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The Journal of Chemical Physics

The supersymmetric diffusional influence functional: General theory

John D. Morgan, Peter Guy Wolynes

Published Jun 15, 1988• 0 citations
Publisher

Abstract

The diffusional influence functional is derived for a general heatbath, using the supersymmetric form of the diffusional path integral. This form has an action which is linear in the potential for the system, rather than quadratic as in the usual diffusional path integral, which allows integration over the heatbath degrees of freedom to form an influence functional. The derivation of the supersymmetric form is reviewed to make the paper more nearly self-contained. The connection between the influence functional and the statistical properties of the heatbath is examined, and some possible applications of the formalism are discussed.

DOI
10.1063/1.454291

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The diffusional influence functional is derived for a general heatbath, using the supersymmetric form of the diffusional path integral. This form has an action which is linear in the potential for the system, rather than quadratic as in the usual diffusional path integral, which allows integration over the heatbath degrees of freedom to form an influence functional. The derivation of the supersymmetric form is reviewed to make the paper more nearly self-contained. The connection between the influence functional and the statistical properties of the heatbath is examined, and some possible applications of the formalism are discussed.
This chapter demonstrates the equivalence between Feynman–Vernon influence functional approach and Keldysh diagram technique developed to describe non-equilibrium phenomena in quantum many-body systems with interactions. Path integrals appearing in both quantum mechanics and quantum field theory may be viewed as a formal representation of the exact infinite series of Feynman diagrams. In other words, path integral and diagram techniques are formally equivalent to each other. The chapter discusses a single degree of freedom, introducing interactions with a bath of harmonic oscillators. In a way, to employ the Caldeira–Leggett model of a bath of harmonic oscillators can be considered just as a physical trick helping to derive the influence functional in the case of linear Ohmic damping. Gaussian influence functionals are exact provided the environment consists of an arbitrary number of harmonic oscillators coupled linearly to the system of interest q .
The recently developed (energy) sudden representation of the Schrödinger propagator is examined in conjunction with the path integral method. The perturbation expansion of the propagator in this representation (with the sudden approximation as the zeroth order term) is easily understood using Feynman diagrams. The zeroth order term is a form involving only sudden paths (i.e., paths resulting from the sudden approximation). Each higher order term is a series of sudden paths which are connected to one another due to the nonsudden effects of the dynamics. The sudden representation, and in particular its perturbation expansion, is also used to study the bath effects in many body systems which are expressed in the form of influence functionals in the path integral formalism. In our treatment, the bath coordinates are assumed to be the slow coordinates. The zeroth order (sudden approximation) influence functionals are expressed in a very simple form; they involve only an ordinary integral over the bath coordinates. Finally, to illustrate the approach, a harmonic oscillator coupled linearly to an unspecified primary system is used as a simple model bath. The reduction of the exact influence functional to the sudden approximation form is also considered.
Feynman path integrals offer a convenient and elegant tool for dealing with the statistical physics of quantum-mechanical systems. By using the Trotter product formula, one can evaluate directly the propagation of many-body quantum systems in imaginary time. The idea of an influence functional arises when one wishes to consider division of the physical system into a primary system and a bath. We present a general approach to the problem of constructing influence functionals, one that is capable of dealing both with Boltzmann baths (classical oscillators) and quantum baths of bosons or fermions. The fermion bath of special interest is generally a submanifold of the electronic state such as the bridge structure connecting chromophores or electron localization subunits which are common in problems of mixed valency and superexchange. The fermion bath is treated by a general rewriting of the influence phase S as a sum of an eigenvalue part SEV{S}_{\mathrm{EV}}SEV​ and a remainder, S0{S}^{0}S0, that describes transitions. Reduction of the original Hamiltonian leads to a reduced Hamiltonian with effective off-diagonal matrix elements and influence functionals which can include memory effects. We present a general formalism for construction of influence functionals, and discuss particular applications to systems of interest, especially in electronic-structure problems. The following paper presents a variational technique for finding effective Hamiltonians.
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.
In this paper the so-called Feynman-Vernon influence functional theory has been generalized in order to include the possibility of dealing with initial conditions of the system plus environment other than the factorizable one. It has been shown that the new influence functional can now be written in terms of paths which represent the coupling of quantum-statistical to quantum-dynamical effects. Once the thermal paths are integrated out of this new influence functional, a generalization of the original Feynman-Vernon expression is obtained.
We establish the path integral approach for the time-dependent heat exchange of an externally driven quantum system coupled to a thermal reservoir. We derive the relevant influence functional and present an exact formal expression for the moment generating functional which carries all statistical properties of the heat exchange process for general linear dissipation. The method is applied to the time-dependent average heat transfer in the dissipative two-state system (TSS). We show that the heat can be written as a convolution integral which involves the population and coherence correlation functions of the TSS and additional correlations due to a polarization of the reservoir. The corresponding expression can be solved in the weak-damping limit both for white noise and for quantum mechanical coloured noise. The implications of pure quantum effects are discussed. Altogether a complete description of the dynamics of the average heat transfer ranging from the classical regime down to zero temperature is achieved.