We present general and rigorous results showing that the microcanonical and
canonical ensembles are equivalent at all three levels of description
considered in statistical mechanics - namely, thermodynamics, equilibrium
macrostates, and microstate measures - whenever the microcanonical entropy is
concave as a function of the energy density in the thermodynamic limit. This is
proved for any classical many-particle systems for which thermodynamic
functions and equilibrium macrostates exist and are defined via large deviation
principles, generalizing many previous results obtained for specific classes of
systems and observables. Similar results hold for other dual ensembles, such as
the canonical and grand-canonical ensembles, in addition to path ensembles used
for describing nonequilibrium systems driven in steady states.

在统计力学的三个层面（热力学、平衡宏态和微态测量）中，只要在热力学极限下微正则熵是能量密度的凹函数，就可以证明所有经典多粒子系统的微正则系综和正则系综是等价的，这适用于具有大偏离原理定义的热力学函数和平衡宏观状态的任何系统，并推广了以前针对特定类别系统和可观察量获得的许多结果。对于其他二元系综，如正则系综和巨正则系综，以及用于描述稳态驱动的非平衡系统的路径系综，类似的结果也成立。

整体系（ensemble）的等价性和非等价性：热力学、宏观态和测量层面

Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate,  and measure levels

We consider numerical methods for thermodynamic sampling, i.e. computing
sequences of points distributed according to the Gibbs-Boltzmann distribution,
using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A
wide variety of numerical methods for Langevin dynamics may be constructed
based on splitting the stochastic differential equations into various component
parts, each of which may be propagated exactly in the sense of distributions.
Each such method may be viewed as generating samples according to an associated
invariant measure that differs from the exact canonical invariant measure by a
stepsize-dependent perturbation. We provide error estimates a la Talay-Tubaro
on the invariant distribution for small stepsize, and compare the sampling bias
obtained for various choices of splitting method. We further investigate the
overdamped limit and apply the methods in the context of driven systems where
the goal is sampling with respect to a nonequilibrium steady state. Our
analyses are illustrated by numerical experiments.

本文研究热力学采样的数值方法，其中涉及 Langevin 动力学和过阻尼 Langevin 动力学，并分析了拆分方法的采样偏差和误差估计方法。同时，作者还探讨了过阻尼极限并将该方法应用于驱动系统的采样。

平衡和非平衡 Langevin 分子动力学求平均数的计算

The computation of averages from equilibrium and nonequilibrium Langevin  molecular dynamics

A system responding to a stochastic driving signal can be interpreted as
computing, by means of its dynamics, an implicit model of the environmental
variables. The system's state retains information about past environmental
fluctuations, and a fraction of this information is predictive of future ones.
The remaining nonpredictive information reflects model complexity that does not
improve predictive power, and thus represents the ineffectiveness of the model.
We expose the fundamental equivalence between this model inefficiency and
thermodynamic inefficiency, measured by dissipation. Our results hold
arbitrarily far from thermodynamic equilibrium and are applicable to a wide
range of systems, including biomolecular machines. They highlight a profound
connection between the effective use of information and efficient thermodynamic
operation: any system constructed to keep memory about its environment and to
operate with maximal energetic efficiency has to be predictive.

研究表明，系统动力学响应随机信号的过程可以被解释为其计算环境变量的隐含模型；而模型的效率可以用熵来衡量，因此该研究揭示了信息使用的效率与热力学操作的关联。