Entropy production in quantum Yang–Mills mechanics in the semiclassical approximation

Abstract

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 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 the simultaneous application of the two methods ensures the reliability of the results of the Husimi-Wehrl entropy at a given time.

Chaos in Chiral Condensates in Gauge Theories

“…Let us note that our Lyapunov exponent L can be written as an out-of-time-ordered correlator e 2Lt ∼ E, M|[Q(t), P (0)] 2 |E, M N 1,λ 1 (11) where Q(t) ≡ψψ(t) is the chiral condensate operator inserted at time t, and P is for its shift, [Q(x), P (x )] = iδ(x−x ). The state |E, M is an energy eigenstate of the supersymmetric QCD Hamiltonian, with a degeneracy index M. [62] The original out-of-time-ordered correlator uses a thermal partition [16,17], while ours is an energy eigenstate, so the temperature scale of the former roughly corresponds to our energy E. Our method can assign a Lyapunov exponent to quantum dynamics of gauge theories and opens broad applications of chaos to particle physics.…”

“…The Lyapunov exponent can be defined only in classical systems -once the systems are quantized, because the strong dependence on initial values is lost due to the quantum effect. Then how can one measure the chaos of purely quantum phenomena, such as the chiral condensate of QCD?In the history concerned with this issue, the chaos of the classical limit of the Yang-Mills theory was first found [1][2][3][4][5][6], and was applied to an entropy production process of heavy ion collisions [7][8][9][10][11][12] together with a color glass condensate [13][14][15]. However, the produced quark gluon plasma is strongly coupled, and a transition from the classical Yang-Mills to quantum states is yet an open question.…”

Entropy production from chaoticity in Yang-Mills field theory with use of the Husimi function

“…Then, we can define the Boltzmann-like entropy in terms of f H as S HW = −Tr f H log f H , where Tr means the integral over the phase space. This entropy was first introduced and called the classical entropy by Wehrl [36], and we call it Husimi-Wehrl (HW) entropy [37,38]. In the previous work [38], the present authors examine thermalization of isolated quantum systems by using the HW entropy evaluated in the semiclassical approximation.…”

“…This entropy was first introduced and called the classical entropy by Wehrl [36], and we call it Husimi-Wehrl (HW) entropy [37,38]. In the previous work [38], the present authors examine thermalization of isolated quantum systems by using the HW entropy evaluated in the semiclassical approximation. It was shown that the semiclassical treatment works well in describing the entropyproduction process of a couple of quantum mechanical systems whose classical counter systems are known to be chaotic.…”

What lattice theorists can do for superstring/M-theory

“…117,118 (See also Ref. 120 for an attempt to include quantum effects.) It has been found 118 that the Lyapunov exponent is of order N 0 and satisfies the proposed bound, and the entire Lyapunov spectrum has a nice large-N limit which is consistent with the argument in the gravity side.…”