A number of years ago, Feynman and Hibbs derived an insightful expression for a quasiclassical effective potential in order to estimate the free energy for a quantum-mechanical system that obeys Boltzmann statistics. In the present paper, it is demonstrated how these ideas can be extended to calculate an approximate equilibrium average of any general coordinate-dependent operator. The central feature of this analysis is the replacement of the operator by an effective Gaussian-averaged quasiclassical function.
A number of years ago, Feynman and Hibbs derived an insightful expression for a quasiclassical effective potential in order to estimate the free energy for a quantum-mechanical system that obeys Boltzmann statistics. In the present paper, it is demonstrated how these ideas can be extended to calculate an approximate equilibrium average of any general coordinate-dependent operator. The central feature of this analysis is the replacement of the operator by an effective Gaussian-averaged quasiclassical function.
We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs, but perform the mean-field treatment of quantum fluctuations about the path starting point rather than the path centroid. Furthermore, rather than performing a full variational optimization of the potential, we explore approximate functional forms that yield a numerical robustness. The resulting closed-form potential is exact in the classical and harmonic limits; benchmarks against exact position distributions for one-dimensional quartic, Morse, and double-well potentials, show good agreement for potentials with harmonic support.
We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs but perform the mean-field treatment of quantum fluctuations about the path starting point rather than the path centroid. Furthermore, rather than performing a full variational optimization of the potential, we explore approximate functional forms that yield numerical robustness. The resulting closed-form potential is exact in the classical and harmonic limits; benchmarks against exact position distributions for one-dimensional quartic, Morse, and double-well potentials show good agreement for potentials with harmonic support.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
A formulation of quantum statistical mechanics is discussed in which the Feynman path centroid density in Feynman path integration is recast as the central statistical distribution used to average equilibrium and dynamical quantities. In this formulation, the path integral centroid density occupies the same role as the Boltzmann density in classical statistical mechanics. Therefore, the statistical ensemble of imaginary time path centroid configurations provides the distribution which is used to average the appropriately formulated effective operators and imaginary time correlation functions. An accurate renormalized diagrammatic perturbation theory for the centroid density and centroid-constrained imaginary time propagator will also be described with particular emphasis given to the mathematical advantages arising from the centroid-based formulation. The present paper is concerned with the calculation of equilibrium properties from the centroid perspective, while the companion paper describes a centroid-based formalism for calculating dynamical time correlation functions.
We consider non-equilibrium open statistical systems, subject to potentials and to external “heat baths” (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann’s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomialsHn’s). The moments of non-equilibrium classical distributions, implied by the Hn’s, fulfill a hierarchy: for long times, the lowest moment dominates the evolution towards thermal equilibrium, either with dissipation or without it (but under certain approximation). We revisit that hierarchy, whose solution depends on operator continued fractions. We review our generalization of that moment method to classical closed many-particle interacting systems with neither a hb nor ab initio dissipation: with initial states describing thermal equilibrium at T at large distances but non-equilibrium at finite distances, the moment method yields, approximately, irreversible thermalization of the whole system at T, for long times. Generalizations to non-equilibrium quantum interacting systems meet additional difficulties. Three of them are: (i) equilibrium distributions (represented through Wigner functions) are neither Gaussian in momenta nor known in closed form; (ii) they may depend on dissipation; and (iii) the orthogonal polynomials in momenta generated by them depend also on positions. We generalize the moment method, dealing with (i), (ii) and (iii), to some non-equilibrium one-particle quantum interacting systems. Open problems are discussed briefly.
We consider quantum nonlinear many-body systems with dissipation described within the Caldeira-Leggett model, i.e., by a nonlocal action in the path integral for the density matrix. Approximate classical-like formulas for thermodynamic quantities are derived for the case of many degrees of freedom, with general kinetic and dissipative quadratic forms. The underlying scheme is the pure-quantum self-consistent harmonic approximation (PQSCHA), equivalent to the variational approach by the Feynman-Jensen inequality with a suitable quadratic nonlocal trial action. A low-coupling approximation permits us to get manageable PQSCHA expressions for quantum thermal averages with a classical Boltzmann factor involving an effective potential and an inner Gaussian average that describes the fluctuations originating from the interplay of quanticity and dissipation. The application of the PQSCHA to a quantum phi(4) chain with Drude-like dissipation shows nontrivial effects of dissipation, depending upon its strength and bandwidth.