In this article we propose a new theory for collective modes of a many-body system which contains linear response theory and supplies us with systematic methods for the analysis of Brownian motion. The essential idea is to develop a method to find the possible justification for Zubarev's nonequilibrium ensemble from the new standpoint which is to draw attention to the usage of the dynamical invariance. The present theory has many interesting features which are relevant to irreversible processes. In this article we are more interested in problems of principle than in problems of applicability.
In this article we propose a new theory for collective modes of a many-body system which contains linear response theory and supplies us with systematic methods for the analysis of Brownian motion. The essential idea is to develop a method to find the possible justification for Zubarev's nonequilibrium ensemble from the new standpoint which is to draw attention to the usage of the dynamical invariance. The present theory has many interesting features which are relevant to irreversible processes. In this article we are more interested in problems of principle than in problems of applicability.
In the spirit of the linear response theory, a simple derivation is given of a Langevin equation for collective modes of a macroscopic system. Unlike in the usual treatment, we separate a weak interaction of individual particles with collective modes which is allowed to depend on time. In nonequilibrium cases, we have shown that a damping constant of the collective modes can be expressed in terms of a time correlation function, in which its average is taken with respect to a nonequilibrium density matrix rather than a usual canonical density matrix. A microscopic expression for a Langevin random force is obtained in terms of fluctuations of a density matrix.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
Slow collective motions in physical systems, particularly those near the critical point, can be best described in the framework of Langevin equations. We may set up Langevin equations when the timescales of slow and fast dynamical variables are distinctly separated. This framework originates from the classical Brownian motion and is justified microscopically via the projection operator formalism. First, in Sections 5.1–5.2, these general aspects will be discussed with a summary of the projection operator method in Appendix 5B. Second, in Section 5.3, we will examine simple Langevin equations in critical dynamics (models A, B, and C) and introduce dynamic renormalization group theory. These models have been used extensively to study fundamental problems in critical dynamics and phase ordering. Third, in Section 5.4, we will review the general linear response theory, putting emphasis on linear response to thermal disturbances. Langevin equation for a single particle Brownian motion Most readers will be aware of the zig-zag motions of a relatively large particle, calleda Brownian particle, suspended in a fluid. When its mass m 0 is much larger than those of the surrounding particles, appreciable changes of the velocity of the Brownian particle can be caused as a result of a large number of collisions with the surrounding molecules.
One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.
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.