Equilibrium reduced-density operators for subsystems of the universe are derived from the generalised Schrödinger variational principle, additionally assuming that the values of von Neumann entropy for the subsystems are fixed and higher than zero.The obtained reduced-density operator may be useful for description of the properties of an arbitrary system (macroscopic, mesoscopic, nanoscopic, or microscopic) in the equilibrium mixed state.
Equilibrium reduced-density operators for subsystems of the universe are derived from the generalised Schrödinger variational principle, additionally assuming that the values of von Neumann entropy for the subsystems are fixed and higher than zero.The obtained reduced-density operator may be useful for description of the properties of an arbitrary system (macroscopic, mesoscopic, nanoscopic, or microscopic) in the equilibrium mixed state.
In this contribution we prove that the entropy of an N-body isolated system can not decrease and the entropy production should be non-negative provided the system possesses an equilibrium state. We define the entropy as a functional of the set of n-particle reduced density operators (n=0,....,N) generalizing the von Neumann fine-grained entropy formula. Additionally, as a consequence of our analysis we find the expression of the equilibrium n-particle reduced density operators which enter the definition of the entropy as well as the dissipated energy in an irreversible process.
The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.
In many cases, measurements are performed on “mixed” ensembles of realizations of a system, which cannot be associated with a single vector in its Hilbert space. In these cases the state of the system is represented by a proper “density operator.” It is instructive to associate density operators with state vectors in a vector space, termed Liouville’s space, where the Schrödinger equation is reformulated as the Liouville–Von Neumann equation. An important consequence of this equation is that the density operator of a system at equilibrium must commute with its Hamiltonian. Namely, the matrix representation of the density operator in the basis of Hamiltonian eigenstates is diagonal, where the diagonal elements are the relative populations of the system’s Hamiltonian eigenstates. The equilibrium populations are revealed by maximizing the (Von Neumann) entropy, subject to given constraints. We derive the equilibrium density operator explicitly for the cases of canonical and grand canonical ensembles.
On y szk i ewic z Ê Q uan tu m Ph ysi cs D i vi sion, Facul ty of Physi cs A. Mi cki ewi cz Uni versity , Um ul towsk a 85, 61-614 Pozna¥, Po l an d ( Recei ved Ju ly 22, 2 002) Retarded for m s of the none quili bri um redu ced-densi ty operator are deri ved from the generali zed Schr odinger variational principl e of a system w ith b oth mechanical and thermal perturbations , ta king into account the additional conditio n that the values of generalized thermo dynamic coordinates are Ùxed at inÙnitel y remote past .T his reduced density op erator may b e usef ul in describi ng nonequili bri um prop erties of nanoscopic and mesoscopic systems, as w ell as ultrathin Ùlms.With the help of this reduced-densi ty op erator the generalized Green { K ub o formulae are obtained.
The nonlinear equation of motion that accomplishes a self-consistent unification of quantum mechanics (QM) and thermodynamics conceptually different from the (von Neumann) foundations of quantum statistical mechanics (QSM) and (Jaynes) quantum information theory (QIT), but which reduces to the same mathematics for the thermodynamic equilibrium (TE) states, and contains standard QM in that it reduces to the time-dependent Schrodinger equation for zero entropy states is discussed in full mathematical detail. By restricting the discussion to a strictly isolated system (noninteracting, disentangled and uncorrelated), we show how the theory departs from the conventional QSM/QIT rationalization of the second law of thermodynamics, which instead emerges in QT (quantum thermodynamics) as a theorem of existence and uniqueness of a stable equilibrium state for each set of mean values of the energy and the number of constituent particles. To achieve this, the theory assumes Trρ lnρ B −k for the physical entropy and is designed to implement two fundamental ansatzs: (1) that in addition to the standard QM states described by idempotent density operators (zero entropy), a strictly isolated and uncorrelated system admits also states that must be described by non-idempotent density operators (nonzero entropy); (2) that for such additional states the law of causal evolution is determined by the simultaneous action of a Schrodinger-von Neumann-type Hamiltonian generator and a nonlinear dissipative generator which conserves the mean values of the energy and the number of constituent particles, and in forward time drives the density operator in the 'direction' of steepest entropy ascent (maximal entropy generation). The resulting dynamics is well defined for all non-equilibrium states, no matter how far from TE. Existence and uniqueness of solutions of the Cauchy initial value problem for all density operators implies that the equation of motion can be solved not only in forward time, to describe relaxation towards TE, but also in backward time, to reconstruct the 'ancestral' or primordial lowest entropy state or limit cycle from which the system originates. Zero entropy states as well as a well defined family of non-dissipative states evolve unitarily according to pure Hamiltonian dynamics and can be viewed as unstable limit cycles of the general nonlinear dynamics.
This chapter begins with the discussion of various meanings of the entropy: Boltzmann entropy, Shannon entropy, von Neumann entropy. It then presents the basic properties of the von Neumann entropy such as concavity, sub-additivity, strong sub-additivity, and continuity. It proves the existence of mean entropy for shift-invariant states on quantum spin chains, and then derives the expression for the mean entropy for quasi-free Fermions on a chain. The chapter defines the quantum relative entropy and presents its basic properties, including behaviour with respect to completely positive maps. The maximum entropy principle defines thermal equilibrium states (Gibbs states). This variational principle is illustrated by the Hartree–Fock approximation for a model of interacting Fermions. The entropy of an equilibrium state for a free quantum particle on a compact Riemannian manifold is also estimated. Finally, the notion of relative entropy is formulated in the algebraic setting using the relative modular operator.