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Progress of Theoretical Physics

Quantum Statistical Mechanics of a Many-Body System with Square-Well Interaction

Raj K. Pathria, M. P. Kawatra

Published Apr 1, 1962• 8 citations
Publisher

Abstract

Following the binary collision approach to the many-body problem in Quantum Statistical Mechanics, as developed recently by Lee and Yang, the equation of state, to the order a2/λ2, is derived for a system of particles interacting through a square-well potential and obeying Bose-Einstein or Fermi-Dirac statistics. Here, λ is the mean thermal wavelength of the particles and a represents the range of the two-body interaction. Expressions are obtained for the second and the third virial coefficients of the system and also for its various thermodynamical properties, such as energy, specific heat and entropy, at finite but low temperatures.

DOI
10.1143/ptp.27.638

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Physical Review•Mar 1, 1959•210 citations
Following the binary collision approach to the many-body problem in Quantum Statistical Mechanics, as developed recently by Lee and Yang, the equation of state, to the order a2/λ2, is derived for a system of particles interacting through a square-well potential and obeying Bose-Einstein or Fermi-Dirac statistics. Here, λ is the mean thermal wavelength of the particles and a represents the range of the two-body interaction. Expressions are obtained for the second and the third virial coefficients of the system and also for its various thermodynamical properties, such as energy, specific heat and entropy, at finite but low temperatures.
The binary collision method of Lee and Yang is applied to investigate the equilibrium properties of a many-body system of particles interacting through a generalized two-body potential. It is found that, to the order a2/λ2 where a and λ represent respectively the range of the two-body interaction and the mean thermal wavelength of the particles, the final results can be written down explicitly in terms of the relevant phase shift for the S-waves. Ground state properties of a few systems of particles interacting through certain typical potentials and obeying Fermi-Dirac or Bose-Einstein statistics are studied.
The binary collision method of Lee and Yang is extended in order to investigate the statistical mechanics of a multicomponent mixture of interacting gases obeying Bose-Einstein or Fermi-Dirac statistics. First, the case of a mixture of two kinds of hard-sphere spinless bosons is studied and then the generalization is carried over to the case of a mixture of several components consisting of particles with arbitrary spin and statistics. An explicit derivation of the fugacity expansion, correct to the second order in the interaction parameters, is given. Expressions are derived for the second and third virial coefficients of the system and a discussion is given of the salient features of the results obtained. Finally, the case of a binary mixture, with one component consisting of bosons and the other of fermions, is considered and the phenomenon of Bose-Einstein condensation in this system is studied.
A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation ("two-body approximation") corresponds to the exact resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applications of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at Tc [ALMOST EQUAL TO] 2[pi]n/[m log(2[pi]/mg )], where n is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of [MICRO SIGN]/T ) and computed from the numerical solutions. For bosons in three spatial dimensions, we present evidence that the gas undergoes 3 a strongly interacting version of Bose-Einstein condensation at n[lamda]T [ALMOST EQUAL TO] 1.3, where n is the number density and [lamda]T is the thermal wavelength. Finally, we look at the ratio of shear viscosity to entropy density [eta]/s of the unitary quantum gas, which has a conjectured lower bound of [eta]/s [GREATER-THAN OR EQUAL TO] /4[pi]kb based on the AdS/CFT correspondence of a strongly coupled Yang-Mills theory.
Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.
We develop a statistical mean-field theory of finite quantum systems in thermal equilibrium. Our formulation employs the canonical ensemble of statistical mechanics, and it enables us to analytically determine the occupation number distributions of interacting particles obeying Bose–Einstein or Fermi–Dirac statistics. We have also developed a numerical procedure that enables us to obtain a universal scaled occupation number distribution that, for a given total number of interacting particles in a finite system, makes it possible to determine the occupation number distribution for any temperature. The developed mean-field theory is applicable to a wide range of atomic, nuclear and condensed matter systems for which finite-size effects can play an important role. In particular, the present approach makes it possible to formulate a finite temperature mean-field theory for a specific ion in a dense plasma.
A formulation is given whereby the grand partition function of a many-body system satisfying Bose-Einstein or Fermi-Dirac statistics is expressed in terms of certain UUU functions defined for the same system with Boltzmann statistics. It is then shown that these UUU functions can be evaluated in successive approximations in terms of a binary kernel BBB which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail.