Kubo Fluctuation Relations in the Generalized Elastic Model
The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. In this paper we show that the Fractional Langevin Equation (FLE) is a suitable framework for the study of the tracer (probe) particle dynamics, when an external force acts only on a single point (tagged probe) belonging to the system. With the help of the Fox function formalism we study the scaling behaviour of the noise- and force-propagators for large and short times (distances). We show that the Kubo fluctuation relations are exactly fulfilled when a time periodic force is exerted on the tagged probe. Most importantly, by studying the large and low frequency behaviour of the complex mobility we illustrate surprising nontrivial physical scenarios. Our analysis shows that the system splits into two distinct regions whose size depends on the applied frequency, characterized by very different response to the periodic perturbation exerted, both in the phase shift and in the amplitude.
The generalized elastic model (GEM) has been firstly introduced in  by the following equation:The general formulation in (1) is given for the -dimensional stochastic field defined in the -dimensional infinite space . The white noise satisfies the fluctuation-dissipation (FD) relation; that is,(), where corresponds to the hydrodynamic friction kernel whose Fourier transform isif . The fractional derivative , defined via its Fourier transform by  has another common definition given in terms of the Laplacian as . GEM (1) governs the dynamics of polymers [4–7], elastic chains [8–10], membranes [5, 11–15], and rough surfaces [16–21] among others. Equation (1) also reproduces the anomalous diffusive behaviour of systems such as crack propagation  and contact line of a liquid meniscus . Each one of the above-mentioned physical systems corresponds to a given set of the parameters defining GEM (1), namely, , , and , with in the case of ( in (3)).
In [24, 25] we have considered the case of GEM (1) with an external local perturbation, namely, a force acting only on the probe in , hereafter called the tagged probe. Under such condition, (1) transforms to the following stochastic evolution equation:Here is a force functional of the stochastic fields and of the time : it represents the external perturbation applied to the probe particle placed at the position .
In this paper we show that the Kubo fluctuation relations rigorously hold when the applied perturbation is a time periodic force. To the best of my knowledge, the problem of the study of the response of a generalized elastic system to a time periodic perturbation has never been treated, not in the context of polymers or membranes nor when one is concerned with the dynamics of rough surfaces. The performed analysis demonstrates that the macroscopic effects are persistent in time. Indeed the system splits into two macroregions whose size is defined by the value of the applied frequency : the responses of these regions differ markedly in amplitude and phase, allowing us to make experimentally testable predictions about the viscoelastic properties of the system. We show this by adopting the Fox -function formalism, which constitutes an excellent tool for the compact representation and the systematic analysis of the scaling properties of any correlation function and drift. Our analysis shows that the low and high frequency behaviours of the system response to the external perturbation reveal an extremely rich physical scenario in both the response amplitude and the phase. We also show that the -functions describe properly the scaling of the noise- and force-propagators in the limit of large and short times (distances).
Fox -functions are defined through the Mellin transformwith and . is given bywhere and are positive numbers, while and are complex. Empty products are interpreted as being unity. For convenience throughout this paper we adopt the following short notation:In general, Fox -functions gain more and more popularity among the scientific community, for their very general nature, which allows tackling different phenomena in a unified and elegant framework. Applications include non-Debye relaxation processes [26, 27], anomalous diffusion [28–31], reaction-diffusion equations , relaxation and reaction processes in disordered systems , and fractional Schrödinger equation , to name a few. The book  serves a deep analysis of properties of the Fox -functions. Recent monograph  lists many useful properties of the Fox -functions, together with some applications, for example, in astrophysics. The handbook  contains the list of useful properties and integrals of the Fox -functions.
This paper is organized according to the following scheme. In Section 2 we recall the FLE framework in the absence of any applied force. We also report the two-point two-time correlation functions making use of the Fox -function formalism. In Section 2.2 we study the scaling short and long times behaviour of the noise-propagator in case of long range and local hydrodynamics. Section 3 is devoted to the FLE under the action of a localized potential; in particular in Section 3.1 we derive the scaling limits of the noise-propagator. In Section 4 we deal with the situation where the force applied to the probe particle in is time periodic: we study the scaling properties of the probe average velocity drifts, demonstrating the validity of the Kubo fluctuation relations (KFR) and Green-Kubo relation. We also derive exact results for the single file model and compare them with the outcomes of numerical simulations. In Appendix A we list the Fox -functions properties that we use in our analysis; in Appendix B we report the technique for the asymptotic solution of Fourier integrals; in Appendix C Olver's theorem for solving integrals in the asymptotic limit, using the Laplace's method, is shown; in Appendix D we derive the exact expression for the Riemann-Liouville fractional derivative applied to an exponential function.
2. Fractional Langevin Equation
Equation (1) governs the dynamics of a linear stochastic system immersed in a viscous fluid. Let us take a different point of view: let us consider the dynamics of a tagged point (hereafter named probe or tracer without distinction) on the system described by (1). The probe can represent a spot on a membrane surface , or a tracer particle among those composing a single file system [38, 39], or a tagged monomer in polymeric chains . In  it was shown that the stochastic equation describing the motion of the probe coordinate placed at position is the following Fractional Langevin Equation: whereThe pseudodifferential operatorrepresents the left side Riemann-Liouville derivative with lower bound [3, 41]. We remind the reader that the derivative (13) is equivalent to the Caputo  derivative; that is,if the lower bound in (13) is set as .
The tracer dynamical representation provided by (9) is only valid whenever ; that is, (). In the case of local hydrodynamics () it is sufficient to set and . The noise appearing on the RHS of (9) is defined aswhere we have introduced the noise-propagator as [24, 25]From definition (15), it appears clear that defines the spatiotemporal retarded effects that the Gaussian white noise acting on the point at time has on the probe in at time . The scaling of for large and short times will be discussed in Section 2.2, resorting to its compact expression furnished in terms of the Fox -functions :where the correlation time has been defined as [24, 25, 43] The correlation properties of the fractional Gaussian noise (fGn) are encapsulated in its -function expression Setting in (19), one recovers the (generalized) fluctuation-dissipation relation; that is,
2.1. Correlation Functions
Any correlation function can be obtained within the FLE framework in (9) . Moreover, the correlation functions can be casted in a compact and elegant expression by using the -function formalism, which presents the practical advantage to study the limiting behaviours in a straightforward way. By instance, the position-position correlation function is expressed as We notice that this expression presents a divergence in the limit of , as can be checked by expanding the Fox -function using the property (A.10). The same limit, however, allows extracting the position-position autocorrelation function:The divergence can be avoided by defining the time difference , whose correlation function reads where has been introduced in . It isin the regime, for i) long range, (ii) local (), and (iii) local () hydrodynamic interactions, respectively. When we have invariably
The velocity correlation function gets the following expression:If we want to calculate the autocorrelation function we must set , achievingthanks to (A.10).
Finally we report the Fox function expression for the position-velocity correlation functionand its autocorrelation counterpart
2.2. Scaling Behaviour
In this subsection we analyze the scaling behaviours of the noise-propagator for times smaller and larger than the diffusion time . Our analysis includes both long ranged and local hydrodynamic systems.
We start by considering the long range hydrodynamic systems.
The same result is obtained from expression (16). Indeed, by applying the following change of variable , the integral can be solved by the method furnished in Appendix B. In particular, thanks to (B.2), we have
We can obtain the same result studying the long time limit of the integral expression (16). Using the Laplace method for asymptotic integrals, as reported in Appendix C, we obtain that the integral in (16) can be solved:
We now turn to the study of systems with local hydrodynamic interactions.
Performing the change of variable in (16) we solve the ensuing integral according to (B.2), recalling that and for systems characterized by local hydrodynamic interactions. Therefore we obtainwhich is reported to (36) once we apply (32).
(ii) Consider . The long time limit expression for local hydrodynamic interactions can be achieved from the second order expansion (A.11). However a simple but tedious analysis reveals that different terms may lead the asymptotic expansion whether or . In particular one has
The integral expression (16) can be handled to obtain the former result. Using Laplace’s method outlined in Appendix C, one haswhich coincides with the first of (38), once we perform the Riemann-Liouville fractional derivative according to (32). However, the range of applicability of expression (32) is limited to systems satisfying the condition . In order to reproduce the result, one has to handle expression (16) by applying the change of variables , getting
In Appendix D we calculate the long time limit of the fractional derivative applied to an exponential function. Hence, thanks to (D.4), we end up with the following equation:which corresponds to the second of (38), once the integral is solved .
The physical interpretation of the results obtained in this section can be explained as follows. Consider a tracer in at time : the fGn acting on it is given by expression (15). From the former analysis it follows that the space-time roughly splits into two regions, and , where the propagator scaling behaviour is correspondingly different. Points lying within the first region are characterized by a propagator whose expressions are given by (30) or (36) whether the system presents long ranged or local hydrodynamic interactions, respectively. Roughly speaking, the contributions of these points to the fGn (15) are very small, asymptotically zero. To the contrary, the main contributions are given by the points in the second region, as it is apparent from (33) and (38). Remarkably, we notice that if the system is identified by local hydrodynamics, the contribution given by probes in the inner region can be drastically different whether or . This class of systems are usually termed Family-Vicsek ()  or superrough () systems [46, 47], in correspondence with the different anomalous roughening properties.
3. Fractional Langevin Equation with Applied Force
We consider the case of a localized applied potential as expressed by the GEM constitutive equation (5). In [24, 43] it was shown that the FLE ruling the motion of a probe at a generic position iswhere the force-propagator has the same meaning of the force-propagator , representing the convolution kernel by which an external perturbation exerted at the point at time is transferred to the point at time . The -function expression of the force-propagator is  while the corresponding integral form readsThe fGn on the RHS of (42) satisfies the same correlation properties in (19). Finally, the FLE expression for the probe particle placed at is
3.1. Scaling Behaviour
In analogy to Section 2.2, we study the scaling behaviours of the force-propagator in the short and long time limits for long ranged and local hydrodynamic systems, respectively.
It is possible to recover the same result studying the long time limit of the integral expression (44). Using the Laplace method for asymptotic integrals, as reported in Appendix C, we obtain that the integral in (44) can be solved:
Plugging this result into (44) and performing the time fractional derivative (32), we obtain the first of (47). The result for is recovered by first changing the variable in the integral appearing in (44); then the fractional derivative is applied to the exponential according to the rule (D.4), getting as intermediate result
The scaling behaviour of the local hydrodynamic force-propagator can be inferred from the corresponding behaviour of the force-propagator. As a matter of fact both expressions coincide but for a constant factor , as it is immediately apparent by a direct comparison of (16) and (44).
Any perturbation exerted on the tagged probe is therefore transferred to a generic probe placed in by the action of . As in the case of , however, the contributions are very different if the probes' distance is such that or . In particular we notice that, in case of long ranged hydrodynamics, if the tagged probe is in the inner region , its influence on the motion of the untagged probe might be considerably different according to whether we are considering Family-Vicsek or superrough systems (see (47)). This contrasts with the noise-propagator expression (33), where the strength of the hydrodynamic interactions appearing in the FD relation (2) suppresses any elastic effect inherent to the system dynamics.
4. Kubo Fluctuation Relations
The Kubo fluctuation relation (KFR) expresses the mean response of the variable at time to a perturbation applied to a variable at time 0 [24, 25, 48–50]. If the dynamics of the system is given, for instance, in the form , the mean linear response of the th degree of freedom to a small perturbation of the th component of the vector field can be expressed as where represents the two-time correlation computed in the unperturbed system.
When the perturbation consists in a constant force applied along one direction (say ), we can perform the following substitution in (5), (42), and (45):In [25, 43] we have studied the average drift of the untagged and tagged probes to such an external force, that is, and , showing by a direct computation that KFR (50) is exactly fulfilled. In particular we have demonstrated the following equalities: where the unperturbed correlation functions are given in (23) and (28), respectively. Notice that the KFR in presence of a constant force coincides with the Generalized Einstein Relation (GER). In the following section, we demonstrate the validity of the KFR in case of an applied perturbation of the type
4.1. Time Periodic Force
Upon applying the time periodic force (53) to the tagged probe, FLEs (42), (45) take the form Our attention is on the average drifts and ; hence it is useful to firstly consider the complex positions and to get the real quantities afterwards as and . The FLEs ruling out the dynamics of the complex positions are Thus performing the Fourier transform in time and taking the average values one readily obtains where we explicitly reported the Fourier transform of the force-propagator expression (44). The complex velocities are achieved by performing the time derivative of the complex positions and can be casted in the following form: owing to the definitions of the complex mobility or admittance
Finally, taking the real part and keeping Kubo's notation , we obtain the linear relations To complete the demonstration of the validity of the KFR also in presence of a time oscillatory force, we need to relate the complex mobility to the velocity correlation function in absence of perturbations. This is done according to the Green-Kubo relation that connects the transport coefficients to the equilibrium two-time correlations of a system:Let us first calculate the LHS of (60). The Fourier transform in time of the velocity correlation function in (26) reads By using the properties (A.5), (A.2), and (A.6) in sequence, one arrives atWe now take the complex mobility in (58) and evaluate its -function expression. Applying the change of variable we obtain for its real and imaginary parts Then, recalling that , it follows that We apply the properties (A.7) and (A.5) to and (A.7), (A.5), and (A.6) to ; then we get the sought expressions: The Green-Kubo relation (60) follows by comparison of the real part of (65) with (62).
4.2. Complex Admittance Scaling Behaviour
Thanks to the short and long term expansion of the Fox -functions, we can study the high and low frequency limits of the complex admittance . We do it by considering the case of long range hydrodynamics and local one, respectively.
(i) Consider . In the low frequency limit the property (A.11) entails
We point out that the same result is easily obtained by handling the mobility integral representation (63) . The former result can be casted in the following compact form:which has a simple physical interpretation on the basis of (59). It means that the response signal is amplified by a factor with respect the input signal, and it presents a constant phase shift equal to . The fact that both amplitude and phase are independent of the distance is worth mentioning, since it signifies that no difference exists between the behaviour of a generic probe and the tagged one’s.
(ii) Consider . By adopting the expansion (A.10) we gain
The phase shift can be calculated as
Therefore in the large limit the imaginary part is negligible (). It follows that the response amplitude is dominated by the real part which has no frequency dependence.
We now consider the scaling behaviour of for systems presenting local hydrodynamics interactions.
(ii) Consider . The second-order term in the expansion (A.10) gives for from which it turns out that the phase isIn the very high frequency limit then the phase shift is if or and if . As a consequence, the response amplitude is given by .
The case is analytically very hard to tackle, since there is no expansion of -function work in this case nor the integral expression (63) can help. Nevertheless we can say that is exponentially small and : for instance, the exact solution for , reads (see Section 4.3 for a practical example).
The physical scenario emerging from this analysis can be illustrated with the help of Figure 1. Let us say that we perturb the system at a frequency : thanks to definition (18) the system splits into two spatial regions, (I) and (II), characterized by well distinct dynamical phases. For long-range or local hydrodynamics, the response of the inner region (I), that is, that closer to the tagged probe, shows a universal dependence and a phase shift with respect to the applied oscillatory force. On the other hand, in the outer region (II) in long ranged hydrodynamic systems, the response amplitude decays as , but almost no phase delay is displayed when compared to the external force. When the hydrodynamic interactions are local, the amplitude of the response is smaller and decays faster, namely, for and for ; the phase shift instead is when , and it grows like if .
4.3. Single File Systems
From  we know that the effective equation for the particle position in single file systems with applied periodic force is given byIn (73) represents the damping term and with file’s density. Hereafter we will choose