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arXiv:nucl-th/0406025

Damped collective motion of isolated many body systems within a variational approach to functional integrals

Christian Rummel, Helmut Hofmann

Published Jun 9, 2004• 0 citations
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Abstract

Two improvements with respect to previous formulations are presented for the calculation of the partition function $\mathcal{Z}$ of small, isolated and interacting many body systems. By including anharmonicities and employing a variational approach quantum effects can be treated even at very low temperatures. A method is proposed of how to include collisional damping. Finally, our approach is applied to the calculation of the decay rate of metastable systems.

DOI
10.48550/arxiv.nucl-th/0406025

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Two improvements with respect to previous formulations are presented for the calculation of the partition function Z\mathcal{Z}Z of small, isolated and interacting many body systems. By including anharmonicities and employing a variational approach quantum effects can be treated even at very low temperatures. A method is proposed of how to include collisional damping. Finally, our approach is applied to the calculation of the decay rate of metastable systems.
We evaluate the exponential suppression of the quantum tunneling rate due to the coupling with a thermal reservoir. The exponential dependence is calculated by a variational method, in which an approximate saddle point of the effective action is found. The effect of finite temperatures is accounted for. We calculate the exponential part of the rate for several potentials, some of which have not been considered previously. The method gives excellent agreement with numerically calculated values, where they exist. We also consider the effect of Ohmic damping mechanisms of Drude form (memory damping). This latter case may be of importance for experiments on rf superconducting quantum interference devices and Josephson junctions where frequency-dependent damping mechanisms have been observed. The trends exhibited by the variational approximation can be understood qualitatively on the basis of physical arguments.
Abstract This chapter presents a short review of path integrals in quantum mechanics. This is followed by a detailed discussion of the evaluation of the semi-classical propagator, involving quadratic expansions of the action and the calculation of determinants. The concept of functionals is introduced for multi-dimensional cases, using a system linearly coupled to many oscillators as an example. The use of path integrals in statistical mechanics is described and quantum corrections to the classical limit are obtained. The connection with Green functions and level density is presented, with an excursion to periodic orbit theory and to the Gutzwiller formula for the oscillating part of the single particle level density. Many-body systems with separable interactions are treated by the Hubbard-Stratonovich transformation. For quantum corrections to the partition function, the static- and perturbed static path approximations are developed. The breakdown for unstable modes is discussed and a possible extension to damped motion is sketched.
We consider the decay of a metastable state, coupled to a thermal reservoir, by quantum tunnelling. We calculate the leading exponential term in the W.K.B. expansion of the decay rate through a variational approximation. The approach is simple, quick and versatile, when compared with the alternative method which involves the numerical solution of non-linear, non-local, integro-differential equations. We have demonstrated the versatility of the variational approach by applying it to a few types of potentials, most of which have not been previously studied. In the few cases where numerical results are available, the variational approximation gives excellent agreement. In the cases where no numerical results are available for comparison, the variational approximation reproduces trends that are either physically reasonable or that may be inferred by interpolation between exactly soluble limits. We also address the question: What are the effects of frequency dependent friction, or a finite noise correlation time, on the quantum tunnellling rate? This question is one that has been recently broached in the discussion of experiments on highly damped r.f. SQUIDs and Josephson junctions. We find that the effect of the memory time is to reduce the lifetime of the metastable state, towards its undamped value. Again, we find that the variational approximation reproduces the exact variation of the action to a surprisingly good degree of accuracy.
It is shown that, by means of a parametric representation, a variational calculation for a system of Bose particles can be carried through using explicit trial wave functions which describe a state with a fixed number of particles and include pair correlations. For the ground state the results of previous work are obtained as the first term in a steepest descent calculation. An effective Schrodinger equation is obtained for the relative motion of two particles. The application of the method to excited states is discussed; in particular for the case of a vortex line in the system it is shown that the method leads to an analysis which is similar to that for the ground state and could be carried through in the same way.