Thermodynamics of a subensemble of a canonical ensemble

Abstract

Two approaches to describe the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E S is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z S is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E S = −d(ln Z S )/dβ is used to obtain the mean system energy. Employing both classical and quantum mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.

On Work and Heat in Time-Dependent Strong Coupling

“…If F B is the (constant) free energy of the bath, theñ F s = F TOT − F B is the thermodynamic potential (free energy) of the system appropriate for strong coupling [11][12][13]22], and the quantity appearing in the exponent of the right-hand side of (15) is ∆F s . Since we assume that the system and the bath are uncoupled at the initial and the finite time, ∆F TOT in (1) is however in fact here equal to ∆F, the free energy change of the system only.…”

Stochastic and Macroscopic Thermodynamics of Strongly Coupled Systems

“…A similar factorization arises in the canonical-rather than isothermal-isobaric-setting [11,12,[14][15][16][17]33,34]. Now imagine that the composite system begins in equilibrium at t ¼ 0, with λ 0 ¼ A; then it evolves in time as the work parameter is varied from λ 0 ¼ A to λ τ ¼ B according to a protocol λ t ≡ λðtÞ.…”

“…(12)] was made for convenience. Alternative choices (canonical, microcanonical, grand) would lead to the same solvated ensemble for S, as discussed in greater detail in the Appendix.…”

“…A number of authors have proposed and investigated precise definitions of thermodynamic quantities for strongly coupled systems [11][12][13][14][15][16][17], with Seifert's approach [15] being the closest in spirit to the present work. In these earlier papers, the condition of macroscopic consistency was not imposed.…”

Exact quantum master equation for a molecular aggregate coupled to a harmonic bath

“…The problem of bath correlations can be analyzed from a microscopic perspective [57]. If the harmonic potentials in the Hamiltonians H SB and H B are considered as linearizations of anharmonic interaction potentials between the particles of the system and the bath, one arrives at the Hamiltonian (4), because the identical particles of the system should interact via the same (in our case, harmonic) potentials with the bath particles.…”