The existence of thermodynamic limits of non-equilibrium reduced density matrix and distribution function is proved on the following assumptions on initial states and temporal evolution of the system: The initial state is described by a canonical distribution, an effective Hamiltonian of which is the sum of local Hamiltonians and some averages of “local” operators are bounded. This proof yields an extension of Kubo's Ansatz on the extensive property of a macrovariable to quantal systems and to non-Markoffian macrovariables in stochastic models. A generating function (generalized thermodynamic potential) formalism is used effectively for discussing the above non-equilibrium problems. Non-linear response and fluctuation are also discussed in the above-mentioned scheme.
The existence of thermodynamic limits of non-equilibrium reduced density matrix and distribution function is proved on the following assumptions on initial states and temporal evolution of the system: The initial state is described by a canonical distribution, an effective Hamiltonian of which is the sum of local Hamiltonians and some averages of “local” operators are bounded. This proof yields an extension of Kubo's Ansatz on the extensive property of a macrovariable to quantal systems and to non-Markoffian macrovariables in stochastic models. A generating function (generalized thermodynamic potential) formalism is used effectively for discussing the above non-equilibrium problems. Non-linear response and fluctuation are also discussed in the above-mentioned scheme.
We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory . The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.
Without assuming detailed balance, the linear response theory of systems, in which the notation of a quantum dynamical semigroup can be defined, is discussed. This theory is used to prove a fluctuation-dissipation theorem for nonstationary nonequilibrium states. We propose an extended version of the Liouville equation in which we incorporate the stochastic behavior of the subsystem of the macroscopic motions. For this we introduce a non-Hermitian Hamiltonian and set a Liouville equation for a density matrix which is not Hermitian. We show that a response to external perturbations reduces to a sum of two contributions, one for which an improved fluctuation-dissipation theorem is proved and the other due to stochastic excitations that does not satisfy a reciprocity relation of Onsager type.
This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic index q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value qc which depends on the rate by which the eigenvalues of the Hamiltonian diverge. Existence of a unique equilibrium density matrix is proved if in addition q<2 holds. For q between 0 and 1, such that 2 <q+ qc , the free energy has at least one minimum in the set of trial density matrices. If a unique equilibrium density matrix exists then it is necessarily one of the trial density matrices. Note that this is a finite rank operator, which means that in equilibrium high energy levels have zero probability of occupancy. 1. Introduction The formalism of non-extensive thermodynamics started more than 10 years ago with the introduction by C. Tsallis [1] of a family of entropies, parameterized with a parameter q called the entropic index. It has developed gradually to a collection of mostly phenomenological results with a large number of applications, some more convincing than others. Nevertheless, part of the physics community is still sceptic about the need and physical relevance of the theory. By providing mathematical proofs for some of the fundaments of the formalism this paper tries to improve its credibility and to provide the necessary base for further extension. Some of the results presented in the present paper, in particular concerning q>1-statistics, have already been published in [2]. Here, missing details are filled in and the q<1-case is added. The structure of the paper is as follows. In the...
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
The overview of the principles of quantum statistical mechanics are given, emphasizing the fundamental differences with respect to classical statistical mechanics, as well as the analogies prevailing for the formulation of the properties. A functional time-reversal symmetry relation is presented, allowing the deduction of response theory. The Kubo formula is obtained for the linear response properties and the fluctuation–dissipation theorem is established. For weakly coupled systems, the quantum master equation and the corresponding stochastic Schrödinger equation are deduced. The slippage of initial conditions is discussed in relation to the positivity of the reduced statistical operator. The results are illustrated with the spin-boson model.
Descriptions of molecular systems usually refer to two distinct theoretical frameworks. On the one hand the quantum pure state, i.e., the wavefunction, of an isolated system is determined to calculate molecular properties and their time evolution according to the unitary Schrödinger equation. On the other hand a mixed state, i.e., a statistical density matrix, is the standard formalism to account for thermal equilibrium, as postulated in the microcanonical quantum statistics. In the present paper an alternative treatment relying on a statistical analysis of the possible wavefunctions of an isolated system is presented. In analogy with the classical ergodic theory, the time evolution of the wavefunction determines the probability distribution in the phase space pertaining to an isolated system. However, this alone cannot account for a well defined thermodynamical description of the system in the macroscopic limit, unless a suitable probability distribution for the quantum constants of motion is introduced. We present a workable formalism assuring the emergence of typical values of thermodynamic functions, such as the internal energy and the entropy, in the large size limit of the system. This allows the identification of macroscopic properties independently of the specific realization of the quantum state. A description of material systems in agreement with equilibrium thermodynamics is then derived without constraints on the physical constituents and interactions of the system. Furthermore, the canonical statistics is recovered in all generality for the reduced density matrix of a subsystem.