Phase Space Homogenization of Noisy Hamiltonian Systems

Birrell, Jeremiah; Wehr, Janek

doi:10.1007/s00023-018-0646-x

Cited by 5 publications

(6 citation statements)

References 28 publications

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“…The homogenization heuristics proposed under Assumption 1 applies here as well: the limiting Langevin equation can be interpreted as a result of averaging over the conditional stationary distribution of the z variable. A rigorous result, corroborating this picture has recently been proven in [6].…”

Section: Small Mass Limit-a Perturbative Approachsupporting

confidence: 72%

A Homogenization Theorem for Langevin Systems with an Application to Hamiltonian Dynamics

Birrell

Wehr

2019

Springer Proceedings in Mathematics &Amp; Statistics

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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.Mathematics Subject Classification (2010). 60H10, 82C31.

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Section: Small Mass Limit-a Perturbative Approachsupporting

confidence: 72%

A Homogenization Theorem for Langevin Systems with an Application to Hamiltonian Dynamics

Birrell

Wehr

2019

Springer Proceedings in Mathematics &Amp; Statistics

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“…We will also need the convergence result from [49], concerning the joint distribution of q m t and z m t , where…”

Section: Background and Previous Resultsmentioning

confidence: 99%

Entropy Anomaly in Langevin–Kramers Dynamics with a Temperature Gradient, Matrix Drag, and Magnetic Field

Birrell

2018

J Stat Phys

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We investigate entropy production in the small-mass (or overdamped) limit of Langevin-Kramers dynamics. The results generalize previous works to provide a rigorous derivation that covers systems with magnetic field as well as anisotropic (i.e. matrix-valued) drag and diffusion coefficients that satisfy a fluctuation-dissipation relation with state-dependent temperature. In particular, we derive an explicit formula for the anomalous entropy production which can be estimated from simulated paths of the overdamped system.As a part of this work, we develop a theory for homogenizing a class of integral processes involving the position and scaled-velocity variables. This allows us to rigorously identify the limit of the entropy produced in the environment, including a bound on the convergence rate.Keywords Langevin equation · entropy anomaly · small-mass limit · homogenizationwhere γ and σ are the dissipation (i.e. drag) and diffusion coefficients respectively and W t is a Wiener process. Smoluchowski [1] and Kramers [2] pioneered the study of such diffusive systems in the small-mass (or overdamped) limit; see [3] for more

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“…In particular, when γ (σ if the Stratonovich integral is used) is state-dependent, the limiting equation can be shown to involve an additional drift term that was not present in the original system. This noise-induced drift phenomenon was first derived in [17] and has been studied in numerous subsequent works [18,19,20,21,22,23,24]. See [20] for further references and discussion.…”

Section: Introductionmentioning

confidence: 83%

“…We will also need the following bound on the difference between the fundamental solutions corresponding to two linear ODEs. See the Appendix to [24].…”

Section: Hierarchy Of Approximations and The Convergence Proofmentioning

confidence: 99%

Langevin equations in the small-mass limit: Higher-order approximations

Birrell¹,

Wehr²

2018

Preprint

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We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and statedependent drift and diffusion. The present work generalizes prior derivations of the homogenized equation for the position degrees of freedom in the m → 0 limit. Specifically, we develop a hierarchy of approximate equations for the position degrees of freedom that achieves accuracy of order m ℓ/2 over compact time intervals for any ℓ ∈ Z + . The results cover bounded forces, for which we prove convergence in L p norms, and unbounded forces, in which case we prove convergence in probability.

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Phase Space Homogenization of Noisy Hamiltonian Systems

Cited by 5 publications

References 28 publications

A Homogenization Theorem for Langevin Systems with an Application to Hamiltonian Dynamics

A Homogenization Theorem for Langevin Systems with an Application to Hamiltonian Dynamics

Entropy Anomaly in Langevin–Kramers Dynamics with a Temperature Gradient, Matrix Drag, and Magnetic Field

Langevin equations in the small-mass limit: Higher-order approximations

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