Abstract

Diffusion-reaction phenomena are generally described by parabolic differential equations (PDEs), and I am interested in those possessing solutions that fail at large time. A sophisticated method to study the large-time behavior is the Renormalization Group usually encountered in Particles-Physics and Critical Phenomena. In this paper, I review the application of such an approach. In particular, attention is paid to Quantum Field Theory techniques used for the extraction of the asymptotic solutions to PDEs. Finally, I extend discussion to the fractional-time PDEs and with noise.

1. Introduction

Diffusion-reaction equations are generally PDEs; that is, they contain a simple derivative with respect to time and spatial derivatives of arbitrary order [1, 2]. In fact, these equations model a large class of natural systems. Among these, I can quote thermal transfer with absorption, flow in porous media, pressure of a gas in the chock-waves problem, pressure of a liquid in a porous medium, growth of an interface, diffusion-reaction phenomena, or diffusion fronts.

PDEs are very often nonlinear and may rarely be solved analytically. When an analytic solution exists, it can be obtained transforming the original equation to a linear one that is easy to solve. Usually, use is made of Lie-algebra techniques [3–6] if this equation possesses a certain symmetry.

Two fundamental questions to ask are the study of the stability of the solution of PDEs and its behavior at large time. The most sophisticated method for the study of the time-asymptotic behavior is the Renormalization Group (RG) techniques. In the past, this approach has been successfully used, first in High-Energy Physics [7, 8], and second, in Critical Phenomena [9–11]. For the first domain, RG has been very useful to studying the infrared behavior and the asymptotic freedom. For the second, RG was introduced in a series of brilliant papers by Wilson [12, 13], for the extraction of the critical behavior of those systems exhibiting a second order phase transition.

Since few years, it turned out that RG could be applied to investigate the time-asymptotic behavior of the solution of PDEs [14–22]. The fundamental hypothesis is that the solution asymptotically behaves as , where the exponent and the scaling function depend generally on the nature of the problem. Here, is the spatial variable and is time. Therefore, searching the asymptotic behavior necessitates the knowledge of these two quantities. To this end, one first constructs a renormalization transformation. The fixed point of this transformation is nothing else but the scaling function .

Recently, use was made of Quantum Field Theory (QFT) for the study of the time-asymptotic behavior of solutions of PDEs [23]. This is precisely the purpose of this paper. A special attention will be paid to PDEs with noise and time-fractional PDEs.

The remaining of presentation proceeds as follows. In Section 2, we briefly recall the obtaining of the more general diffusion-reaction equations. Then, we place the renormalization theory in its historical context, in Section 3. The important notion of self-similarity is recalled in Section 4. Application of real-space RG for the determination of the time-asymptotic behavior of solutions of PDEs is described in Section 5. Field-theoretical approach to large-time behavior is the aim of Section 6. Finally, some concluding remarks are drawn in the last section.

2. Diffusion-Reaction Equations

The diffusion mechanism models the motion of individuals in an environment or in a medium. These individuals may be small, as particles in physics, bacteria, molecules and cells, or very large objects like humans, animals, insects, organisms, plants, or certain types of events (epidemics, rumors…).

I assume that individuals reside in some region, , which is an open domain of the Euclidean space , with . In particular, I shall be concerned with ,  , and cases. But the formalism presented here applies for any space dimensionality.

The basic mathematical variable I consider is the population density function: , where is the position and is time. This function is defined as the number of particles per unit length (for ), per unit area (for ), or per unit volume (for ). For example, the human population density is often expressed as the number of individuals per square kilometer.

But as for other mathematical models, I assume that the function has good mathematical properties, such as continuity and differentiability. This is more reasonable, when one considers a population with high number of individuals.

Technically, one defines the population density function as follows: let be a point of habitat , and let be a sequence of spatial regions surrounding the point . Here, the subdomains are chosen so that the spatial measures (length, area, volume, or mathematically a Lesbegue measure) tend to zero, as , and ; then if the limit exists. It is clear that the total population in any region of , at time , is

The question of interest is how the function changes, when varying time and position . In fact, the population can change in two ways: one is that individual particles can move in their field; and the second is that they can give birth to new individuals, or kill existing individuals for physical, chemical, or biological reasons. These two different phenomena can be separately modeled.

Now, the question is how particles can move? Generally, this is a high-complicated process that may be attributed to several reasons. Usually, the population moves from high-density regions to low-density ones. This fact is similar to many physical phenomena, such as heat transfer (from the hottest place to the coldest), or a chemical dilution in water.

The motion of is called population density flow that is a vector. This flow points to the direction of rapid decay of , which is a negative gradient of . This is the Fick law, mathematically traduced by where is the flow of , is the diffusion coefficient at point , and is the gradient operator: .

On the other hand, the number of individuals at any point may change, because of some reasons as the birth, death, hunting, or chemical reactions. I suppose that the change rate of the population density function is , termed reaction rate. Now, the aim is the obtaining of a differential equation solved by the population density, by using the so-called balance law. I choose an arbitrary region , where the population is . Then, the change rate of the total population is The net growth of the population in domain is and the total flow is where is the boundary of domain and is the exterior normal direction at point . Therefore, the balance law implies From the standard divergence theorem, I obtain Combining formulae (3), (7), and (8) and interchanging the order of integration and derivation yields Since the choice of region is arbitrary, then, we have the following differential equation: at any point and any time . The above equation is called diffusion-reaction equation. Here, accounts for the diffusion term governing the motion of individuals, and for the reaction term that describes the birth/death or the reaction producing inside habitat or reactor.

The diffusion coefficient is not generally constant, since the environment is usually heterogenous. But when the diffusion region is approximately homogenous, one can suppose that , at any point . Therefore, (10) becomes where is the Laplace operator.

3. Brief Overview of the Renormalization Theory

The renormalization is an old concept that first appeared in elementary physics. So, for vacuum, the dielectric constant is , but for a dielectric medium (glasses, plastic materials…), this constant is rather , where is the relative permittivity. It is then a multiplicative renormalization of the dielectric constant. In solids, for example, the free electrons, ensuring the electric conduction, interact with the underlying crystalline structure. To simplify the study, one can replace the real mass of electrons by an effective mass , where the factor reflects the crystalline structure-electrons interactions.

Actually, the renormalization appears in QFT [7–11]. Such a theory is constructed with an action, , which is a functional of field , and function of microscopic parameters, namely, the mass of particles, , and the coupling constant, . According to the diffusion (in High-Energy Physics language), one may have many fields (then many masses and coupling constants). Mathematically, a field is a temperate distribution. Here, represents a -vector, where is the -vector, is the time variable, and  m/s is the light speed in vacuum. The set of all -vectors constitutes the so-called Minkowski space-time   , which is a differential manifold.

For -theory, for instance, the associated action reads with Here, the free action is quadratic in field and its derivatives, and accounts for the interaction action, which is nonlinear in this field.

The theory has been developed through the Green functions, , which are the vaccum average of a product of field , where is the chronological order operation. Thanks to these functions, one calculates a basic quantity, which is the -matrix allowing the deduction of the experimentally measured cross section [7, 8]. These Green functions cannot be calculated exactly and are developed in series in the coupling constants ’s and appear short-distances (or ultraviolet) divergences that first necessitate a regularization procedure (introducing some cut-off). There are many ways to regularize the theory, but the most commode regularization is the dimensional one [24–26], with the regulator , where is the space dimensionality. Within the framework of this regularization, short-distances divergences are poles in . Hence, regularized (or bare) Green functions present as Laurent series in -variable and entire series in the bare coupling constant .

It is not sufficient to regularize the theory but this must be renormalized removing all short-distances divergences. It has been shown [7, 8] that QFT is renormalized at any order of the perturbation series in the coupling constant. This renormalization theorem stipulates that there exists a renormalized square mass, , a renormalized coupling constant, , and a renormalization factor, , which are functions of bare parameters and regulator , such as the bare Green function is directly proportional to the renormalized one, , The function is finite in the limit , at fixed renormalized parameters and . Therefore, a part of divergences has been absorbed redefining the bare parameters, and the remaining divergent part has been multiplicatively factorized (factor ). One can say that a renormalization is a change of parameters.

From the multiplicative renormalization (15), one can derive the RG-equation or Callan-Symanzik equation [7, 8], satisfied by the renormalized Green function . The resolution of this equation informs on the infrared (or short-distances) of Green functions, that is, when .

Thereafter, Wilson was extended RG originally known in High-Energy Physics to Critical Phenomena [9–11]. In this case, the order parameter plays the role of -field, and the shift to the critical temperature , the role of square mass , and the correlation functions are the analog of Green functions . Here, accounts for the critical temperature. Thus, the approach of the transition , for Critical Phenomena, corresponds to the infrared limit , for QFT.

For PDEs of interest, these various limits are the analog of the asymptotic limit of the solution.

4. Notion of Self-Similarity

I consider a given property, , which depends on the point . For dynamic systems, is time and are the freedom degrees. For magnetic materials, for example, is the distance from the critical temperature and is the magnetic field .

I assume that there exists some domain , such as the restriction of property to , denoted by , satisfies the generalized homogeneity property (scale invariance or self-similarity) Here, the scale dilatation is an arbitrary integer and is the homogeneity degree. The exponents are real numbers. The above definition is equivalent to the standard Euler formula

I rewrite now relation (17) as Of course, the RHS of this equality must not depend on the scale . Then, I decide to choose this scale, such that or . With this choice, relation (19) becomes with the notation

It is straightforward to check that the function is scale invariant, called scaling function. Therefore, the solution of the functional equation (19) reads with the principal exponent and crossover exponents

For dynamic systems, the self-similarity relation (21) constitutes the time-asymptotic behavior. Thus, the knowledge of this behavior necessitates that of exponents and and the scaling function . The universality, when it is present, means that exponents and , and scaling function are independent of the details of the problem; that is, they are the same for a large class of phenomena of different nature.

5. Asymptotic Behavior from Real-Space RG

I am interested in the following PDE, solved by the dynamic variable , I consider here the one-dimensional problem , but the analysis I present may be extended to any space dimensionality . In the above equation, represents the Laplace operator, and the “reaction” is a nonlinear function of solution and its first and second derivatives and .

5.1. Case

In this situation, the solution is trivial and is given by the Gaussian Such a solution is exactly self-similar; that is, This corresponds to the quantities On the other hand, solution (25) rewrites on the scaling form

5.2. Case

Generally, in this case, the asymptotic solution cannot be determined in an exact way. I propose that, in the limit , the solution of (24) has the asymptotic form Therefore, asymptotically, the solution has the scaling-invariance property

The aim is then the search of this asymptotic solution applying real-space RG. This is precisely the objective of the next paragraph.

5.3. Renormalization Transformations

I start by the introduction of the function Here, is the solution at scale . It is easy to see that and that is solution to the following PDE: with the nonlinearity at scale

I start with the initial data: . It is the solution at the initial time . The set of all initial data, , is a Banach-space. By definition [15–17], a renormalization transformation (RT) is the application of , such that Thus, RT depends on scale and the form of the nonlinearity .

It is easy to convince that RT fulfils the semigroup property RG is then the set of all possible ’s, with . If I apply times a RT at scale , then, I shall obtain Taking into account the definition (35) of RT gives This result must be compared to the asymptotic form (20a), and one finds that The scaling function then constitutes a fixed point of RT. In this limit, the nonlinearity , at scale , goes to the fixed value . Therefore, it is sufficient to obtain the asymptotic solution replacing in the initial PDE the nonlinearity by . One then writes , where the deformation informs on corrections to the leading asymptotic behavior.

Universality means the independence of exponent and scaling function on the initial data and the nature of the problem.

The next step consists in constructing some relevance criterion for the search of the time-asymptotic behavior of solution. To this end, we first recall the Gaussian fixed point.

5.4. Gaussian Fixed Point

In the absence of any reaction , the solution is defined in relation (25). The initial data and the corresponding RT are given by Then, the renormalization transformation is exact and possesses a line of fixed points, namely, the multiples of , with

5.5. Relevance Criterion

I start from a general nonlinearity, which is a function that maps into . I assume that this function is analytic in the vicinity of origin . It will be sufficient to make a reasoning about only one monomial, . After a scaling transformation, one obtains with the exponent In fact, the analysis shall depend on the sign of this exponent, and one has the criterion

5.5.1. Example 1: Heat Equation with Absorption

The associated PDE is [15–17] The corresponding exponent is . Therefore, one has the criterion It should be noted that the general solution of (45) is exactly self-similar, where the nontrivial fixed point solves an ordinary differential equation, It has been shown that, asymptotically, one has [15–17]

5.5.2. Example 2: Burgers Equation

This equation is such that

The nonlinearity is then irrelevant. Although this equation is nonlinear, it possesses, however, an exact solution (using the Hopf transformation) [15–17], which is where is a known amplitude and denotes the error function [27]. The fixed-point is then nontrivial, since it differs from the Gaussian one.

6. Asymptotic Solution from QFT

The starting point is the following PDE: where the dynamic variable is a scalar field. Here, is the diffusion coefficient, and is a nonlinear function of this field and its first and second derivatives with respect to the space variable . The above PDE must be complected by the initial condition

I set . This function may be written as the double functional integral: with the effective action I shall suppose, thereafter, that the nonlinearity has the polynomial form where the coefficients of this series, , represent the coupling constants.

The free propagator (with ) is Here, is the frequency and is the -dimensional wave-vector.

To select the relevant monomials of the above series, I shall make a reasoning using a naive powers counting.

6.1. Powers Counting

First, I introduce time and distance dimensions of an arbitrary , where and are time and distance scales. So, I have with the notation

The dimension suggests that the effective coupling constants are Hence, the criterion of relevance is such that I note that, for , I recover the relevance criterion presented in the last section.

For a given monomial, that is, at fixed , the reasoning may be done in terms of space dimensionality . To this end, I write as follows: where

accounts for the critical dimension. Then, I have an equivalent criterion

For example, for nonlinearities and , the respective critical dimensions are and .

6.2. Applied Renormalization Theory

Without further details concerning the application of the renormalization theory to extract the large-time behavior of solutions of PDEs, I simply draw the strategy as follows.(1)Make a perturbative expansion of the propagator with respect to the coupling constants , and appear short-distances divergences that present as poles in .(2)Renormalize the theory; that is, , where is the renormalized propagator, is the renormalization factor, and are the renormalized coupling constants.(3)Write, then, the RG-equation satisfied by .(4)Its solution using the characteristics method is then the asymptotic behavior.

For the nonlinearity , for instance, it has been shown that the solution is exactly self-similar [17]; that is,

7. Concluding Remarks

In this paper, I presented a pedagogical review of the application of the renormalization techniques to extract the large-time solutions of PDEs.

I state that the same techniques may be applied to coupled PDEs, that is, with more than one dynamic variable. In this case, the corresponding field theory is constructed with several fields   . This means that one has kinds of populations (e.g., chemotaxis and morphogenesis in biology) that move and react. The associated PDEs are Here, stands for the diffusion coefficient of species . I easily show that the associated effective action is which generalizes that defined in relationship (55). As before, it is assumed that ’s are polynomials in ’s fields and their derivatives and .