for the further development in this field.

2.8. Plasma Collective Oscillations
2.8.1. Microfield Separation on “Individual” and “Collective” Components

As it is evident, the “individual” and “collective” components of microfield are consequences of the same Hamiltonian of plasma charged particles [48]. Thus, many papers, dealing with the problem of microfield distributions of plasma collective oscillations, followed Bohm and Pines [30, 31], attempting to split the system of Coulomb particles with the help of canonical transformations of variables into the two weakly interacting subsystems, which could be considered independently [17 ,  Section  2], [48, 49]. In addition, plasma could usually be considered as uniform and isotropic.

Then, the total distribution function 𝑊(𝐹) of summary microfield 𝐹 under the condition that its components are additive, and the characteristic time scales of their variation are of the same order could be expressed as a convolution [4850]:

𝑊𝐹=𝑑𝐹𝑐𝑑𝐹𝑖𝑊𝑐𝐹𝐹𝑖𝑊𝑖𝐹𝑖,𝐹𝐹=𝑐+𝐹𝑖,(39) where 𝑊𝑐(𝐹), 𝑊𝑖(𝐹) are distribution functions of the collective 𝐹𝑐 and individual 𝐹𝑖 microfield components correspondingly. The microfield distribution function of individual component more or less approaches the Baranger-Mozer-type function, whereas the distribution function of collective component is practically Gaussian [48].

However, the final result of such approach happens to depend on the choice of phenomenological parameter, which controls the separation of subsystems, while the satisfactory methods for its exact determination was not ever found [30, 31, 48, 49].

The determination of this parameter invoked certain difficulties already in Bohm and Pines [30, 31] papers. Firstly, it is rather not a parameter but a function in the space of oscillations wave vectors 𝑘, and secondly the possibility of such separation is valid only in a quite narrow range of plasma parameters [30, 31]. Thereby, this procedure of collective variables extraction is not regular and universal.

In spite of that, it was declared in two publications (see [17, Section  2]), [49] about the realization of such separation after two canonical transformations although no explicit and proving demonstration of this statement was provided. These difficulties, of course, are due to strong interaction between subsystems, when, for example, the usage of the formal technique like the Zwanzig method of projection operators could not be rigorously justified, while the corresponding subsystems of quasiparticles, “dressed by interaction with each other”, could not be managed reasonably in order to separate them by transformations of initial Hamiltonian (see, e.g., [14, 79, 96]).

Basing on sound sense, the separation on “collective” and “individual” subsystems should be possible when the resonance interaction of plasma waves with plasma particles is not essential [6, 50, 82].

So, for these specially stipulated conditions in fact without proof, it was conventionally accepted to consider that the two practically independent and noninteracting subsystems of plasma waves and quasiparticles exist, for which it is possible to introduce independent distributions of microfields.

Per se it means the construction of distribution function of collective oscillations on the basis of independent models. Evidently, this question has sense only if the conditions of quasistatics or large modulation depth are fulfilled:

𝑑𝐸0𝜔𝑐,(40) where 𝑑 is the dipole moment of the emitter (test particle), 𝐸0, 𝜔𝑐 are the amplitude and characteristic frequency of collective plasma electric microfield component.

2.8.2. Rayleigh Distribution

In the assumption of isotropy, multimode property, additivity and randomness of phases of collective oscillations, the distribution of collective microfields is described by Rayleigh function [50]:

𝑊𝑐𝐹6=3𝜋1/2𝐹2𝐹23/2exp3𝐹22𝐹2,𝐹2=0𝑑𝐹𝐹2𝑊𝑐(𝐹).(41) This function is known also under the name of “distribution of random vector,” and by definition, it corresponds to nonpolarized summary field of oscillations.

In one-dimensional case, this distribution has the form

𝑊𝑐2(𝐹)=𝜋1/21𝐹21/2𝐹exp22𝐹2,(42) and then the electric field of oscillations has definite polarization.

2.8.3. Regular Oscillations

For linearly polarized, one mode, and sinusoidal field, it is possible to introduce instantaneous distribution function [51, 52] in the so-called dynamic case, when the atomic state dipole precession frequency in the electric field is much larger than the frequency of oscillations Ω and the reciprocal life time of atomic state 𝜏1e:

𝑑𝐸0Ω𝜏1e.(43) This distribution function has the form [4850]

𝑊𝑐1(𝐹)=𝜋1𝐸20𝐹2,(44) where 𝐸0 is the amplitude of sinusoidal oscillations.

2.9. Joint Distributions

In many problems, the information about distribution of the electrical field strength vector only is insufficient, and it is necessary to consider much more complex joint distribution functions of several scalar, vector, or tensor random variables at once [1921, 5370, 7481]. These variables could have some limitations on the intervals of their variation, as, for example, it happens in APEX. Seemingly, the first works, where the joint distributions in ideal gas of Coulomb (gravitating) particles were considered in application to problems of stellar dynamics, belong to Chandrasekhar and von Neuman [1921].

2.9.1. Distribution of Microfield and Its Space Derivatives

Let us consider the low-frequency ion joint distribution function 𝑊(𝐹;{𝜕𝐹𝛼/𝜕𝑥𝛽}) of the ion electric microfield vector 𝐹 and its spacial derivatives 𝜕𝐹𝑖/𝜕𝑥𝑘, forming the symmetric second rank tensor, following [5362] (compare [63, 64]). The “Spur” of this tensor is not equal to zero for the shielded ions, but is nullified in the case of Coulomb field. The values of arguments are sums of corresponding values of separate field ions, that is, the additivity condition is fulfilled:

𝐹=𝑗𝐹𝑗,𝜕𝐹𝛼𝜕𝑥𝛽=𝑗𝜕𝐹𝑗𝛼𝜕𝑥𝛽.(45) In general form, it is rather complex function in 9-dimensional space of variables: the 3 components of electric field vector and the 6 independent components of symmetric tensor of the second rank.

For arbitrary plasma ionization composition, the quasineutrality condition could be expressed as

𝑁=𝑁𝑒=𝑠𝑍𝑠𝑁𝑠,(46) where 𝑍𝑠, 𝑁𝑠 are the charge and partial concentration of field ions of 𝑠 species correspondingly.

The general expression for the joint distribution function then could be presented in the form

𝑊𝐹;𝜕𝐹𝛼𝜕𝑥𝛽=1(2𝜋)9𝑑3𝜌6𝑚=1+𝑑𝜎𝑚exp𝑖𝜌𝐹𝑖6𝑚=1𝜎𝑚𝜕𝐹𝛼𝜕𝑥𝛽𝑚𝜎𝐴𝜌;𝑚.(47) For the distribution characteristic function 𝐴(𝑘), the following general exponential representation is valid:

𝐴𝜎𝜌;𝑚𝜎=exp𝑁𝐶𝜌;𝑚.(48) Using the generalization of Baranger-Mozer cluster expansion [5358] (compare [63, 64]) on the case of arbitrary plasma composition, the index of exponent of characteristic function could be presented with the accuracy of up to the second-order terms over density in the following recording:

𝐶𝜎𝜌;𝑚=𝐶(𝑜)𝜎𝜌;𝑚𝑁2!𝐶(1)𝜎𝜌;𝑚,𝐶(𝑜)𝜎𝜌;𝑚=𝑠𝐶𝑠𝑑3𝑟𝑔𝑠𝑟𝑟𝜑𝑠𝜎𝜌;𝑟;𝑚,𝐶(1)𝜎𝜌;𝑚=𝑠,𝑠𝐶𝑠𝐶𝑠𝑑3𝑟1𝑑3𝑟2𝜑𝑠𝜌;𝑟1;𝜎𝑚𝜑𝑠𝜌;𝑟2;𝜎𝑚𝑔𝑠𝑠𝑟1;𝑟2𝑔𝑠𝑟𝑟1𝑔𝑠𝑟𝑟2,𝜑𝑠𝜎𝜌;𝑟;𝑚=1exp𝑖Φ𝑠𝜎𝜌;𝑟;𝑚,Φ𝑠𝜎𝜌;𝑟;𝑚𝐸=𝜌𝑠+𝑟6𝑚=1𝜎𝑚𝐸𝜕(𝑠)𝛼𝜕𝑟𝛽𝑚.(49) Here, 𝐶𝑠𝑁𝑠/𝑁, 𝑔𝑠𝑟(𝑟) is the pair correlation function of field ion from 𝑠 species with charge 𝑍𝑠 and the test ion with charge 𝑍0, immersed in the origin of reference frame, 𝑔𝑠𝑠(𝑟1;𝑟2) is the pair correlation function of field ions between each other with charges 𝑍𝑠 and 𝑍𝑠in the field of the test ion with charge 𝑍0, 𝐸𝑠(𝑟) is the elementary electric field, produced by any field ion (quasiparticle) of “s” species in the origin of the reference frame.

This field is determined by the effective interaction potential for such species in plasmas and could be described by the following equations:

𝐸𝑠𝑟=𝑒𝑍𝑠𝑟𝑟31𝜅𝑠,𝐸(𝑟)div𝑠=𝑟𝑒𝑍𝑠𝑟2𝜕𝜅𝑠(𝑟)𝜕𝑟4𝜋𝑒𝑍𝑠𝛿,𝑟𝒱𝑑3𝐸𝑟div𝑠𝑟=0.(50) The latter equations are followed from the properties of screening function 𝜅𝑠(𝑟), connected with its definition: 𝜅𝑠(0)=0, 𝜅𝑠()=1, so that the excess charge of free electrons around the ion 𝑍𝑠 is determined by the expression

𝛿𝑛𝑒(𝑠)1(𝑟)=𝑍4𝜋𝑠𝑟2𝜕𝜅𝑠(𝑟)𝜕𝑟.(51) Then the nonuniformity tensor components of elementary electric field are determined from

𝐺(𝑠)𝑘𝑖𝜕𝐸𝑟𝑠𝑘𝜕𝑥𝑖=𝑒𝑍𝑠𝑟53𝑥𝑖𝑥𝑘𝛿𝑖𝑘𝑟21𝜅𝑠𝑟(𝑟)+3𝜕𝜅𝑠(𝑟)+𝛿𝜕𝑟𝑖𝑘3𝑒𝑍𝑠𝑟2𝜕𝜅𝑠(𝑟).𝜕𝑟(52) Hence, it follows that the screening function 𝜅𝑠(𝑟)0 could be found, for example, on the basis of the recent DFT approach receipts [46, 47], and 𝐺(𝑠)𝑘𝑖(𝑟)𝐺(𝑠)𝑖𝑘(𝑟). In assumption that the field ions are bare nuclei here, the equations that determine the bound electrons distributions are not considered. It is assumed that quantum effects [1417, 46, 47, 71] are not essential in microfield distribution.

The joint distribution obtained earlier provides the instantaneous distribution function of the low-frequency individual ion component of plasma microfield and its spacial derivatives, which per se are defined on time scales 𝜏 of the order 𝜔1pe𝜏(𝑣𝑖𝑁𝑖1/3)1, where 𝜔pe is the plasma electron frequency, 𝑣𝑖 is the relative thermal ion velocity with respect to the test particle, and 𝑁𝑖 is the total ion density.

The basic ideas of this derivation were proposed by Baranger and Mozer and did not undergo any essential changes since that time, in spite of certain differences in posterior papers [2729, 3481, 96, 97], as they are inherent in microfield formalism.

It is important to underline that plasma polarization effects [46, 47, 5362] (or in other words appearance of nonuniformity in distribution of plasma electron density) are included in general form in this consideration from the very beginning via screening function and its derivatives. The integration over 𝐹 or over 𝜕𝐹𝑖/𝜕𝑥𝑘 components leads to separate distributions of microfield or its tensor of nonuniformity, and after implementation of appropriate approximations recovers known earlier results.

One of the most interesting properties of the joint distributions follows from the analysis of its moments 𝜕𝐹𝑖/𝜕𝑥𝑘𝐹 for a given value of 𝐹 that represent itself the averages of 𝜕𝐹𝑖/𝜕𝑥𝑘 over the joint distributions for the fixed vector value of 𝐹

𝑊𝐹𝜕𝐹𝑖𝜕𝑥𝑘𝐹=𝑁(2𝜋)3𝑑3𝐹𝐺𝜌exp𝑖𝜌𝐴𝜌𝑖𝑘,𝐺𝜌𝑖𝑘=𝐺𝜌(𝑜)𝑖𝑘+𝐺𝜌(1)𝑖𝑘,𝐺𝜌(𝑜)𝑖𝑘=𝜌𝑠𝐶𝑠𝐺(𝑠)𝑖𝑘,𝐺𝜌(1)𝑖𝑘𝑁𝜌=2𝑠,𝑠𝐶𝑠𝐶𝑠𝐺(𝑠𝑠)𝑖𝑘,𝐺𝜌(𝑠)𝑖𝑘=𝑑𝜌3𝑟𝑔𝑠𝑟𝑟exp𝑖Φ𝑠𝜌;𝑟𝐺(𝑠)𝑖𝑘,𝐺𝑟(𝑠𝑠)𝑖𝑘=𝑑𝜌3𝑟1𝑑3𝑟2𝑔𝑠𝑠𝑟1;𝑟2𝑔𝑠𝑟𝑟1𝑔𝑠𝑟𝑟2𝐺(𝑠)𝑖𝑘𝑟1exp𝑖Φ𝑠𝜌;𝑟11exp𝑖Φ𝑠𝜌;𝑟2+𝐺(𝑠)𝑖𝑘𝑟2exp𝑖Φ𝑠𝜌;𝑟21exp𝑖Φ𝑠𝜌;𝑟1.(53) It was found that the expressions for the first moments of the nonuniformity tensor could be presented via microfield distribution functions in general form [5255, 59]

𝑊𝐹𝜕𝐹𝑖𝜕𝑥𝑘𝐹=𝑁𝑠𝐶𝑠𝑑3𝑟𝑔𝑠𝑟𝑟𝐺(𝑠)𝑖𝑘𝐸𝑟𝑊𝐹𝑠𝑁𝑟2𝑠𝑠𝐶𝑠𝐶𝑠𝑑3𝑟1𝑑3𝑟2𝑔𝑠𝑠𝑟1;𝑟2𝑔𝑠𝑟𝑟1𝑔𝑠𝑟𝑟2𝐺(𝑠)𝑖𝑘𝑟1𝑊𝐸𝐹𝑠𝑟1𝐸𝑊𝐹𝑠𝑟1𝐸𝑠𝑟2+𝐺(𝑠)𝑖𝑘𝑟2𝑊𝐸𝐹𝑠𝑟2𝐸𝑊𝐹𝑠𝑟1𝐸𝑠𝑟2.(54) To carry out expressions that could be processed in numerical calculations, it is necessary to apply additional simplifications and approximations for correlation functions in the aforedescribed general formulas. For this it is presumed that the pair correlation function depends only on the module of particles radii-vectors difference and the Kirkwood approximation is used for disentanglement [710] of the three-particle correlations. This yields [5558, 80, 81]

𝑔𝑠𝑟𝑟𝑔𝑠𝑟𝑔(𝑟),𝑠𝑠𝑟1;𝑟2𝑔𝑠𝑠||𝑟1𝑟2||𝑔𝑠𝑟𝑟1𝑔𝑠𝑟𝑟2,𝑠𝑠||𝑟1𝑟2||𝑔𝑠𝑠||𝑟1𝑟2||1.(55) Then, it is possible to obtain the following general representations of correlation functions in the form of series over harmonics:

𝑠𝑠||𝑟1𝑟2||=𝑛=0(2𝑛+1)𝑃𝑛cos𝑟1𝑟2𝑠𝑠𝑛;𝑟1;𝑟2,𝑠𝑠𝑛;𝑟1;𝑟2=0𝑑𝑘𝑘2𝑗𝑛𝑘𝑟1𝑗𝑛𝑘𝑟2𝑠𝑠(𝑘),𝑠𝑠1(𝑘)=(2𝜋)3𝑑3𝑖𝑟exp𝑘𝑟𝑠𝑠(𝑟).(56) Here, 𝑃𝑛(𝑧) are the Legendre polynomials depending on cosine of the angle between vectors 𝑟1 and 𝑟2, where 𝑗𝑛(𝑦)- is the spherical Bessel function.

This allows to simplify general results, mentioned earlier, and obtain, for example, the distribution function of reduced microfield values 𝛽𝐹/𝐹0 (where 𝐹0 is the value of normal microfield [18]), more general expression than those known before [24, 25] (compare [72, 73, 164]):

𝑊𝐹=4𝜋𝐹2𝐹𝑊(𝐹),𝐴𝜌=𝐴(𝜌),0=Λ𝑒𝑁2/3,Λ2𝜋(4/15)2/3,(57)𝑊(𝛽)=2𝛽𝜋0Ψ𝑑𝑘𝑘sin𝑘𝛽𝐴(𝑘),𝐴(𝑘)=exp0(𝑘)+Ψ1(,Ψ𝑘)(58)0(𝑘)=4𝜋Λ3/2𝑠𝐶𝑠𝐼𝑠Ψ(𝑘),1(𝑘)=8𝜋2Λ3𝑠𝑠𝐶𝑠𝐶𝑠𝐼𝑠𝑠(𝑘),𝑟0𝑒𝐹01/2,𝐼(59)𝑠(𝑘)=0𝑑𝑥𝑥2𝑔𝑠𝑟𝑟0𝑥1sin𝑘𝜖𝑠(𝑥)𝑘𝜖𝑠,𝜖(𝑥)𝑠𝑍(𝑥)=𝑠𝑥21𝜅𝑠𝑟0𝑥,𝐼(60)𝑠𝑠(𝑘)=0𝑑𝑥1𝑥21𝑥10𝑑𝑥2𝑥22𝑔𝑠𝑟𝑟0𝑥1𝑔𝑠𝑟𝑟0𝑥2𝑛=0(1)𝑛𝑗(2𝑛+1)𝑛𝜖𝑠𝑥1𝛿𝑜𝑛𝑗𝑛𝜖𝑠𝑥2𝛿𝑜𝑛𝑠𝑠𝑛;𝑟0𝑥1;𝑟0𝑥2.(61) Here, the function Ψ1(𝑘) describes ion-ion correlations.

The explicit representation for distribution function allows to obtain analytical expressions for the first moments of nonuniformity tensor, describing its fundamental properties for the fixed value of the ion electric microfield vector:


𝜕𝐹𝑍𝜕𝑍𝐹=4𝜋𝑁𝑒3𝐵𝐷(𝛽)𝑃2(cos𝜃)+𝐵𝐷𝑂,(𝛽)(64)𝜕𝐹𝑌𝜕𝑋𝐹=𝜋𝑁𝑒3𝐵𝐷(𝛽)𝑃||2||2(cos𝜃)sin2𝜙2,(65)𝜕𝐹𝑍𝜕𝑋𝐹=2𝜋𝑁𝑒3𝐵𝐷(𝛽)𝑃||1||2(cos𝜃)cos𝜙,(66)𝜕𝐹𝑍𝜕𝑌𝐹=2𝜋𝑁𝑒3𝐵𝐷(𝛽)𝑃||1||2(cos𝜃)sin𝜙,(67) where 𝜃 and 𝜙 are the polar and azimuthal angles of vector 𝐹 in the laboratory Cartesian reference frame XYZ, 𝑃𝑛|𝑚|(𝑥) is the generalized Legendre polynomial.

The universal function 𝐵𝐷𝑂(𝛽) is due to plasma polarization effects [5559].

The universal functions 𝐵𝐷(𝛽) and 𝐵𝐷𝑂(𝛽) with an account of ion-ion correlations are determined by the expressions, where the terms with upper subindex (1) are connected with ion-ion correlations:

𝐵𝐷(𝛽)=𝐵𝐷(𝑜)(𝛽)+𝐵𝐷(1)𝐵(𝛽),𝐷𝑂(𝛽)=𝐵(𝑜)𝐷𝑂(𝛽)+𝐵(1)𝐷𝑂𝐵(𝛽),𝐷(𝑜)(𝛽)=12𝜋𝛽2𝑊(𝛽)𝑠𝐶𝑠𝑍𝑠𝑏𝑠(𝐵𝛽),𝐷(1)(𝛽)=12𝜋𝛽2𝑊(𝛽)𝑠𝑠𝐶𝑠𝐶𝑠𝑏𝑠𝑠𝐵(𝛽),(𝑜)𝐷𝑂2(𝛽)=𝜋𝛽2𝑊(𝛽)𝑠𝐶𝑠𝑍𝑠𝑏𝑠(𝑜)𝐵(𝛽),(1)𝐷𝑂2(𝛽)=𝜋𝛽2𝑊(𝛽)𝑠𝑠𝐶𝑠𝐶𝑠𝑏(𝑜)𝑠𝑠(𝛽),(68) The functions, describing the first terms of expansion and connected with quadrupolar tensor 𝑏𝑠(𝛽) and scalar 𝑏𝑠(𝑜)(𝛽) correspondingly, could be transformed to the following form:

𝑏𝑠(𝛽)=0𝑑𝑘𝑘2𝐴(𝑘)𝑗2(𝑘𝛽)Φ𝑠𝑏(𝑘),𝑠(𝑜)(𝛽)=0𝑑𝑘𝑘2𝐴(𝑘)𝑗0(𝑘𝛽)Φ𝑠(𝑜)(𝑘),(69) where the Fourier-components of nonuniformity tensor Φ𝑠(𝑘) and its trace Φ𝑠(𝑜)(𝑘) of the field ion for “s” species enters the integrands:

Φ𝑠(𝑘)=0𝑑𝑥𝑥2𝑔𝑠𝑟𝑟0𝑥𝑗2𝑘𝜖𝑠(𝑥)Φ𝑠Φ(𝑥),𝑠(𝑜)(𝑘)=4𝜋0𝑑𝑥𝑥2𝑔𝑠𝑟𝑟0𝑥𝑗0𝑘𝜖𝑠𝑟(𝑥)30𝛿𝑛𝑒(𝑠)𝑟0𝑥𝑍𝑠,Φ𝑠1(𝑥)𝑥31𝜅𝑠𝑟0𝑥+𝑥3𝜕𝜅𝑠𝑟0𝑥.𝜕𝑥(70) It is convenient to represent the next-order functions 𝑏𝑠𝑠(𝛽) and 𝑏(𝑜)𝑠𝑠(𝛽) due to ion-ion correlations in the following form using the same designations:

𝑏𝑠𝑠(𝛽)=0𝑑𝑘𝑘2𝐴(𝑘)𝑗2(𝑘𝛽)𝑏𝑠𝑠𝑏(𝑘),(𝑜)𝑠𝑠(𝛽)=0𝑑𝑘𝑘2𝐴(𝑘)𝑗0(𝑘𝛽)𝑏(𝑜)𝑠𝑠(𝑘),(71) where the corresponding Fourier components of correlation contributions are represented in the series

𝑏𝑠𝑠(𝑘)=0𝑑𝑥1𝑥21𝑥10𝑑𝑥2𝑥22𝑔𝑠𝑟𝑟0𝑥1𝑔𝑠𝑟𝑟0𝑥2𝑏𝑠𝑠𝑘;𝑥1;𝑥2,𝑏𝑠𝑠𝑘;𝑥1;𝑥2=𝑍𝑠Φ𝑠𝑥1𝑗2𝑘𝜖𝑠𝑥1𝑠𝑠0;𝑟0𝑥1;𝑟0𝑥2𝑛=0(1)𝑛(2𝑛+1)3𝑛(𝑛1)2𝑘2𝜖2𝑠𝑥1𝑗1𝑛𝑘𝜖𝑠𝑥1+3𝑘𝜖𝑠𝑥1𝑗𝑛+1𝑘𝜖𝑠𝑥1𝑗𝑛𝑘𝜖𝑠𝑥2𝑠𝑠𝑛;𝑟0𝑥1;𝑟0𝑥2,𝑏(𝑜)𝑠𝑠(𝑘)=0𝑑𝑥1𝑥21𝑥10𝑑𝑥2𝑥22𝑔𝑠𝑟𝑟0𝑥1𝑔𝑠𝑟𝑟0𝑥2𝑏(𝑜)𝑠𝑠𝑘;𝑥1;𝑥2,𝑏(𝑜)𝑠𝑠𝑘;𝑥1;𝑥2=4𝜋𝑟30𝛿𝑛𝑒𝑟0𝑥1𝑗0𝑘𝜖𝑠𝑥1𝑠𝑠0;𝑟0𝑥1;𝑟0𝑥2𝑛=0(1)𝑛(2𝑛+1)𝑗𝑛𝑘𝜖𝑠𝑥1𝑗𝑛𝑘𝜖𝑠𝑥2𝑠𝑠𝑛;𝑟0𝑥1;𝑟0𝑥2.(72) Now, it is useful to present substitutions for obtaining previous results [24, 25] in the linearized Debye-Hückel approximation for correlation functions of field particles from expressions, derived earlier:

𝜅𝑠𝑟0𝑥𝜅𝐷𝑠[]𝑟(𝑥)1exp𝑎𝑥(1+𝑎𝑥),𝑎0𝑟𝐷,𝑟𝐷𝑇𝑒4𝜋𝑒2𝑁𝑒,𝑔𝑠𝑟𝑟0𝑥exp𝑍0𝑍𝑆Θ𝑎2Λ3/2[]4𝜋exp𝑎𝑥𝑥,𝑇Θ𝑒𝑇𝑖,𝑛;𝑥1;𝑥2Θ𝑎3Λ3/24𝜋𝑓>𝑛𝑎𝑥1𝑓<𝑛𝑎𝑥2,𝑓>𝑛(𝑧)(1)𝑛𝑧𝑛𝑑𝑧𝑑𝑧𝑛𝑒𝑧𝑧,𝑓<𝑛(𝑧)𝑧𝑛𝑑𝑧𝑑𝑧𝑛sinh(𝑧)𝑧.(73) In these formulas, the conventional designations from original works [2426, 72, 73, 164] are used, while 𝑇𝑖,𝑇𝑒 are the electron and ion temperatures correspondingly.

After performing the pointed out simplifications for one sort of field ions, the equations (57)–(61) reproduce the Baranger-Mozer results for low-frequency ion component of plasma microfield distribution function.

In Figure 11 the Baranger-Mozer (BM) functions 𝑊(𝛽) for several 𝑎 values, calculated along with this section derivation in the charged point, and the results of corresponding MC calculations are presented. The comparison has shown that within the accuracy of the figure drawing BM and MC data are indistinguishable from each other. It should be noted that the direct calculations of general joint distribution functions is very complicated task even for the current powerful supercomputers. Moreover, sometimes even a definition of such functions is difficult to accomplish. That is why the most accessible approximation is characterizing these functions with the help of its moments of various ranks over different variables.

In practical application, it is important to keep in mind that even in the case of calculations the simplest distribution functions, depending only on the module of reduced field, the known difficulties exist with arising oscillations in results at small and especially at large 𝛽 due to Fourier transform. That is why the most accepted method of introduction of various distribution functions in calculations is connected with the use of their tables. As a rule under calculations of the sum of terms with ion-ion correlations, the convergence is rather rapid, and it is quite enough to include only 3-4 first terms of the sum [2226].

In Figure 12, the universal function 𝐵𝐷(𝛽) in the charged point 𝑍𝑟=𝑍1=1 for different values of parameter 𝑎 is presented [58]. The dashed lines show the results of calculations using only the first two terms of cluster expansion and three terms of expansion over 𝑙. The nearest neighbor result is also shown as 2𝛽3/2. Solid lines represent MC results. In Figure 13, the illustration of the universal function 𝐵𝐷𝑂(𝛽) for different values of parameter 𝑎 in the charged point 𝑍𝑟=𝑍1=1, according to [58] similarly to 𝐵𝐷(𝛽) is presented. Only the two first terms of cluster expansion and three terms of expansion over 𝑙 are used. Solid lines show MC results (see [58]). The analysis of 𝐵𝐷𝑂(𝛽) asymptotic for small 𝛽 discovers that this function stems to constant at 𝛽0. Moreover, from graphs, generally speaking, the presence of another constant is evident in asymptotic for large 𝛽. These properties have principal significance and signalize on the necessity of simultaneous correct account for electron contribution under consideration of quadrupole interaction, for example, in the spectral lines broadening [59, 60, 62].

The described-here Baranger-Mozer cluster expansion approach for joint distribution function of ion microfield and its nonuniformity tensor with Debye-Hückel correlation functions was firstly proposed by the author of this review in [54] and completely realized in [55, 56], where the functions 𝐵𝐷(𝛽) and 𝐵𝐷𝑂(𝛽) were defined and its asymptotics was described. Two years later and with much less generality, similar results appeared in [64]. Interestingly, the designations in [64] coincide with the corresponding from [55].

In order to obtain results for strongly coupled plasmas, it is necessary to apply MC [5762], molecular dynamics or APEX approaches. However, the APEX scheme allows only some reformulation of general expression on the basis of (54) relations, and then derivation from it the expressions for the first moments [65, 68, 69]. However, it is not possible to construct with APEX namely the joint distributions and then to derive the first moment from such a function, if it would exist in APEX.

Indeed, within the APEX these operations do not commute (see [58, 65, 68, 69]), and APEX authors avoid the attempts of construction joint distribution functions [165]. Resultantly, they attempt to generalize the relation for the first moment, basing on [165] (where the correlation function in the given electric field 𝜖 was introduced 𝑔(𝑟;𝜖)) and miss the partial derivative from 𝐶(𝜌;𝜎𝑚) over one of 𝜎𝑚 components, that during reduction of microfield function to APEX form provides the effective density distribution 𝑔e.APEX(𝑟) [59, 60], diverging at large values of argument.

For the APEX distribution function itself and the “quadrupole” part of the first moment of nonuniformity tensor, this increase is damped by corresponding decrease of function in (60) (see [57, 58, 80, 81]):

1sin𝑘𝜖𝑠(𝑥)𝑘𝜖𝑠(𝑥),𝜖𝑠𝑍(𝑥)=𝑠𝑥21𝜅𝑠𝑟0𝑥(74) and function in (70):

𝑗2𝑘𝜖𝑠(𝑥),(75) but in the scalar part of first moment, this invokes divergence [68, 69, 80, 81].

Thus, pointed out noncommutativity is connected on one hand with the behavior of effective screening 𝛼 in APEX, which is stronger than the Debye one (see Figure 14), and on the other hand with the outcome to constant for small reduced field values of the polarization (scalar) part of the first moment of nonuniformity tensor (see Figures 12 and 13), which was not taken into account in the first APEX work on the first moment of nonuniformity tensor calculation [65]. These factors both lead to divergence at upper limit in the polarization part of the first moment of nonuniformity tensor if to derive it from expression for the joint distribution function of microfield and its spacial derivatives within APEX (see [57, 58, 65, 68, 69, 76, 80, 81]).

In order to obtain the finite result, APEX authors in fact calculate field derivative, averaged over the APEX distribution function. This way means that such an average could be performed over any microfield distribution, and consequently, the given microfield is associated with the derivative as “if of quite other microfield”. So, the presence of unequivocal connection between the field and its derivative is not requested, which does not correspond to the setting of a problem under consideration, and from the logic point of view is absurd.

Nevertheless, the general derivation in [65] contains several new interesting formal results. Indeed, in [65], the constrained distributions of the nonuniformity tensor components at the fixed value of microfield are introduced instead of joint distribution functions, which formally allows to avoid the approximation of preaveraged Hamiltonian over components of microfield nonuniformity tensor. However, to our opinion, the numerical calculation of such functions is not simpler than the full-joint distribution itself and avoiding the approximation of averaged Hamiltonian has only an illusive character.

The described difficulties are due to the fact that the APEX distribution itself already is derived under certain limitations, imposed by fulfillment of the (24) condition for the second moment of microfield in the charged point. So, trying to preserve the natural asymptotic for large field values on one hand, and on the other hand comprehending well that 𝑔e.APEX(𝑟) is divergent al large 𝑟, the authors of [65, 165] decided not to use the first moment of the joint distribution function, but to use the mean value of derivative over APEX distribution. ( The presence of divergence and non-normalization of 𝑔e.APEX(𝑟) and the information on real values of screening parameter 𝛼 in comparison with the Debye reciprocal length was not mentioned in previous APEX papers till [68, 69].) Per se introduced in [165], the definitions of averaged values deviate from the conventional approach of Chandrasekhar-von Neuman [1921] and represent itself some additional approximation that is not connected with formalism of joint distribution functions, and which region of validity is at least unclear.

Besides the fact that APEX in this case does not allow to construct namely joint distribution function of the electric field strength vector and its nonuniformity tensor in the conventional “Chandrasekhar” sense [1921], this problem seemingly is connected with inadmissibility of separate consideration of the electron and ion contributions to polarization interaction. Indeed, simultaneous consideration in real physical problems of the ion and electron contributions to polarization interaction lead to conversion to zero, at infinite distances, the constant in summary polarization interaction, and in this way remove the problem of pointed out divergence (see [59, 60]). The physical sense of this is that the distribution of ion charge also becomes nonuniform in response to the nonuniform distribution of electron density (see [59, 60]), and both effects compensate each other at sufficient distances from test charge according to general plasma quasineutrality condition. The alike outlook is presented in the interesting paper of Ortner, Valuev, and Ebeling, where such model is called as OCP on polarizable background (POCP).

On the other hand, it is obvious that up to now, not all variants of joint distribution function construction, using APEX, were analyzed.

The comparison of results for the moments of total nonuniformity tensor (without separation of contributions on tensor and scalar parts) for fixed value of the field 𝜖 using MD method, APEX (on the basis of relation (54)), and MC (from [58]) for 𝑇=800 eV and 𝑁=1024 cm3 is presented according to [68, 69] in Figure 14 versus the reduced field values. The designations are taken from original work [68, 69]. The designations in the figure are connected with the conventional ones in the present paper in the following way:

𝑑𝑥𝐸𝑥𝜖𝜕𝐹𝑥𝜕𝑥𝛽,𝑑𝑧𝐸𝑧𝜖𝜕𝐹𝑧𝜕𝑧𝛽,(76) and the results are obtained after average over angles of microfield vector in the expressions (62)–(67).

In Figure 14, the dependencies presented are obtained specially for distribution of nearest neighbor (NN) with screening by plasma electrons and without it, but neglecting the scalar part of nonuniformity tensor. It is seen that the APEX version for average values of nonuniformity tensor [68, 69] noticeably deviates at large reduced microfield values from the results of the nearest neighbor distribution. The MD results practically coincide with MC ones, while APEX curves are located inside MC curves in Figure 14. The presence of constant at small values of reduced microfield is confirmed. This comparison shows that the APEX calculation of averaged components of nonuniformity tensor in principle gives sound results in the context of coincidence with values of the first moments of nonuniformity tensor, although its derivation within APEX could not be recognized as completely correct and justified.

It seems instructive to demonstrate how with the increase of plasma coupling the differences of radial distribution functions within Debye and 𝐻𝐶𝑁 approximations become ever more pronounced, which is illustrated in Figure 15. Partially, namely, the implementation of 𝐻𝐶𝑁 correlation functions provides the APEX success in description of microfield distribution functions for strongly coupled plasmas. As was already mentioned with the increase of plasma coupling, the considerable changes of pair correlation function 𝑔(𝑟) occur, which acquires oscillations versus 𝑟 for large values of Γ, which are due to the formation of the short-range ordering [166, 167]. This is explicitly demonstrated in Figures 16 and 17 within one component plasma model (OCP) for the different values of plasma coupling Γ. The presented results are obtained by the different authors with ten years interval [166, 167]. These data demonstrate visually qualitative changes of 𝑔(𝑟) versus variation of plasma coupling in the range Γ=0.11, Γ=1020, Γ100. At the same time, the special study, done in [58], showed that using in the Baranger-Mozer scheme 𝐻𝐶𝑁 correlation functions and MC correlation functions does not eliminate completely the noticeable discrepancies between APEX and Baranger-Mozer microfield distribution functions for strongly coupled plasmas.

It should be taken into account that the linearization approximation for Debye-Hückel correlation functions is not inalienable part of Baranger-Mozer scheme, which allows the usage of any arbitrary accurate correlation function, including of the nonlinearized Debye-Hückel ones as well.

2.9.2. Distribution of Microfield and Its Time Derivatives

In the most general form, the joint distribution functions of microfield and its time derivatives could be written as Fourier-transform of its characteristic functions analogously to the previous section [1921, 7481], if following Chandrasekhar to consider the values of derivatives at initial time 𝑡=0. These functions [7481] describe microfield evolution in time on sufficiently small time intervals.

Without loss of generality, as an example, we present the function ̇̈𝑊(𝐹;𝐹;𝐹) [5456, 80, 81] (compare with [1921]):

𝑊̇̈𝐹=1𝐹;𝐹;(2𝜋)9𝑑3𝑑𝜌3𝑑𝜎3𝜉̇̈𝐹𝐴𝜉𝜉.𝜉exp𝑖𝜌𝐹+𝜎𝐹+𝜉𝜌;𝜎;=exp𝑁𝐶𝜌;𝜎;𝐴𝜌;𝜎;,(77) It is worthy to underline that in spite of the presence of the first and the second derivatives of microfield over time, this is per se the instantaneous static distribution function. At the same time, of course, it is very complex function in 9-dimensional space of its variables. Using the same designations and plasma composition, we express ln𝐴(𝜌;𝜎;𝜉) as in the previous section:

𝐶𝜉𝜌;𝜎;=𝐶(𝑜)𝜉𝑁𝜌;𝜎;𝐶2!(1)𝜉,𝐶𝜌;𝜎;(𝑜)𝜉=𝜌;𝜎;𝑠𝐶𝑠𝑑3𝑢𝑠𝑤𝑠𝑢𝑠𝑑3𝑟𝑔𝑠𝑟𝑟𝜑𝑠𝜉,𝐶𝜌;𝑟;𝜎;(1)𝜉=𝜌;𝜎;𝑠,𝑠𝐶𝑠𝐶𝑠𝑑3𝑢𝑠𝑤𝑠𝑢𝑠𝑑3𝑢𝑠𝑤𝑠𝑢𝑠𝑑3𝑟1𝑑3𝑟2𝜑𝑠𝜌;𝑟1𝜉𝜑;𝜎;𝑠𝜌;𝑟2𝜉𝑔;𝜎;𝑠𝑠𝑟1;𝑟2𝑔𝑠𝑟𝑟1𝑔𝑠𝑟𝑟2,𝜑𝑠𝜉𝜌;𝑟;𝜎;=1exp𝑖Φ𝑠𝜉,Φ𝜌;𝑟;𝜎;𝑠𝜉𝐸𝜌;𝑟;𝜎;=𝜌𝑠̇𝐸𝑟+𝜎𝑠+̈𝐸𝑟𝜉𝑠.𝑟(78) In distinction from the previous section, these expressions contain the additional integration over thermal velocities 𝑢𝑠 of field ions with the velocity distribution function 𝑤𝑠(𝑢𝑠).

Moreover, ̇𝐸𝑠(𝑟) and ̈𝐸𝑠(𝑟) determine the first and the second time derivatives of elementary electric field, produced by arbitrary field ion of 𝑠 species in the origin of reference frame and having the same value of relative velocity at 𝑡=0: ̇𝐸𝑠=𝑟𝑒𝑍𝑠𝑟3𝑣3𝑛𝑛𝑠𝑣𝑠1𝜅𝑠𝑟(𝑟)+3𝜕𝜅𝑠(𝑟)+𝑣𝜕𝑟𝑠3𝑒𝑍𝑠𝑟2𝜕𝜅𝑠(𝑟),̈𝐸𝜕𝑟𝑠𝑣𝑟;𝑠;̇𝑣𝑠=̈𝐸(1)ṡ𝑣𝑟;𝑠+̈𝐸𝑠𝑣𝑟;𝑠,̈𝐸(1)ṡ𝑣𝑟;𝑠=𝑒𝑍𝑠𝑟3̇𝑣3𝑛𝑛𝑠̇𝑣𝑠1𝜅𝑠(𝑟𝑟)+3𝜕𝜅𝑠(𝑟)+𝜕𝑟𝑒𝑍𝑠3𝑟2𝜕𝜅𝑠(𝑟)̇𝑣𝜕𝑟𝑠,̈𝐸𝑠𝑣𝑟;𝑠=3𝑒𝑍𝑠𝑟42𝑣𝑠𝑣𝑛𝑠+𝑛𝑣2𝑠1𝜅𝑠𝑟(𝑟)+3𝜕𝜅𝑠(𝑟)𝑣𝜕𝑟5𝑛𝑛𝑠21𝜅𝑠7(𝑟)+𝑟15𝜕𝜅𝑠(𝑟)𝑟𝜕𝑟2𝜕152𝜅𝑠(𝑟)𝜕2𝑟,(79) where 𝑛𝑟/𝑟, and 𝑣𝑠=𝑢𝑠𝑢𝑟 is the relative thermal velocity of field ion of “s” species with respect to the test particle with velocity 𝑢𝑟. It is seen that there are summands, containing ̇𝑢𝑠, that is,

̇𝑣𝑠=̇𝑢𝑠̇𝑢𝑟=𝑒𝑍𝑠𝑚𝑠𝐹𝑟𝑒𝑍𝑟𝑚𝑟𝐹(0),(80) where 𝐹(𝑟) is the microfield in the location point of field ion of “s” species, and 𝑚𝑠,𝑚𝑟 are masses of the field ion and test particle correspondingly. These terms cause nonlinearity and loss of locality of joint distribution, if to include them into the expression for the second derivative.

Indeed, the microfield distribution at the origin of reference frame becomes dependent on microfield values in the total space. This controversy could be removed, assuming that the thermal velocities of field ions are constant due to stationarity conditions, as it was done in Chandrasekhar papers [19, 20], namely, ̇𝑢𝑠=0 for all 𝑠. In the opposite case, the back reaction of field ions requests special study, which is beyond the frames of the present work.

On the other hand, there are terms due to polarization effects as well. That is why the results of Chandrasekhar and von Neuman could be reproduced only after discarding the neutralization background of electrons by setting 𝑁𝑒=0. Here, it is supposed as before that all field ions are bare nuclei.

This joint distribution provide instantaneous low-frequency distribution function of individual (but many-body) ion component of plasma microfield and its time derivatives, which are defined on time scales 𝜏 of the order of 𝜔1pe𝜏(𝑣𝑖𝑁𝑖1/3)1, where 𝜔pe is the electron plasma frequency, 𝑣𝑖 is the relative thermal velocity of field ions with respect to the test particle.

We note that this distribution in distinction from Chandrasekhar results includes effects of neutralizing background of plasma electrons and its polarization (or in the other words the appearance of nonuniformity in distribution of free electrons). The convolution over components 𝐹 or ̇𝐹 leads to separate distributions of the field and its derivatives, and after corresponding simplifications reproduces known results.

As it was already pointed out, the computation of joint distributions is very complex problem and it is possible now to present only some unique examples of such calculations [58, 70, 76], which contain as a rule many additional approximations and simplifications (only the projections of such functions are calculated with fixed values of a part of variables).

So, one of few methods to characterize these distributions is the calculations of their moments. This is achieved by the convolution over 𝑑3̈𝐹. After that, it is possible to obtain the following expressions for the first moment of ̇𝐹 for a given value of 𝐹