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Progress of Theoretical Physics

Thermal Effect in the Lipkin Model. III: Dynamical Fluctuation of Thermal Equilibrium State

A. Kuriyama, João da Providência, Y. Tsue, M. Yamamura

Published Jul 1, 1996• 1 citation
Publisher

Abstract

The thermal effects in the Lipkin model manifest themselves mainly through the change in the seniority quantum number. Within the linear approximation, we describe the dynamical fluctuation of the thermal equilibrium state which does not disturb the thermal state, i.e., which does not affect the particle distribution. Further, we discuss the damping effect on the dynamical fluctuation induced by the dynamical coupling with the auxiliary field.

DOI
10.1143/ptp.96.125

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The thermal effects in the Lipkin model manifest themselves mainly through the change in the seniority quantum number. Within the linear approximation, we describe the dynamical fluctuation of the thermal equilibrium state which does not disturb the thermal state, i.e., which does not affect the particle distribution. Further, we discuss the damping effect on the dynamical fluctuation induced by the dynamical coupling with the auxiliary field.
We study the thermal effect with the use of Lipkin model. We define the density and entropy operator associated with the mixed state representation of Lipkin model, which has been developed with the aid of auxiliary fermion field. We investigate the thermal equilibrium state and its phase transition. In super phase, the thermal effect breaks the particle-hole pairs with coupled angular momentum 0 and does not lift up nucleons from the lower level to upper one, contrary to the case of normal phase.
The Lipkin model is the simplest algebraic model in which we can investigate a ground state transition in a fermion system. This model is also applied to the description of the thermal equilibrium state. In this paper, we investigate its stability with respect to changes in the particle number and the volume. In order to enable us to take the thermal limit, an appropriate number dependence is introduced into the force strength. We show with the use of a mean field approximation that the thermal equilibrium state in the condensed phase of particle-hole pairs becomes unstable.
Abstract We have previously developed this framework in the context of static response, here we will generalize this perspective to dynamic response, and the evolution between distinct thermal equilibrium states. Of particular importance will be the derivation of the fluctuation-dissipation theorem, which relates the response of a system to its time dependent fluctuations, clarifying an equivalence between how a system responds to a small perturbation and how it regresses back to equilibrium in the absence of a perturbation. With this framework we can develop linear models of relaxation, classically and quantum mechanically, and derive equa- tions of motion suitable for describing the dynamics of thermal ensembles held at fixed temperature rather than fixed energy. These models are Gaussian, and approximate the dynamical correlations of interacting systems. These effective equations of motion are rendered stochastic due to the incomplete description of the microscopic degrees of freedom that evolve the system.
We apply the mixed state representation to the description of the thermal properties of the pairing model. We derive an extended form of the boson realization of the quasi-spin operators in a classical image. We introduce a density operator and entropy combined with the mixed state representation. Then we define a thermal equilibrium state with use of the variation principle. In this representation, the seniority number is specified by a distribution of the Fermi type as a function of temperature. We study the pairing collective modes that induce no change in the entropy.
A large number of multifaceted quantum transport processes in molecular systems and physical nanosystems can be treated in terms of quantum relaxation processes which couple to one or several fluctuating environments. A thermal equilibrium environment can conveniently be modelled by a thermal bath of harmonic oscillators. An archetype situation provides a two-state dissipative quantum dynamics, commonly known under the label of a spin-boson dynamics. An interesting and nontrivial physical situation emerges, however, when the quantum dynamics evolves far away from thermal equilibrium. This occurs, for example, when a charge transferring medium possesses nonequilibrium degrees of freedom, or when a strong time-dependent control field is applied externally. Accordingly, certain parameters of underlying quantum subsystem acquire stochastic character. Herein, we review the general theoretical framework which is based on the method of projector operators, yielding the quantum master equations for systems that are exposed to strong external fields. This allows one to investigate on a common basis the influence of nonequilibrium fluctuations and periodic electrical fields on quantum transport processes. Most importantly, such strong fluctuating fields induce a whole variety of nonlinear and nonequilibrium phenomena. A characteristic feature of such dynamics is the absence of thermal (quantum) detailed balance.