25 research outputs found
A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics
This paper studies homogenization of stochastic differential systems. The
standard example of this phenomenon is the small mass limit of Hamiltonian
systems. We consider this case first from the heuristic point of view,
stressing the role of detailed balance and presenting the heuristics based on a
multiscale expansion. This is used to propose a physical interpretation of
recent results by the authors, as well as to motivate a new theorem proven
here. Its main content is a sufficient condition, expressed in terms of
solvability of an associated partial differential equation ("the cell
problem"), under which the homogenization limit of an SDE is calculated
explicitly. The general theorem is applied to a class of systems, satisfying a
generalized detailed balance condition with a position-dependent temperature.Comment: 32 page
Stratonovich-to-Ito transition in noisy systems with multiplicative feedback
Cataloged from PDF version of article.Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system's state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Ito calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation-convention pair must be determined from the available experimental data before being able to predict the system's behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Ito calculus. We track such a transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We discuss possible implications of our conclusions, supported by numerics, for biology and economics
The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and Positive Friction
A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to three physically realizable examples where the coefficients defining the Langevin equation for these examples grow unboundedly either at a boundary, such as a wall, and/or at the point at infinity. This unboundedness violates the assumptions of previous limit theorems in the literature. The main result of this paper proves convergence for such examples. © 2016, Springer Science+Business Media New York
Small Mass Limit of a Langevin Equation on a Manifold
We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as m→ 0 , its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems. © 2016, Springer International Publishing
Phase space reduction of the one-dimensional Fokker-Planck (Kramers) equation
A pointlike particle of finite mass m, moving in a one-dimensional viscous
environment and biased by a spatially dependent force, is considered. We
present a rigorous mapping of the Fokker-Planck equation, which determines
evolution of the particle density in phase space, onto the spatial coordinate
x. The result is the Smoluchowski equation, valid in the overdamped limit,
m->0, with a series of corrections expanded in powers of m. They are determined
unambiguously within the recurrence mapping procedure. The method and the
results are interpreted on the simplest model with no field and on the damped
harmonic oscillator.Comment: 13 pages, 1 figur
Relation of a New Interpretation of Stochastic Differential Equations to Ito Process
Stochastic differential equations (SDE) are widely used in modeling
stochastic dynamics in literature. However, SDE alone is not enough to
determine a unique process. A specified interpretation for stochastic
integration is needed. Different interpretations specify different dynamics.
Recently, a new interpretation of SDE is put forward by one of us. This
interpretation has a built-in Boltzmann-Gibbs distribution and shows the
existence of potential function for general processes, which reveals both local
and global dynamics. Despite its powerful property, its relation with classical
ones in arbitrary dimension remains obscure. In this paper, we will clarify
such connection and derive the concise relation between the new interpretation
and Ito process. We point out that the derived relation is experimentally
testable.Comment: 16 pages, 2 figure
Stratonovich-to-Itô transition in noisy systems with multiplicative feedback
Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system's state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Itô calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation-convention pair must be determined from the available experimental data before being able to predict the system's behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Itô calculus. We track such a transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We discuss possible implications of our conclusions, supported by numerics, for biology and economics. © 2013 Macmillan Publishers Limited. All rights reserved