The Smoluchowski-Kramers Limit of Stochastic Differential Equations with Arbitrary State-Dependent Friction

Abstract

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient. arXiv:1404.2330v4 [math-ph] 24 Oct 2014 II. SMOLUCHOWSKI-KRAMERS APPROXIMATION For the main theorem, we assume x m t , x t ∈ U ⊂ R d , an open set, and v m t ∈ R d for all 0 ≤ t ≤ T . For an arbitrary vector a ∈ R d , |a| is the Euclidean norm and, for a d × d matrix A ∈ R d×d , |A| is the induced operator norm. We now state the assumptions and main theorem.

The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and Positive Friction

“…Nevertheless this heuristic, first employed by Smoluchowski in [17] and later by Kramers in [10], serves as a good first step in understanding how some parts of (2.1) arise. See [7] for further, more specific details in how the noise-induced drift term, i.e. S(x(t)), in equation (2.1) is produced.…”

“…In essence, provided the friction matrix γ(x) is positive-definite for each x ∈ X , our main result shows that we can still extract convergence of x m (t) as m → 0 pathwise on bounded time intervals in probability, without making strong boundedness assumptions on the coefficients F, γ, σ and their derivatives. These boundedness assumptions were made in the earlier work [7] and have also been made previously in other, related works [2,3,16]. From a physical standpoint, however, there are many natural model equations that do not satisfy these strong boundedness requirements, and therefore the use of the small-mass approximation of the dynamics above is in question.…”

“…Assumption 2.3 assures that the presumed limiting dynamics x(t) remains in its domain of definition X for all finite times t ≥ 0 almost surely. Different from the previous references [2,7,16], we will not assume that the solution of (1.1) is non-explosive or, more importantly, that either x(t) is contained in a compact subset of X or the coefficients F, γ, σ are bounded on X . We do, however, need control over an additional derivative of γ.…”

Nondiffusive fluxes in a Brownian system with Lorentz force

“…Whereas the Langevin equations are convenient for simulations, a statistical description is often preferred for theoretical analysis. To this end one derives the Fokker-Planck equation for the position degrees of freedom, which for a Brownian particle subject to inhomogeneous Lorentz force, is given as [2,5] ∂P (r, t) ∂t = ∇ · D(r)∇P (r, t) ,…”

Lorentz forces induce inhomogeneity and flux in active systems

“…Due to the nonconservative nature of the Lorentz force, the overdamped equation of motion cannot be obtained by simply setting the mass of the particles to zero [4,42]. Though there exists a nontrivial limiting procedure [43][44][45] that yields the small-mass limit of the (velocity) Langevin equation, the resulting overdamped equation is not suitable for determining velocity dependent variables such as flux and entropy [4,5,[46][47][48] despite the fact that it captures the position statistics accurately. In this work, we have avoided these problems by performing simulations using the (velocity) Langevin equation with a finite but small mass.…”