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International Journal of Modern Physics B

THERMO-STATISTICAL THEORY OF KINETIC AND RELAXATION PROCESSES

Fabio Stucchi Vannucchi, Aurea Rosas Vasconcellos, Roberto Luzzi

Published Oct 30, 2009• 18 citations
Publisher

Abstract

We describe, in a short overview, the construction of a Nonequilibrium Statistical Mechanics Ensemble Formalism, providing a thermo-statistical theory of kinetic and relaxation processes. Such construction has been approached along the recently past 20th century by a pleiad of distinguished scientists, a work that can be subsumed in a large systematization in the form of a physically sound, general and useful, theoretical framework. We briefly comment on the main questions associated to that construction. Among them are the relevant ones of choice of the basic variables, and of historicity and irreversibility. The derivation of a nonequilibrium grand-canonical statistical operator and a brief description of the all-important accompanying Nonlinear Quantum Kinetic Theory of relaxation processes are presented. The aspect of validation of the theory (comparison of theory and experiment) is reviewed in compact form, and its use is illustrated in a study of a nonequilibrium system of quantum oscillators embedded in a thermal bath and under the action of an external force, showing how a far-reaching generalization of Mori–Langevin equations arises.

DOI
10.1142/s0217979209054284

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We describe, in a short overview, the construction of a Nonequilibrium Statistical Mechanics Ensemble Formalism, providing a thermo-statistical theory of kinetic and relaxation processes. Such construction has been approached along the recently past 20th century by a pleiad of distinguished scientists, a work that can be subsumed in a large systematization in the form of a physically sound, general and useful, theoretical framework. We briefly comment on the main questions associated to that construction. Among them are the relevant ones of choice of the basic variables, and of historicity and irreversibility. The derivation of a nonequilibrium grand-canonical statistical operator and a brief description of the all-important accompanying Nonlinear Quantum Kinetic Theory of relaxation processes are presented. The aspect of validation of the theory (comparison of theory and experiment) is reviewed in compact form, and its use is illustrated in a study of a nonequilibrium system of quantum oscillators embedded in a thermal bath and under the action of an external force, showing how a far-reaching generalization of Mori–Langevin equations arises.
The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.
The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.
The nonlinear quantum kinetic theory for many-body systems either near or far from equilibrium that a nonequilibrium ensemble formalism provides is revisited. In this communication we consider an important limit of such transport equations, consisting of the memoryless approximation, which leads to the so-called Markovian kinetic equations. They are derived in Zubarev's approach to the method, and next applied to a particular model of a spin system in interaction with a thermal bath of lattice vibrations. The limitations of the approach, as well as some criticism it has received, are discussed.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
We present basic equations of nonequilibrium thermo field dynamics of dense quantum systems. A formulation of nonequilibrium thermo field dynamics has been performed using the nonequilibrium statistical operator method by D.N.Zubarev. Generalized transfer and hydrodynamic equations of a consistent description of kinetics and hydrodynamics have been obtained in thermo field representation. To demonstrate how obtained results do work at the description of kinetics and hydrodynamics of a dense nuclear matter we consider quantum system with strongly coupled states.
Using the general framework of nonequilibrium statistical mechanics for relativistic quantum field systems we derive the basic equations of quantum field kinetics. The main aim of the approach is calculation of observables associated with quark-gluon plasma which is out of thermal and chemical equilibrium. We show that in the regime of high rate of the phase and statistical mixing the perturbation theory for many-body quantum field system should have the form which much differs from the standard expansion in powers of coupling constant. ( Figures can be obtained from the author )