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.
A new cluster development for the logarithm of the grand partition function of a system of interacting particles is derived. The leading term in this expansion is the pressure exerted by an ideal Bose or Fermi gas at the same temperature and absolute activity
Z as the actual system. Succeeding terms involve quantum cluster integrals which themselves depend upon
Z, unlike their classical analogs. The definition of these cluster integrals follows in a natural fashion using techniques illustrated by construction (in closed form) of the successive
Z derivatives of the Bose and Fermi ideal gas grand partition functions. It is not possible (except in the classical limit) to eliminate
Z explicitly between the pressure and density series, so that the equation of state must remain in parametric form.
A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and DeDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble for the case of a nondegenerate gas. In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through short-range forces.
For a system of point charges that interact through the three-dimensional electrostatic Coulomb potential (without any regularization) and obey the laws of nonrelativistic quantum mechanics with Bose or Fermi statistics, the static correlations are expressed in terms of Mayer-like diagrams. The exchange effects are taken into account systematically and the long-range Coulomb divergencies are exactly resummed in order to get finite diagrams. For this purpose, in the framework of the grand canonical ensemble, the matrix elements of the imaginary-time evolution operator are represented by the Feynman-Kac functional integral according to Ginibre's idea [J. Math. Phys. 6, 238 (1965); 6, 252 (1965), 6, 1432 (1965)] and we exhibit a correspondence between the correlations in the quantum system of point particles and the distribution functions in a classical fluid of "exchange" loops with Brownian random shapes. The size of a loop, which corresponds to the number of particles involved in the corresponding cyclic permutation, and the shape of a loop, which describes the quantum fluctuations, play the part of internal degrees of freedom that must be integrated over when calculating the distributions relative to the quantum point charges. The loops interact through a two-body potential that is different from the electrostatic interaction between two charged curves: each line element of a loop interacts only with a discrete number of line elements in every other loop. The linear response theory to an infinitesimal external charge distribution can be written in this formalism by a formula analogous to that of classical statistics, and the loop-fugacity and loop-density expansions of the loop-distribution functions are derived by the usual techniques of Mayer diagrams generalized to extended objects. At large distances
r, the loop potential behaves as
r1 and, as in the classical case, every Mayer diagram diverges. However, this nonintegrable tail is independent of the shape of the loops and it is explicitly and exactly resummed by a generalization of the method developed by Meeron [J. Chem. Phys. 28, 630 (1958); Plasma Physics (McGraw-Hill, New York, 1961)], for classical fluids of point entities. Auxiliary
r1 bonds are introduced and subdiagrams involving chains of
r1 bonds are integrated over first in a systematic way. The new diagrams contain bonds between loops that decay either exponentially or algebraically, with a
r31 leading term, and the new diagrams are at least conditionally integrable. The part of the quantum particle-particle correlation arising directly from exchange, which is derived from the density of loop, decays faster than any inverse power law, whereas, as shown in the following paper [Phys. Rev. E 53, 4595 (1996)], the whole quantum particle-particle correlation, which also involves the correlations between loops, decays only as
r61.
The quantum principles that establish the energy levels and degeneracies needed to evaluate the partition functions are explored. The uncertainty principle is associated with the dual wave-particle nature of the model used to describe quantized gas particles. The Schroedinger wave equation is presented as a generalization of Maxwell's wave equation; the former applies to all particles while the Maxwell equation applies to the special case of photon particles. The size of the quantum cell in phase space and the representation of momentum as a space derivative operator follow from the uncertainty principle. A consequence of this is that steady-state problems that are space-time dependent for the classical model become only space dependent for the quantum model and are often easier to solve. The partition function is derived for quantized free particles and, at normal conditions, the result is the same as that given by the classical phase integral. The quantum corrections that occur at very low temperatures or high densities are derived. These corrections for the Einstein-Bose gas qualitatively describe the condensation effects that occur in liquid helium, but are unimportant for most practical purposes otherwise. However, the corrections for the Fermi-Dirac gas are important because they quantitatively describe the behavior of high-density conduction electron gases in metals and explain the zero point energy and low specific heat exhibited in this case.
By means of a quantum mechanical phase space distribution function
introduced by von Roos, the Schroedinger equation for a non-relativistic
system of N identical particles with scalar interactions is transformed
into a quantum mechanical generalization of the Liouville equation,
thereby formulating the problem in terms of a generalized density in
phase space, a quantity of primary interest in most treatments of the
corresponding classical system (or plasma). This transformation permits
a parallel development of the theories of classical and quantum plasmas
and thus allows the quantum many-body problem to be discussed virtually
completely in classical terms. In particular, a kinetic theory of
quantum plasmas is obtained by deriving the quantum analogue of the BBGKY
hierarchy, and applying thereto approximation techniques similar to those
of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from
similar previous studies based on the Wigner distribution function is that
the proper exchange symmetry can be tractably introduced into the formalism. Attention is first focused on the Hartree and Hartree-Fock approximations,
in which case the quantum BBGKY system reduces to a simple quantum
generalization of the Vlasov equation. This equation is used to study
the response of spatially homogeneous systems to weak external forces,
and the associated problems of plasmon and spin-wave excitations. It is
also used to derive the quantum and exchange corrected equations of
inviscid hydrodynamical transport which are then applied to the problem
of sound propagation in the degenerate electron gas. The second part of the study is concerned with the theory of the
many-electron atom in the Hartree and Hartree-Fock approximations.
The relevant quantum Vlasov equations lead naturally to a statistical
theory of the atom which reduces to the Thomas-Fermi-Amaldi and Thomas Fermi
models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and
exchange corrections to these models are simultaneously generated. The
quantum hydrodynamical theory developed earlier is used to determine
the influence of these corrections on the boundary conditions of the
model, and a theory of the compressed atom is consequently obtained.
Considered in somewhat less detail are the effects of non-zero temperature,
net orbital angular momentum, relativity and correlations, as well as
time dependent processes. The final part deals with the problem of the degenerate electron
gas with a uniform neutralizing background. Going beyond the Hartree-Fock approximation, the pair functions for particles with
parallel and anti-parallel spin are obtained by neglecting three particle
correlations. From these functions, a quantum-mechanical collision
integral is derived which differs from that obtained by Silin
and Guernsey and conjectured by Wyld and Pines in that dynamical exchange
effects are included. Also obtained from the pair function
is an expression for the correlation which reduces in the high
density limit to the result of Gell-Mann and Brueckner. At intermediate
densities an additional term appears in the energy due to the screening
of the exchange interaction by the dielectric properties of the medium.
It is evaluated in the high density limit and found to be -0.151 r_s ln r_s
Ryd/electron in marked disagreement with the corresponding value obtained
by DuBois.