Effective Gibbs state for averaged observables

Abstract

We consider the effective Gibbs state for averaged observables. In particular, we perturbatively calculate the correspondent effective Hamiltonian. We show that there are a lot of similarities between this effective Hamiltonian and the mean force Hamiltonian. We also discuss a thermodynamic role of the information loss due to restriction of our measurement capabilities to such averaged observables.

Hamiltonian of mean force in the weak-coupling and high-temperature approximations and refined quantum master equations

“…In order to express the result in the quasi-Gibbsian form (22) and to find the Hamiltonian of mean force, we first express the corrections to the Gibbs state in the exponential form (with the condition that the expansion up to the second order in β should coincide with the expressions obtained above):…”

“…MF (β) to the Hamiltonian of mean force. 1 It is worthwhile to mention paper [22], where the effective Hamiltonian with corrections to the Gibbs state in the weak-coupling regime was derived for the time-averaged observables.…”

Effective Heisenberg equations for quadratic Hamiltonians

“…Using an analogy with Ref. [1] we define the averaging projector in the following way. Definition 1.…”

“…In Ref. [1] we have discussed the effective Gibbs state, which could be obtained from the exact Gibbs state by applying an averaging projector with respect to free dynamics. Here we consider a dynamical analog of such a procedure.…”

“…Namely, in section 2 we define an averaging projector which acts on dynamical maps and we regard the result as an effective dynamical map. Let us remark that the fact that such an operator acts on dynamical maps rather than states or observables differs it both from equilibrium situation [1] and from the projectors which are usually discussed in Nakajima-Zwanzig projection formalism. [2,3] In spite of the fact, that very different projection superoperators [4,5,6,7] are used in Nakajima-Zwanzig projection formalism, they usually act on states or observables, so the direct analog of our projector is a "hyperoperator" (map from superoperators to superoperators) that acts on a dynamical map by Nakajima-Zwanzig projection superoperators on both sides of the dynamical map.…”