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Journal of the Physical Society of Japan

Time Development of Distribution Function in Coarse-Grained Mechanics

Kin’ya Takahashi

Published Feb 15, 1988• 11 citations
Publisher

Abstract

Relationship between Husimi function ρ H and distribution function ρ CC in coarsegrained classical mechanics is studied where ρ CC is introduced by same manner of coarse-graining in Husimi function. Defining the product of dynamical variables and Poisson brackets in coarse-grained classical mechanics, we reduce the equations for time development of the quantities. Particularly, ρ CC shows good correspondence to ρ H in early stage of time development, but shows poor conformity after quantum interference appeared.

DOI
10.1143/jpsj.57.442

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Relationship between Husimi function ρ H and distribution function ρ CC in coarsegrained classical mechanics is studied where ρ CC is introduced by same manner of coarse-graining in Husimi function. Defining the product of dynamical variables and Poisson brackets in coarse-grained classical mechanics, we reduce the equations for time development of the quantities. Particularly, ρ CC shows good correspondence to ρ H in early stage of time development, but shows poor conformity after quantum interference appeared.
The quantum dynamics in Husimi representation is studied for the commutation relation, time development of distribution function and eigen distribution function. In the integrable system, the Husimi eigen distribution function is confined almost on the classical torus and can be approximated by Gaussian distribution with the variance (Δξl)2∼ℏ/2(\Delta\xi_{l})^{2}{\sim}\hbar/2(Δξl​)2∼ℏ/2 around the torus. Further, considering the correspondence between quantum and classical mechanics, the Husimi distribution function is a better representation than the Wigner distribution function, because coarse graining is usually involved in observational process.
The correspondence between classical and quantum mechanics is an important subject for the better understandings of “quantum chaos”. In particular, it is very important to investigate the correspondence between distribution functions in classical mechanics and in phase space representation of quantum mechanics. This is the review of our recent progresses in the study of distribution functions in classical and quantum mechanics, namely distribution functions in classical mechanics and in coarse-grained classical mechanics as well as the Wigner function and the Husimi function. Topics dealt with include formulations of the Wigner representation, the Husimi representation and coarse-grained classical mechanics, and their applications to the analyses of the eigenstates and time developments of the distribution functions.
We study the correspondence between quantum and classical dynamics with Wigner and Husimi distribution functions, respectively. The nature of quantum-mechanical regular and irregular behaviors is discussed with the use of Wigner and Husimi distribution functions. By numerical studies of the chaotic state, the Husimi distribution function is found to be a better representation than the Wigner distribution function, because coarse graining is usually involved implicitly in an observational process using the Wigner distribution function.
This paper deals with how a classical mechanics relates to the quantum mechanics, comparing the stationary distribution function on a constant energy in a coarse-grained classical system with the Husimi function ρ H of the eigenstate of quantum system. In the domain of a torus with one degree of freedom, these two functions are quite similar to each other. On the other hand, around a hyperbolic fixed-point the functions satisfy the equations of the similar type. However the solutions admissible for classical and quantum distribution functions have very different forms. In this paper we discuss the necessity for the problem of the correspondence between classical and quantum mechanics to investigate not only the similarity of the equations but also the geometrical structure of the solutions.
The Husimi distribution provides for a coarse-grained representation of the phase-space distribution of a quantum system, which may be used to track the growth of entropy of the system. We present a general and systematic method of solving the Husimi equation of motion for an isolated quantum system, and we construct a coarse-grained Hamiltonian whose expectation value is exactly conserved. As an application, we numerically solve the Husimi equation of motion for two-dimensional Yang-Mills quantum mechanics (the x-y model) and calculate the time evolution of the coarse-grained entropy of a highly excited state. We show that the coarse-grained entropy saturates to a value that coincides with the microcanonical entropy corresponding to the energy of the system.
Time development of quantum wave packet is studied in the Hisumi representation for regular and irregular case. The Husimi distribution function spreads on tori or into irregular region for regular or irregular case, respectively and coincides to classical probability distribution function. However by the quantum mechanical interference effects, small packets appear in the process of time development and thus the Husimi distribution function distributes inhomogeneously in contrast to the classical uniform probability distribution function. These small packets are shown to be more stable than classical one.