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Entropy production and isotropization in Yang-Mills theory using a quantum distribution function

Hidekazu Tsukiji, Teiji Kunihiro, Akira Ohnishi A, TORU T. TAKAHASHI

Published Jan 1, 2018• 7 citations
OpenAlex

Abstract

We investigate the thermalization process from a glasma-like initial condition in a static box in terms of the Husimi–Wehrl (HW) entropy defined with the Husimi function, a quantum distribution function in a phase space. We calculate the semiclassical time evolution of the HW entropy in Yang–Mills field theory with the phenomenological initial field configuration given by mimicking the McLerran–Venugopalan model in a static box, which has instability triggered by initial field fluctuations. Husimi–Wehrl entropy production implies the thermalization of the system and it reflects the underlying dynamics such as chaoticity and instability. By comparing the production rate with the Kolmogorov–Sinaï rate, we find that the HW entropy production rate is significantly larger than that expected from chaoticity. We also show that the HW entropy is finally saturated when the system reaches a quasi-stationary state. The saturation time of the HW entropy is comparable with that of pressure isotropization, which is around 1 fm/c in the present calculation.

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We investigate the thermalization process from a glasma-like initial condition in a static box in terms of the Husimi–Wehrl (HW) entropy defined with the Husimi function, a quantum distribution function in a phase space. We calculate the semiclassical time evolution of the HW entropy in Yang–Mills field theory with the phenomenological initial field configuration given by mimicking the McLerran–Venugopalan model in a static box, which has instability triggered by initial field fluctuations. Husimi–Wehrl entropy production implies the thermalization of the system and it reflects the underlying dynamics such as chaoticity and instability. By comparing the production rate with the Kolmogorov–Sinaï rate, we find that the HW entropy production rate is significantly larger than that expected from chaoticity. We also show that the HW entropy is finally saturated when the system reaches a quasi-stationary state. The saturation time of the HW entropy is comparable with that of pressure isotropization, which is around 1 fm/c in the present calculation.
We investigate the thermalization process from a glasma-like initial condition in a static box in terms of the Husimi–Wehrl (HW) entropy defined with the Husimi function, a quantum distribution function in a phase space. We calculate the semiclassical time evolution of the HW entropy in Yang–Mills field theory with the phenomenological initial field configuration given by mimicking the McLerran–Venugopalan model in a static box, which has instability triggered by initial field fluctuations. Husimi–Wehrl entropy production implies the thermalization of the system and it reflects the underlying dynamics such as chaoticity and instability. By comparing the production rate with the Kolmogorov–Sinaï rate, we find that the HW entropy production rate is significantly larger than that expected from chaoticity. We also show that the HW entropy is finally saturated when the system reaches a quasi-stationary state. The saturation time of the HW entropy is comparable with that of pressure isotropization, which is around 1 fm/c in the present calculation.
We investigate the thermalization process from a glasma-like initial condition in a static box in terms of the Husimi–Wehrl (HW) entropy defined with the Husimi function, a quantum distribution function in a phase space. We calculate the semiclassical time evolution of the HW entropy in Yang–Mills field theory with the phenomenological initial field configuration given by mimicking the McLerran–Venugopalan model in a static box, which has instability triggered by initial field fluctuations. Husimi–Wehrl entropy production implies the thermalization of the system and it reflects the underlying dynamics such as chaoticity and instability. By comparing the production rate with the Kolmogorov–Sinaï rate, we find that the HW entropy production rate is significantly larger than that expected from chaoticity. We also show that the HW entropy is finally saturated when the system reaches a quasi-stationary state. The saturation time of the HW entropy is comparable with that of pressure isotropization, which is around 1 fm/|ccc| in the present calculation.
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.
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.
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.
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.