2011/12/12 by Scott Hottovy, Giovanni Volpe, Jan Wehr · 52 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Brownian motion #Diffusion #Diffusion and Search Dynamics #Einstein relation #Langevin dynamics #Langevin equation #Limit (mathematics) #Limiting #Stochastic differential equation #Stratonovich integral #cond-mat.stat-mech #math-ph #math.MP #stochastic dynamics and bifurcation
paper · pdf · doi:10.1007/s10955-012-0418-9
published in Journal of Statistical Physics 146(4), 762-773 (Springer Science+Business Media) · 11 pages, 5 figures
arxiv created 2011/12/12 · openalex publication_date 2012/01/11 · arxiv updated 2012/08/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 α∈ ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0.5) and anti-Itô (α=1) integrals, we show that position-dependent α= α(x), and even stochastic integrals with α∉ [0,1] arise. Our findings are supported by numerical simulations.