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arXiv:hep-ph/9509302

The dynamics of a thermal non-equilibrium anharmonic oscillator

Herbert Nachbagauer

Published Sep 15, 1995• 0 citations
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Abstract

We propose an non-standard method to calculate non-equilibrium physical observables. Considering the generic example of an anharmonic quantum oscillator, we take advantage of the fact that the commutator algebra of second order polynomials in creation/annihilation operators closes. We solve the von~Neumann equation for the density-operator exactly in the mean field approximation and study the time evolution of the particle number and the expectation value < X^2 > .

DOI
10.48550/arxiv.hep-ph/9509302

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We propose an non-standard method to calculate non-equilibrium physical observables. Considering the generic example of an anharmonic quantum oscillator, we take advantage of the fact that the commutator algebra of second order polynomials in creation/annihilation operators closes. We solve the von~Neumann equation for the density-operator exactly in the mean field approximation and study the time evolution of the particle number and the expectation value < X^2 > .
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.
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We present an approximation of a non-Markovian solution for the time evolution of the mean number of quanta in an anharmonic oscillator linearly and stochastically coupled to a thermal bath. A colored Gaussian bath is explicitly considered, and a quantum treatment of the bath correlation functions, based on the use of a relation between their real and imaginary parts, is applied. The behavior of the oscillator is characterized by a reduced set of (five) parameters referring to the oscillator, the bath, and their mutual interaction. A comparison is made between the prediction of the approach for the mean stationary number of quanta and its exact value, which enables us to analyze its accuracy. When the Markovian limit holds, an explicit expression for a parameter measuring the time scale for the relaxation of the oscillator is obtained in terms of the characteristic parameters of the model. Except for very high temperatures, the anharmonicity has a small influence on this relaxation time. A numerical study of the time dependence of the mean number of quanta is presented on the basis of the reported analytical solution. We find that the anharmonicity is not a significant parameter in the appearance nor in the increase of non-Markovian effects.
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Out of thermal equilibrium state, the vacuum is unstable and evolves in time.Consequently, the annihilation operators associated with the unstable vacuum depend on time.This dissipative time-evolution of quantum systems can be systematically treated, within the canonical operator formalism referred to as non-equilibrium thermo-field dynamics.Given is an alternative route to derive the time-dependent annihilation operators within the formalism.As an example, time-dependent annihilation operators for the systems of bosonic and fermionic semi-free fields are derived.