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Journal of Guangxi Normal University

Dissipation Effect of Quantum Fluctuation on Envelope Soliton in Fermi-Pasta-Ulam Model

Dan Liang

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

In this paper,the time-dependent variational principle(TDVP) is applied to study the one-dimensional Fermi-Pasta-Ulam(FPU) model,and the dynamic equation of particles in the chain is obtained by TDVP.In the semi-quantal study,the considered quantum effect is quantum fluctuation.The Jackiw-Kerman wave function is used for the single particle.The research results show that quantum fluctuation can disperse,blur and even destroy the nonlinear excitation of FPU model,envelope soliton.The effect of quantum fluctuation has the quasi-exponentional relationship with the effective Plank constant.Furthermore,it is found that coupling the effective Plank constant the anharmonic coupling strength has the joint effect on quantum dissipation.The uniquely joint effect may be induced by the unique Hamiltonian of the FPU chain.

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In this paper,the time-dependent variational principle(TDVP) is applied to study the one-dimensional Fermi-Pasta-Ulam(FPU) model,and the dynamic equation of particles in the chain is obtained by TDVP.In the semi-quantal study,the considered quantum effect is quantum fluctuation.The Jackiw-Kerman wave function is used for the single particle.The research results show that quantum fluctuation can disperse,blur and even destroy the nonlinear excitation of FPU model,envelope soliton.The effect of quantum fluctuation has the quasi-exponentional relationship with the effective Plank constant.Furthermore,it is found that coupling the effective Plank constant the anharmonic coupling strength has the joint effect on quantum dissipation.The uniquely joint effect may be induced by the unique Hamiltonian of the FPU chain.
We investigate the mechanism of heat conduction in one-dimensional (1D) quantum FPU chain with quantum fluctuation in the framework of semiquantal molecular dynamics. In the semiquantal many-body simulations, a Hartree-type many-body wave function is adopted for a whole chain and a single-particle state of a particle is represented by a trial wave function with Jackiw–Kerman (JK) form. With the help of the Dirac's time-dependent variational principle (TDVP), a set of equations of the variational parameters contained in the JK wave packet is obtained, and it describes the quantum dynamics of the nonlinear lattices approximately. These equations not only prove highly efficient in recovering dynamics of classical heat conduction, but also allow exploring the case with quantum mechanical characteristics. As a consequence, we find the thermal conductivity diverges with system size as well as in the classical version when considering quantum fluctuation. Furthermore, in three different temperature regions it is observed that the enhancement of the quantum fluctuation increases the heat flux through the 1D quantum FPU chain.
We propose an experimental scheme for studying the emergence of effectively classical nonlinear revivals—the Fermi–Pasta–Ulam (FPU) phenomenon—in interacting quantum many-body systems, e.g. ultracold atoms. Specifically, we suggest and analyze a setup of one-dimensional Bose gases confined into an optical lattice. The strength of quantum fluctuations is controlled by tuning the number of atoms per lattice sites (filling factor). By simulating the real-time dynamics of the Bose–Hubbard model by means of the exact numerical method of time-evolving block decimation, we investigate the effects of quantum fluctuations on the FPU recurrence and show that strong quantum fluctuations cause significant damping of the FPU oscillation.
The phonon dispersion relation of the quantum FPU model is studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction.The single-particle state is taken to be a frozen Jackiw-Kerman wavefunction.Under the condition of minimum uncertainty relation,equations of motion for the particle expectation values are derived to obtain the phonon dispersion relation.It is shown that the harmonically coupling strength is strengthened and the width of the phonon band is broadened due to the quantum fluctuations in comparison with those of the classical FPU model.In addition,the harmonically coupling strength and the width of the phonon band increase with increasing the quantum fluctuations and quartic coupling strength.
We study the statistics and short-time dynamics of the classical and the quantum Fermi-Pasta-Ulam chain in the thermal equilibrium. We analyze the distributions of single-particle configurations by integrating out the rest of the system. At low temperatures, we observe a systematic increase in the mobility of the chain when transitioning from classical to quantum mechanics due to zero-point energy effects. We analyze the consequences of quantum dispersion on the dynamics at short times of configurational correlation functions.
We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact, our scheme is more general and would also cover other one-dimensional Hamiltonian systems, for example, classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode, it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solution has been found so far, and we rely on a one-loop approximation. The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving 3×3 correlator matrix of the locally conserved fields. Long time asymptotics is computed analytically, and finite time properties are obtained through a numerical simulation of the mode-coupling equations.