Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit

Abstract

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). {Using the Itô stochastic integral convention,} we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. {Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation}, which can be parametrized by $\alpha \in \mathbb{R}$. Interestingly, in addition to the classical Itô ($\alpha =0$), Stratonovich ($\alpha =0.5$) and anti-Itô ($\alpha =1$) integrals, we show that position-dependent $\alpha =\alpha (x)$, and even stochastic integrals with $\alpha \notin [0,1]$ arise. Our findings are supported by numerical simulations.

11 pages, 5 figures

References in corpus (3)

• On the Poisson equation and diffusion approximation 3
• Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. IV: Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics
• Force measurement in the presence of Brownian noise: Equilibrium distribution method vs. Drift method

Cited by in corpus (21)

• Nonergodic Subdiffusion from Brownian Motion in an Inhomogeneous Medium
• The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction
• Direct measurement of thermophoretic forces
• Relation of a New Interpretation of Stochastic Differential Equations to Ito Process
• Quantum Brownian Motion with Inhomogeneous Damping and Diffusion
• Martingales for physicists: A treatise on stochastic thermodynamics and beyond
• Anomalous fluxes in overdamped Brownian dynamics with Lorentz force
• Thermophoresis of Brownian particles driven by coloured noise
• Functionals in stochastic thermodynamics: how to interpret stochastic integrals
• Entropy production of a Brownian ellipsoid in the overdamped limit
• The small-mass limit and white-noise limit of an infinite dimensional Generalized Langevin Equation
• Control of anomalous diffusion of a Bose polaron
• Homogenization for a Class of Generalized Langevin Equations with an Application to Thermophoresis
• On the settling of small grains in dusty discs: analysis and formulas
• Efficiency of random search with space-dependent diffusivity
• On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion
• Phase space reduction of the one-dimensional Fokker-Planck (Kramers) equation
• Anisotropic active Brownian particle with a fluctuating propulsion force
• Driven anisotropic diffusion at boundaries: noise rectification and particle sorting
• Intrinsic Langevin dynamics of rigid inclusions on curved surfaces
• A Rayleigh criterion for mechanical instability: inducing activity by chemo-mechanical coupling