Smoluchowski Kramers Approximation Research Articles

Averaging on Macroscopic Scales with Application to Smoluchowski–Kramers Approximation

Averaging on Macroscopic Scales with Application to Smoluchowski–Kramers Approximation

Rate of convergence for the Smoluchowski–Kramers approximation for distribution-dependent SDEs driven by fractional Brownian motions

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the following distribution-dependent SDE driven by fractional Brownian motion [Formula: see text] where [Formula: see text] denotes the law of [Formula: see text], [Formula: see text] is a [Formula: see text]-dimensional fractional Brownian motion with Hurst parameter [Formula: see text]. Based on the techniques of multiple integrals and Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence.

Stochastic wave equation with Hölder noise coefficient: Well-posedness and small mass limit

We construct unique martingale solutions to the damped stochastic wave equationμ∂2u∂t2(t,x)=Δu(t,x)−∂u∂t(t,x)+b(t,x,u(t,x))+σ(t,x,u(t,x))dWtdt, where Δ is the Laplacian on [0,1] with Dirichlet boundary condition, W is space-time white noise, σ is 34+ϵ -Hölder continuous in u and uniformly non-degenerate, and b has linear growth. The same construction holds for the stochastic wave equation without damping term. More generally, the construction holds for SPDEs defined on separable Hilbert spaces with a densely defined operator A, and the assumed Hölder regularity on the noise coefficient depends on the eigenvalues of A in a quantitative way. We further show the validity of the Smoluchowski-Kramers approximation: assume b is Hölder continuous in u, then as μ tends to 0 the solution to the damped stochastic wave equation converges in distribution, on the space of continuous paths, to the solution of the corresponding stochastic heat equation. The latter result is new even in the case of additive noise.

On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable friction

We study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, including equations of Klein-Gordon type, in the joint small mass and small noise limit. The friction term is assumed to be state dependent. We also provide the proof of the Smolchowski-Kramers approximation for the case of variable friction, non-Lipschitz nonlinear term and unbounded diffusion.

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

Smoluchowski–Kramers approximation with state dependent damping and highly random oscillation

<p style='text-indent:20px;'>The small mass limit (Smoluchowski–Kramers approximation) of class systems of ordinary differential equations describing motions of small mass particle with state dependent friction and high oscillation is derived by a diffusion approximation approach. In the small mass limit, due to the state dependent damping, one additional term appears in the limit equation, which leads to a stochastic differential equation (SDE) as the highly random oscillation appears as a multiplicative white noise.</p>

A Smoluchowski–Kramers approximation for an infinite dimensional system with state-dependent damping

We study the validity of a Smoluchowski–Kramers approximation for a class of wave equations in a bounded domain of Rn subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-induced extra drift is created.

Large deviation principles for Langevin equations in random environment and applications

In contrast to the study of Langevin equations in a homogeneous environment in the literature, the study on Langevin equations in randomly varying environments is relatively scarce. Almost all the existing works require random environments to have a specific formulation that is independent of the systems. This paper aims at considering large deviation principles (LDPs) of Langevin equations involving a random environment that is a process taking values in a measurable space and that is allowed to interact with the systems, without specified formulation on the random environment. Examples and applications to statistical physics are provided. Our formulation of the random environment presents the main challenges and requires new approaches. Our approach stems from the intuition of the Smoluchowski–Kramers approximation. The techniques developed in this paper focus on the relation between the solutions of the second-order equations and the associated first-order equations. We obtain the desired LDPs by showing that a family of processes enjoy the exponential tightness and local LDPs with an appropriate rate function.

The Rate of Convergence for the Smoluchowski-Kramers Approximation for Stochastic Differential Equations with FBM

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the equation $$X_t=x+B_t^H+\int _0^t b(X_s)ds$$ where $$\{B_t^H, t\in [0,T]\}$$ is a fractional Brownian motion with Hurst parameter $$H\in \big (\frac{1}{2},1\big )$$ . Based on the techniques of Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence.

An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $\mu$ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $\e>0$. We show that for all small but fixed frictions the small mass limit of $q_{\mu, \e}$ gives the solution $q_\e$ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion $q_\e$ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

A Berry–Esseen Bound in the Smoluchowski–Kramers Approximation

In this paper, we use the Kolmogorov distance to investigate the Smoluchowski–Kramers approximation for stochastic differential equations. We obtain an explicit Berry–Esseen error bound for the rate of convergence. Our main tools are the techniques of Malliavin calculus.

On the convergence of stationary solutions in the Smoluchowski-Kramers approximation of infinite dimensional systems

We prove the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, both with Lipschitz-continuous and with polynomial non-linearities. In particular, we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation. The Wasserstein metric is associated to a suitable distance on the space of square integrable functions, that is chosen in such a way that the dynamics of the limiting stochastic parabolic equation is contractive with respect to such a Wasserstein metric. This implies that the limiting result is a consequence of the validity of a generalized Smoluchowski-Kramers limit at fixed times. The proof of such a generalized limit requires new delicate bounds for the solutions of the stochastic wave equation, that must be uniform with respect to the size of the mass.

A parameter estimator based on Smoluchowski–Kramers approximation

We devise a simplified parameter estimator for a second order stochastic differential equation by a first order system based on the Smoluchowski–Kramers approximation. We establish the consistency of the estimator by using Γ-convergence theory. We further illustrate our estimation method by an experimentally studied movement model of a colloidal particle immersed in water under conservative force and constant diffusion.

Smoluchowski–Kramers approximation for the damped stochastic wave equation with multiplicative noise in any spatial dimension

We show that the solutions to the damped stochastic wave equation converge pathwise to the solution of a stochastic heat equation. This is called the Smoluchowski–Kramers approximation. Cerrai and Freidlin have previously demonstrated that this result holds in the cases where the system is exposed to additive noise in any spatial dimension or when the system is exposed to multiplicative noise and the spatial dimension is one. The current paper proves that the Smoluchowski–Kramers approximation is valid in any spatial dimension when the system is exposed to multiplicative noise.

On the Langevin equation with variable friction

We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski-Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.

On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior

Partially supported by the NSF grant DMS 1407615Partially supported by the NSF grant DMS 1411866. (DMS 1407615 - NSF; DMS 1411866 - NSF)

Smoluchowski–Kramers approximation and large deviations for infinite-dimensional nongradient systems with applications to the exit problem

In this paper, we study the quasi-potential for a general class of damped semilinear stochastic wave equations. We show that as the density of the mass converges to zero, the infimum of the quasi-potential with respect to all possible velocities converges to the quasi-potential of the corresponding stochastic heat equation, that one obtains from the zero mass limit. This shows in particular that the Smoluchowski–Kramers approximation is not only valid for small time, but in the zero noise limit regime, can be used to approximate long-time behaviors such as exit time and exit place from a basin of attraction.

On the Smoluchowski–Kramers approximation for a system with infinite degrees of freedom exposed to a magnetic field

We study the validity of the so-called Smoluchowski–Kramers approximation for a two dimensional system of stochastic partial differential equations, subject to a constant magnetic field. Since the small mass limit does not yield to the solution of the corresponding first order system, we regularize our problem by adding a small friction. We show that in this case the Smoluchowski–Kramers approximation holds. We also give a justification of the regularization, by showing that the regularized problems provide a good approximation to the original ones.

The long time behavior of Brownian motion in tilted periodic potentials

A variety of phenomena in physics and other fields can be modeled as Brownian motion in a heat bath under tilted periodic potentials. We are interested in the long time average velocity considered as a function of the external force, that is, the tilt of the potential. In many cases, the long time behavior–pinning and de-pinning phenomenon–has been observed. We use the method of stochastic differential equation to study the Langevin equation describing such diffusion. In the over-damped limit, we show the convergence of the long time average velocity to that of the Smoluchowski–Kramers approximation, and carry out asymptotic analysis based on Risken’s and Reimann et al.’s formula. In the under-damped limit, applying Freidlin et al.’s theory, we first show the existence of three pinning and de-pinning thresholds of the normalized tilt, corresponding to the bi-stability phenomenon; and second, as noise approaches zero, derive formulas of the mean transition times between the pinning and running states.

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

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.