## Abstract

The principle of dynamic invariance is applied to obtain closed moment equations from the Fokker–Planck kinetic equation. The analysis is carried out to explicit formulae for computation of the lowest eigenvalue and of the corresponding eigenfunction for arbitrary potentials.

This article is part of the themed issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

## 1. Introduction

The Fokker–Planck equation (FPE) is a familiar model in various problems of non-equilibrium statistical physics [1,2]. In this paper, we consider the FPE of the form
1.1Here, *W*(*x*,*t*) is the probability density over the configuration space *x*, at a time *t*, whereas *U*(*x*) and *D*(*x*) are the potential and the positively semi-definite (*y*⋅*D*⋅*y*≥0) diffusion matrix. The dot denotes convolution in the configuration space. The FPE (1.1) is particularly important in the studies of polymer solutions [3–5]. Let us recall the two properties of the FPE (1.1), which are important in what follows: (i) conservation of the total probability: and (ii) dissipation: the equilibrium distribution, , is the unique stationary solution to the FPE (1.1). The entropy,
1.2is a monotonically growing function owing to the FPE (1.1), and it arrives at the global maximum in the equilibrium. These properties follow immediately when the FPE (1.1) is rewritten as follows:
1.3where
1.4is a positive semi-definite symmetric operator with kernel 1 (the latter statement is implied by

In many cases, one is interested in the dynamics of moments of the distribution function *W* rather than in the dynamics of the *W* itself. In the context of polymer dynamics, for example, closed equations for second-order moments (conformation tensor) and higher-order moments are used to derive constitutive equations for polymeric stress (see, for example, [6] and references therein). However, except for the simplest potentials *U* and diffusion matrices *D*, the moment equations, as they follow from the FPE (1.1), are not closed. Therefore, closure procedures are required.

In this paper, we address the problem of closure for the FPE (1.1) in a general setting. First, we review the maximum entropy principle (MaxEnt) as a source of suitable initial approximations for the closures. We also discuss a version of the MaxEnt, valid for a near-equilibrium dynamics and which results in explicit formulae for arbitrary *U* and *D*. The MaxEnt closures are almost never invariants of the true moment dynamics, and corrections to the MaxEnt closures are the focus of this paper. We apply the method of invariant manifold [7], which is carried out (subject to certain approximations explained below) to explicit recurrence formulae for one-moment near-equilibrium closures for arbitrary *U* and *D*. These formulae give a method for computing the lowest eigenvalue of the problem, and which dominates the near-equilibrium FPE dynamics.

## 2. MaxEnt closures

Let us denote as *M* the set of linearly independent moments, {*M*_{0},*M*_{1},…,*M*_{k}}, where , *m*_{0}=1. We assume that there exists a function *W**(*M*,*x*) that extremizes the entropy *S* (1.2) under the constraints of fixed *M*. This quasi-equilibrium distribution function may be written as
2.1where *Λ*={*Λ*_{0},*Λ*_{1},…,*Λ*_{k}} are Lagrange multipliers.

Closed-form equations for moments *M* are derived in two steps. First, the quasi-equilibrium distribution (2.1) is substituted into the FPE (1.1) or (1.3) to give a formal expression
2.2where is given by (1.4). Second, introducing the quasi-equilibrium projector *Π**,
2.3and applying *Π** on both sides, we derive a closed expression for *M* in the quasi-equilibrium approximation. In the following, we do not show the dependence on *x* in order to ease the notation.

Further processing requires an explicit solution to the constraints, to get the dependence of Lagrange multipliers *Λ* on the moments *M*. Although typically the functions *Λ*(*M*) are not known explicitly, one general remark about the moment equations is readily available: the moment equations in the quasi-equilibrium approximation have the form
2.4where *S**(*M*)=*S*[*W**(*Λ*(*M*))] is the quasi-equilibrium entropy, and where is the *M*-dependent (*k*+1)×(*k*+1) matrix
2.5Indeed, applying projector (2.3) on (2.2) gives after integration by part
where we have used (cf. (2.1)). Finally, remembering that *Λ*_{i}=−∂*S**(*M*)/∂*M*_{i} and integrating the above expression with *m*_{i}, we arrive at (2.4) and (2.5).

The matrix is symmetric, positive semi-definite, and its kernel is the vector *δ*_{0i}. Thus, the quasi-equilibrium closure reproduces the gradient structure on the macroscopic level (1.3); the vector field of quasi-equilibrium equations (2.4) is a transform of the gradient of the quasi-equilibrium entropy given by the symmetric positive operator. Applications of quasi-equilibrium approximations to polymer dynamics can be found in [6].

## 3. Triangle MaxEnt approximation

The following version of the quasi-equilibrium closures makes it possible to derive more explicit results in a general case: in many cases, one can split the set of moments *M* into two parts, *M*_{I}={*M*_{0},*M*_{1},…,*M*_{l}} and *M*_{II}={*M*_{l+1},…,*M*_{k}}, in such a way that the quasi-equilibrium distribution can be constructed explicitly for *M*_{I} as *W**_{I}(*M*_{I}). The full quasi-equilibrium problem for *M*={*M*_{I},*M*_{II}} in the ‘shifted’ formulation reads: extremize the functional *S*[*W**_{I}+Δ*W*] with respect to Δ*W*, under the constraints and . Let us denote as Δ*M*_{II}=*M*_{II}−*M*_{II}(*M*_{I}) deviations of the moments *M*_{II} from their values in the quasi-equilibrium state . For small deviations, the entropy is well approximated with its quadratic part
Taking into account the fact that , we come to the following maximization problem:
3.1The solution to the problem (3.1) is always explicitly found from a (*k*+1)×(*k*+1) system of linear algebraic equations for Lagrange multipliers. This triangle maximum entropy method was introduced in the context of the Boltzmann equation in [8].

In the remainder of this paper, we deal with the one-moment near-equilibrium closures: *M*_{I}=*M*_{0} (i.e. *W**_{I}=*W*_{eq}), and the set *M*_{II} contains a single moment , *m*(*x*)≠1. We shall specify notations for the near-equilibrium FPE, writing the distribution function as *W*=*W*_{eq}(1+*Ψ*), where the function *Ψ* satisfies an equation
3.2where . The triangle one-moment quasi-equilibrium function reads
3.3where
3.4Here, brackets denote equilibrium averaging. The superscript (0) indicates that the triangle quasi-equilibrium function (3.3) will be considered as the initial approximation to the procedure that we address below. The projector for the approximation (3.3) has the form
3.5Substituting the function (3.3) into the FPE (3.2), and applying the projector (3.5) on both sides of the resulting formal expression, we derive the equation for *M*,
3.6where 1/λ_{0} is an effective relaxation time of the moment *M* to its equilibrium value, in the quasi-equilibrium approximation (3.3)
3.7

## 4. The invariance equation for the Fokker–Planck equation

Both the quasi-equilibrium and the triangle quasi-equilibrium closures are almost never the invariants of the FPE dynamics. That is, the moments *M* of solutions of the FPE (1.1) and the solutions of the closed moment equations such as (2.4) are different functions of time, even if the initial values coincide. These variations are generally significant even for the near-equilibrium dynamics. Therefore, we ask for corrections to the quasi-equilibrium closures to end up with the invariant closures. This problem falls precisely into the framework of the method of invariant manifold [7], and we shall apply this method to the one-moment triangle quasi-equilibrium closure approximations, as a simple example.

First, the invariant one-moment closure is given by an unknown distribution function, , which satisfies the invariance equation
4.1Here, is the projector, associated with function , and which is also yet unknown. Equation (4.1) is a formal expression of the invariance principle for a one-moment near-equilibrium closure: considering as a one-parametric set of functions in the space of distribution functions, parametrized with the moment Δ*M*, we require that the microscopic vector field be equal to its projection, , onto the tangent space .

Now, we turn our attention to solving the invariance equation (4.1) iteratively, beginning with the triangle one-moment quasi-equilibrium approximation *W*^{(0)} (3.3). We apply the following iteration process to approximate (4.1):
4.2where *k*=0,1,…, and where *m*^{(k+1)}=*m*^{(k)}+*μ*^{(k+1)}, and the correction satisfies the condition 〈*μ*^{(k+1)}*m*^{(k)}〉=0. The projector is updated after each iteration, and it has the form
4.3Applying *Π*^{(k+1)} to the formal expression
we derive the (*k*+1)th update of the effective time (3.7)
4.4Specializing to the one-moment near-equilibrium closures, and following the general argument of Gorban & Karlin [7], solutions to the invariance equation (4.1) are eigenfunctions of the operator , whereas the limit of the iteration process (4.2) is the eigenfunction that corresponds to the eigenvalue with the minimal non-zero absolute value.

## 5. Diagonal approximation

In order to obtain more explicit results, we shall now proceed with an approximate solution to the problem (4.2) at each iteration. The correction *μ*^{(k+1)} satisfies the condition 〈*m*^{(k)}*μ*^{(k+1)}〉=0, and can be decomposed as follows: . Here, *e*^{(k)} is the defect of the *k*th approximation: , where
5.1The function is orthogonal to both *e*^{(k)} and *m*^{(k)} (, and ).

Our diagonal approximation (DA) consists of neglecting the part . In other words, we seek an improvement of the non-invariance of the *k*th approximation along its defect, Δ=*e*^{(k)}. Specifically, we consider the following ansatz at the *k*th iteration:
5.2Substituting the ansatz (5.2) into (4.2), we integrate the latter expression with the function *e*^{(k)} to evaluate the coefficient *α*_{k}
5.3where functions *A*_{k} and *B*_{k} are represented by the following equilibrium averages:
5.4

Finally, putting together (4.4), (5.1)–(5.4), we arrive at the following DA recurrence solution: 5.5and 5.6

Note that the stationary points of the DA process (5.6) are the true solutions to the invariance equation (4.1). What may be lost within the DA is the convergence to the true limit of the procedure (4.2), i.e. to the minimal eigenvalue. In a general situation, this is highly improbable, though.

## 6. Examples

In order to test the convergence of the DA process (5.6), we considered two potentials *U* in the FPE (1.1) with a constant diffusion matrix *D*. The first test was with the quadratic potential *U*=*x*^{2}/2, in the three-dimensional configuration space, because, for this potential, the spectrum is well known. We have considered two examples of the initial one-moment quasi-equilibrium closures with *m*^{(0)}=*x*_{1}+100(*x*^{2}−3) (example 1) and *m*^{(0)}=*x*_{1}+100*x*^{6}*x*_{2} (example 2) in (3.4). The result of performance of the DA for λ_{k} is presented in table 1, together with the error *δ*_{k} which was estimated as the norm of the variance at each iteration: *δ*_{k}=〈*e*^{(k)}*e*^{(k)}〉/〈*m*^{(k)}*m*^{(k)}〉. In both examples, we see a good monotonic convergency to the minimal eigenvalue , corresponding to the eigenfunction *x*_{1}. This convergence is even striking in example 1, where the initial choice was very close to a different eigenfunction *x*^{2}−3, and which can be seen in the non-monotonic behaviour of the variance. Thus, we have an example to trust the DA approximation as converging to the proper object.

For the second test, we have taken a one-dimensional potential , and the configuration space is the segment |*x*|≤1. Potentials of this type (a so-called finitely extensible nonlinear elastic (FENE) potential) are used in applications of the FPE to models of polymer solutions [3–5]. Results are given in table 2 for the two initial functions, *m*^{(0)}=*x*^{2}+10*x*^{4}−〈*x*^{2}+10*x*^{4}〉 (example 3) and *m*^{(0)}=*x*^{2}+10*x*^{8}−〈*x*^{2}+10*x*^{8}〉 (example 4). Both examples demonstrate a stabilization of the λ_{k} at the same value after some 10 iterations.

## 7. Conclusion

In conclusion, we have developed the principle of invariance to obtain moment closures for the FPE (1.1), and have derived explicit results for the one-moment near-equilibrium closures, particularly important to obtain information about the spectrum of the FP operator.

## Competing interests

I declare I have no competing interests.

## Funding

This work was supported by European Research Council advanced grant no. 291094-ELBM.

## Acknowledgements

The author thanks A. Gorban for many discussions, and V. Zmievski for help with numerics.

## Footnotes

One contribution of 17 to a theme issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

- Accepted August 23, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.