Universal ideal behavior and macroscopic work relation of linear irreversible stochastic thermodynamics

Abstract

We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic 'equation of state' of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.which is an OUP with parameters α and ϵ; M and Γ are two n × n constant matrices, B t is standard Brownian motion. We further assume that all the eigenvalues of M are strictly positive and Γ is non-singular. According to the concept of detailed balance, equation (2) can be uniquely written as [29][30][31][32]

Irreversible samplers from jump and continuous Markov processes

“…And the more irreversibility is introduced, the less the acceptance rate α MH will be in the MH algorithm. This gap between α MH and 1 was first discovered in the statistical mechanics literature and relates to the "house keeping heat" [16,28,47]. (30) also gives us insight into the fact that using more accurate numerical integrators could lead to higher acceptance rates.…”

“…And the more irreversibility is introduced, the less the acceptance rate α MH will be in the MH algorithm. This gap between α MH and 1 was first discovered in the statistical mechanics literature and relates to the "house keeping heat" [16,28,47].…”

“…For the reverse proposal g(z|y, y p ) in ( 23), we use the adjoint process P † z|y; t , inverting the irreversible dynamics via Q(z) → −Q(z) [47]:…”

A thermodynamic theory of ecology: Helmholtz theorem for Lotka–Volterra equation, extended conservation law, and stochastic predator–prey dynamics

“…In the classical studies on statistical mechanics, developed by Helmholtz, Boltzmann, Gibbs and others, the dynamical foundation is a Hamiltonian system [10][11][12]. The theory in [9,13] generalized such an approach that requires no a priori identification of variables as position and momentum; it also suggested a possible thermodynamic structure which is purely mathematical in nature, independent of Newtonian particles. In the context of population dynamics, we shall show that the mathematical analysis yields a conservative ecology.…”

Focus on stochastic thermodynamics

“…Ford discusses various measures of thermodynamic irreversibility with a focus on the role of initially non-symmetric velocity distributions [32]. Ma and Qian revisit the well-known multi-dimensional Ornstein-Uhlenbeck process as a paradigm of how to derive an emergent macroscopic description of a complex stochastic dynamics from its mesoscopic law of motion [33]. Last, but certainly not least, Knoch and Speck present a general new method of how to coarse-grain, or renormalize, Markov networks for systems driven into a steady state such that the entropy production of the original network remains preserved [34].…”