Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force

Leimkuhler, Benedict; Sachs, Matthias

doi:10.1007/978-3-030-15096-9_8

Cited by 12 publications

(12 citation statements)

References 61 publications

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“…The matrices Γ M and Σ are such that the operator ∂ t − L GLE is hypoelliptic. (see [28,Proposition 7] for sufficient algebraic conditions on Γ M , Σ for Hypoellipticity of the operator.) In particular,…”

Section: Assumptionmentioning

confidence: 99%

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Efficient Numerical Algorithms for the Generalized Langevin Equation

Leimkuhler¹,

Sachs²

2020

Preprint

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We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on canonical sampling properties of numerical integrators. For this purpose, we cast the GLE in an extended phase space formulation and derive a family of splitting methods which generalize existing Langevin dynamics integration methods. We show exponential convergence in law and the validity of a central limit theorem for the Markov chains obtained via these integration methods, and we show that the dynamics of a suggested integration scheme is consistent with asymptotic limits of the exact dynamics and can reproduce (in the short memory limit) a superconvergence property for the analogous splitting of underdamped Langevin dynamics. We then apply our proposed integration method to several model systems, including a Bayesian inference problem. We demonstrate in numerical experiments that our method outperforms other proposed GLE integration schemes in terms of the accuracy of sampling. Moreover, using a parameterization of the memory kernel in the GLE as proposed by Ceriotti et al [12], our experiments indicate that the obtained GLE-based sampling scheme outperforms state-of-the-art sampling schemes based on underdamped Langevin dynamics in terms of robustness and efficiency.

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Section: Assumptionmentioning

confidence: 99%

“…Under the above stated assumptions exponential convergence of the associated semi-group and a central limit theorem for trajectory averages can be established: Proposition 1.2 ( [28]). Let Assumptions 1 to 3 be satisfied.…”

Section: Assumptionmentioning

confidence: 99%

Efficient Numerical Algorithms for the Generalized Langevin Equation

Leimkuhler¹,

Sachs²

2020

Preprint

Self Cite

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“…The success of the hypoellipticity method inspires us to develop a hypocoercivity theory for analyzing the dynamical behavior of the EMZ equation. The hypocoercivity method was mainly developed by Villani [40], Dolbeault, Mouhot and Schmeiser [4] and many other researchers [12,13,11,20,19] to study the kinetic equations in a pure functional analysis framework. For detailed explorations in this regard, we refer to the above papers and the reference therein.…”

Section: Introductionmentioning

confidence: 99%

A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

Zhu¹

2021

Preprint

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We provide a simple hypocoercivity analysis for the effective Mori-Zwanzig equation governing the time evolution of noise-averaged observables in a stochastic dynamical system. Under the hypocoercivity framework mainly developed by Dolbeault, Mouhot and Schmeiser and further extended by Grothaus and Stilgenbauer, we prove that under the same conditions which lead to the geometric ergodicity of the Markov semigroup e tK , the Mori-Zwanzig orthogonal semigroup e tQKQ is also geometrically ergodic, provided that P = I − Q is a finite-rank, orthogonal projection operator in a certain Hilbert space. The result is applied to the widely used Mori-type effective Mori-Zwanzig equations in the coarse-grained modeling of molecular dynamics and leads to exponentially decaying estimates for the memory kernel and the fluctuation force.

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“…The last term η(t) is the noise which is the fluctuation part of the interaction between the system and the heat bath. Later, the GLE was recovered by dimension reduction from Ford-Kac and Kac-Zwanzig models using Mori-Zwanzig projection ( [16,17,18,19]). In the GLE models, the noise η and the kernel for the friction γ(·) satisfy the so-called fluctuation-dissipation theorem (FDT) E(η(t)η(t + τ )) = kT γ(|τ |), ∀τ ∈ R.…”

Section: Introductionmentioning

confidence: 99%

A Discretization of Caputo Derivatives with Application to Time Fractional SDEs and Gradient Flows

Li¹,

Liu²

2019

SIAM J. Numer. Anal.

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We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the corresponding implicit schemes, are proved by its linkage to Volterra integrals with completely monotone kernels. We then apply the backward scheme corresponding to this discretization to two time fractional dissipative problems, and these implicit schemes are helpful for the analysis of the corresponding problems. In particular, we show that the overdamped generalized Langevin equation with fractional noise has a unique limiting measure for strongly convex potentials and establish the convergence of numerical solutions to the strong solutions of time fractional gradient flows. The proposed scheme and schemes derived using the same philosophy can be useful for many other applications as well. *

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Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force

Cited by 12 publications

References 61 publications

Efficient Numerical Algorithms for the Generalized Langevin Equation

Efficient Numerical Algorithms for the Generalized Langevin Equation

A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

A Discretization of Caputo Derivatives with Application to Time Fractional SDEs and Gradient Flows

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