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1 Towards a Theory of Entropy Production in the Little and Big Bang ∗)

Teiji Kunihiro, Berndt Müller, Akira Ohnishi A, Andreas Schäfer

Published Jan 1, 2009• 19 citations
OpenAlex

Abstract

We propose a broadly applicable formalism for the description of coarse grained en-tropy production in quantum mechanical processes. Our formalism is based on the Husimi transform of the quantum state, which encodes the notion that information about any quan-tum state is limited by the experimental resolution. We show in two analytically tractable cases (the decay of an unstable vacuum state and reheating after cosmic inflation) that the growth rate of the Wehrl entropy associated with the Husimi function approaches the classi-cal Kolmogorov-Sinäı entropy. We also discuss various possible applications of our formalism, including the production of entropy in the early stages of a relativistic heavy ion collision. The production of entropy at the quantum level, i. e. the loss of information about the state of the system under consideration, is a long-standing problem in statistical physics. Many of its conceptual aspects are similar to those encountered in classical physics, but quantum mechanics poses the additional, but closely related,

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We propose a broadly applicable formalism for the description of coarse grained en-tropy production in quantum mechanical processes. Our formalism is based on the Husimi transform of the quantum state, which encodes the notion that information about any quan-tum state is limited by the experimental resolution. We show in two analytically tractable cases (the decay of an unstable vacuum state and reheating after cosmic inflation) that the growth rate of the Wehrl entropy associated with the Husimi function approaches the classi-cal Kolmogorov-Sinäı entropy. We also discuss various possible applications of our formalism, including the production of entropy in the early stages of a relativistic heavy ion collision. The production of entropy at the quantum level, i. e. the loss of information about the state of the system under consideration, is a long-standing problem in statistical physics. Many of its conceptual aspects are similar to those encountered in classical physics, but quantum mechanics poses the additional, but closely related,
Understanding the thermalization process in a pure quantum system is a challenge in theoretical physics. In this work, we explore possible thermalization mechanism in Yang-Mills(YM) theory by using a positive semi-definite quantum distribution function called a Husimi function which is given by a coarse graining of the Wigner function within the minimum uncertainty. Then entropy is defined in terms of the Husimi function, which is called the Husimi-Wehrl(HW) entropy. We propose two numerical methods to calculate the HW entropy. We find that it is feasible to apply the semi-classical approximation with the use of classical YM equation. It should be noted that the semi-classical approximation is valid in the systems of physical interest including the early stage of heavy-ion collisions. Using a product ansatz for the Husimi function, which is confirmed to reproduce the HW entropy within 20% error (overestimate) for a few-body quantum system, we succeed in a numerical evaluation of HW entropy of YM fields and show that it surely has a finite value and increases in time.
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.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
Our everyday descriptions of the universe are highly coarse grained, following only a tiny fraction of the variables necessary for a perfectly fine-grained description. Coarse graining in classical physics is made natural by our limited powers of observation and computation. But in the modern quantum mechanics of closed systems, some measure of coarse graining is inescapable because there are no nontrivial, probabilistic, fine-grained descriptions. This essay explores the consequences of that fact. Quantum theory allows for various coarse-grained descriptions, some of which are mutually incompatible. For most purposes, however, we are interested in the small subset of “quasiclassical descriptions” defined by ranges of values of averages over small volumes of densities of conserved quantities such as energy and momentum and approximately conserved quantities such as baryon number. The near-conservation of these quasiclassical quantities results in approximate decoherence, predictability, and local equilibrium, leading to closed sets of equations of motion. In any description, information is sacrificed through the coarse graining that yields decoherence and gives rise to probabilities for histories. In quasiclassical descriptions, further information is sacrificed in exhibiting the emergent regularities summarized by classical equations of motion. An appropriate entropy measures the loss of information. For a “quasiclassical realm” this is connected with the usual thermodynamic entropy as obtained from statistical mechanics. It was low for the initial state of our universe and has been increasing since.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
Our everyday descriptions of the universe are highly coarse grained, following only a tiny fraction of the variables necessary for a perfectly fine-grained description. Coarse graining in classical physics is made natural by our limited powers of observation and computation. But in the modern quantum mechanics of closed systems, some measure of coarse graining is inescapable because there are no nontrivial, probabilistic, fine-grained descriptions. This essay explores the consequences of that fact. Quantum theory allows for various coarse-grained descriptions, some of which are mutually incompatible. For most purposes, however, we are interested in the small subset of ``quasiclassical descriptions'' defined by ranges of values of averages over small volumes of densities of conserved quantities such as energy and momentum and approximately conserved quantities such as baryon number. The near-conservation of these quasiclassical quantities results in approximate decoherence, predictability, and local equilibrium, leading to closed sets of equations of motion. In any description, information is sacrificed through the coarse graining that yields decoherence and gives rise to probabilities for histories. In quasiclassical descriptions, further information is sacrificed in exhibiting the emergent regularities summarized by classical equations of motion. An appropriate entropy measures the loss of information. For a ``quasiclassical realm'' this is connected with the usual thermodynamic entropy as obtained from statistical mechanics. It was low for the initial state of our universe and has been increasing since.