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Journal of Mathematical Physics

A Perturbation Method for the Classical Time-Dependent Pair Correlation Function

Raphael Aronson

Published Apr 1, 1966• 2 citations
Publisher

Abstract

A perturbation technique useful for computing many-time thermal averages of classical quantities is developed. The canonical distribution function for the system is shown to evolve isothermally from that for a free-particle system as the interaction is switched on slowly. This permits convenient use of an interaction picture in which to perform thermal averaging. The technique is applied to the calculation of the time-dependent pair correlation function in position for a uniform gas. The correlation function is shown to be a sum of two components, one the solution of a kinetic equation and essentially a generalization of the autocorrelation in equilibrium, the other a generalization of the mutual correlation function in equilibrium. The equations arise as sums over diagrams. The equations resulting from the random phase approximation, valid for the short-time behavior, are solved exactly. It is shown directly that the generalized dielectric function given in terms of the correlation function is identical with that found by solution of the kinetic equation in the random phase approximation for all frequencies.

DOI
10.1063/1.1704970

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Cambridge University Press eBooks•Aug 27, 2015•0 citations
A perturbation technique useful for computing many-time thermal averages of classical quantities is developed. The canonical distribution function for the system is shown to evolve isothermally from that for a free-particle system as the interaction is switched on slowly. This permits convenient use of an interaction picture in which to perform thermal averaging. The technique is applied to the calculation of the time-dependent pair correlation function in position for a uniform gas. The correlation function is shown to be a sum of two components, one the solution of a kinetic equation and essentially a generalization of the autocorrelation in equilibrium, the other a generalization of the mutual correlation function in equilibrium. The equations arise as sums over diagrams. The equations resulting from the random phase approximation, valid for the short-time behavior, are solved exactly. It is shown directly that the generalized dielectric function given in terms of the correlation function is identical with that found by solution of the kinetic equation in the random phase approximation for all frequencies.
The pair distribution function (PDF) is a key quantity for the analysis of correlation effects of a quantum system both in equilibrium and far from equilibrium. We derive an expression for the PDF in terms of the single-particle Green functions—the solutions of the Keldysh/Kadanoff-Baym equations in the two-time plane—for a one- or two-component system. The result includes initial correlations and generalizes previous density matrix expressions from single-time quantum kinetic theory. Explicit expressions for the PDF are obtained in second Born approximation.
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.
Formal relations between the free energy, the "dielectric constant" which expresses the response of the system to an external perturbation, and the two-particle Green's function in temperature space are derived. Connection with perturbation expansion for free energy and for general admittance tensors are established. These results are applied to a general discussion of the random-phase approximation at finite temperature and it is shown that the sum of ring diagrams corresponds to the calculation of the dielectric constant in the approximation of the neglect of local field corrections.
As independent variables in the statistical description of processes in systems of interacting particles, use is made of particles in various points of phase space of coordinates and momenta, which for any point of phase space are random functions of time. For these functions there are given the classical and quantum equations, from which, by averaging over their distribution, functions chains of equations are obtained for classical and quantum distribution functions. To illustrate the possible use of this method expressions are obtained for excitation spectrum and the correlation function for a system of particles with central interaction for the case when close interactions are not essential. This method is convenient for considering systems of particles in electromagnetic interaction. This method may be generalized for the relativistic case. (trauth)
A general method which permits the derivation of the equations which describe the approach to equilibrium correct to an arbitrary finite order in the coupling constant is presented. This method is applied in the present paper to normal modes interacting through three-phonon processes. In a subsequent paper the method will be applied to interacting particles. The distribution function is first Fourier-analyzed with respect to the angle variables. All Fourier components, except the distribution function of action variables, describe correlations among the normal modes. The formal solution for the Fourier components is studied in the limiting case of a very large number of degrees of freedom N → ∞, and for large times by means of a diagram technique. Each component ρ3n can be split into 2 parts: ρ3n′ and ρ3n″; one (ρ′) due to ``scattering'' of the normal modes satisfies diagonal differential equations. The other (ρ3n″) contains the direct interaction between the normal modes involved in the corresponding correlation. It is completely determined by the functions ρ3n′. The study of this set of equations enables us to study the approach to equilibrium.
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.