Effective thermodynamics of two interacting underdamped Brownian particles

Abstract

Starting from the stochastic thermodynamics description of two coupled underdamped Brownian particles, we showcase and compare three different coarse-graining schemes leading to an effective thermodynamic description for the first of the two particles: Marginalization over one particle, bipartite structure with information flows and the Hamiltonian of mean force formalism. In the limit of time-scale separation where the second particle locally equilibrates, the effective thermodynamics resulting from the first and third approach is shown to capture the full thermodynamics and to coincide with each other. In the bipartite approach, the slow part does not, in general, allow for an exact thermodynamic description as the entropic exchange between the particles is ignored. Physically, the second particle effectively becomes part of the heat reservoir. In the limit where the second particle becomes heavy and thus deterministic, the effective thermodynamics of the first two coarse-graining methods coincides with the full one. The Hamiltonian of mean force formalism however is shown to be incompatible with that limit. Physically, the second particle becomes a work source. These theoretical results are illustrated using an exactly solvable harmonic model.

Stochastic thermodynamics of all-to-all interacting many-body systems

“…Refs. [72,76]. In the present work, the coarse-graining does not rely on any time-scale separation but results from the all-to-all interactions which do not discriminate energetically between the different microstates leading to the same global occupation.…”

“…Many of these are based on time-scale separation: fast degrees of freedom reach a local stationary state over timescales much shorter than the slow dynamics and can be adiabatically eliminated. The resulting transition rates of the slow dynamics then satisfy a local detailed balance condition which carries the information about the thermodynamic potentials (energetic and/or entropic) [70][71][72] or the driving forces [73][74][75] resulting from the fast dynamics. Some coarse-grainings do not require time-scale separation and the hidden degrees of freedom have been shown to behave as work sources (pure energy no entropy) on the remaining degrees of freedom, see e.g.…”

Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force

“…Therefore, we conclude that all choices fulfilled by Eq. (19) are equally legitimate starting points to construct a theory of nonequilibrium thermodynamics. Experimentally, reconstructingH S (λ t ) can be done in various ways, in particular in the classical case.…”

“…We start by reviewing recent progress in nonequilibrium strong-coupling thermodynamics based on the Hamiltonian of mean force [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] in a unified way. The goal is to find a consistent thermodynamic description for a system, which can be driven far away from equilibrium and which exchanges energy with a single arbitrary strongly coupled bath.…”

Sustaining a temperature difference

“…In the second identity in Eq. 12, { • , • } denotes the Poisson bracket and the dissipative current is given by [19,21] j :…”