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Physical Review Letters

Approach to Equilibrium in Quantum Systems

Pierre M. V. Résibois

Published Nov 1, 1960• 6 citations
Publisher

Abstract

Prigogine and his co-workers developed a genera; theory of approach to equilibrium in classical mechanics. A inethod is presented which indicates the possibility for obtaining similar results for quantum systems. This method makes use of a representation for the density matrix which stresses the formal analogy between the von Neumann equation and the classical Liouville equation. This representation was used in the particular cases of weakly-coupled systems and of quantum plasmas. (B.O.G.)

DOI
10.1103/physrevlett.5.411

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Prigogine and his co-workers developed a genera; theory of approach to equilibrium in classical mechanics. A inethod is presented which indicates the possibility for obtaining similar results for quantum systems. This method makes use of a representation for the density matrix which stresses the formal analogy between the von Neumann equation and the classical Liouville equation. This representation was used in the particular cases of weakly-coupled systems and of quantum plasmas. (B.O.G.)
The methods developed by Prigogine and his colleagues are applied to the study of quantum mechanical systems. The density matrix is divided into a slowly varying and an oscillatory part, and asymptotic equations are established for the elements of this matrix. In the representation in which the unperturbed hamiltonian is diagonal, these equations take a particularly simple form for weakly coupled systems: the matrix elements lying on diagonals parallel to the principal diagonal undergo separate transformations. A particular case is that of the diagonal elements themselves; for these the well-known Pauli equations are obtained. As an illustrative example the problem of the frictional forces on an oscillator immersed in a thermostat is considered in detail. The frictional forces introduced by Langevin and Lorentz are special cases of this.
Abstract For a system consisting of a quantum and a classical subsystem, an equation of the Liouville type is obtained starting with the density matrix equation. The validity condition and the transformation properties of this quantum-classical equation as well as the application of the adiabatic approximation and the perturbation theory within the frame of this equation are discussed. Some general expressions are obtained for the probability of the transition between the quasi-equilibrium states of the system. Special attention is given to the electron-nuclei system, as the general formalism was developed first of all in order to obtain a foundation for the theoretical analysis of elementary processes in liquids. An example is given to demonstrate that the quantum-classical approach in the range of its validity gives the same result as the corresponding quantum solution. It is also shown that both the cases of weak and strong interaction (leading to the transition) can be described by the quantum-classical equation in question.
It is well known that the statistics of closed classical systems evolves according to the Liouville theorem. Here we study the dynamics of the marginal statistics of classical systems coupled to external degrees of freedom, by developing a time-local equation of motion using Green's functions and a series expansion method. We also compare this equation of motion with its supposed quantum counterpart, namely the quantum master equation, which we hope could shed some light on quantum-classical analogy (QCA) from the perspective of "open system dynamics". We notice an apparent exception to QCA in this case, as the first-order classical equation of motion derived herein contains a term that does not appear to have a quantum analogue. We also propose possible ways of getting around this tension, which may help re-establish QCA (in first perturbative order). We do not draw a definitive conclusion about QCA in the context of open system dynamics but hope to provide a starting point for investigations along this line.
A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong-coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand-canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak-coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities and the shell structure of harmonically bound charges.
A bstract In this paper, we investigate the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix defined on a single Cauchy surface. We present the construction of the spacetime density matrix in general quantum systems and its representation via the Schwinger-Keldysh path integral. We further introduce a super-operator framework, within which the spacetime density matrix appears as a special case, and discuss possible generalizations from this perspective. We also show that the spacetime density matrix satisfies a Liouville-von Neumann type equation of motion. When considering subsystems, a reduced spacetime density matrix can be defined by tracing over complementary degrees of freedom. We study the general properties of its moments and, in particular, derive universal short-time behavior of the second moment. We find that coupling between subsystems plays a crucial role in obtaining nontrivial results. Assuming weak coupling, we develop a perturbative method to compute the moments systematically.
A one-dimensional quantum gas with \ensuremath{\delta}-function interaction is known to be integrable for any finite number of particles. We show that in the thermodynamic limit invariants of motion are destroyed in conjunction with a class of singular density matrices that appear commonly in both equilibrium and nonequilibrium statistical mechanics. The system becomes nonintegrable, due to Poincar\'e resonances. However, the eigenvalue problem of the Liouville--von Neumann operator LH{L}_{H}LH​ may be solved in an extended functional space that contains the singular density matrices. We may in this way describe the approach to equilibrium. This is a simple example of the Liouville space extension of quantum mechanics, due to Prigogine and his co-workers. The fundamental quantity in this formulation is the probability (and not the wave function). We obtain indeed an irreducible complex spectral representation of LH.{L}_{H}.LH​. Irreducible means that the eigenstates of LH{L}_{H}LH​ cannot be implemented by wave functions, and complex means that the eigenvalues are complex numbers, whose imaginary parts determine relaxation times of the system. The dynamical evolution is described by a semigroup. Due to dissipative effects the real part of the eigenvalues is no more a difference of energy levels, as is the case in standard quantum mechanics.