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Progress of Theoretical Physics

Phase Transition in Bose Liquid. II

Shokichi Kanno

Published Feb 1, 1973• 1 citation
Publisher

Abstract

We apply the method proposed in a previous paper to the calculation of thermodynamical functions. This model resembles the Landau Fermi liquid theory in the sense that the energy, excitation energy, etc., are expressed by the functionals of occupation numbers of quasi-particles. It is shown that if the excitation energy of a quasi-particle εp is proportional to pr with 2>r>3/2, then the specific heat cp behaves as log (Tc/|T-Tc|). We discuss also the thermal behaviour of condensation density. It is shown that the fluctuation of condensation density becomes infinite on the transition line.

DOI
10.1143/ptp.49.401

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We apply the method proposed in a previous paper to the calculation of thermodynamical functions. This model resembles the Landau Fermi liquid theory in the sense that the energy, excitation energy, etc., are expressed by the functionals of occupation numbers of quasi-particles. It is shown that if the excitation energy of a quasi-particle εp is proportional to pr with 2>r>3/2, then the specific heat cp behaves as log (Tc/|T-Tc|). We discuss also the thermal behaviour of condensation density. It is shown that the fluctuation of condensation density becomes infinite on the transition line.
We show that the Bose liquid can be worked out in a way similar to the Landau Fermi liquid theory by decomposing the diagonal elements of the Boltzmann factor for the system into irreducible diagrams. The energy of the system is expressed by the functional of the occupation number np. The excitation energy is defined by the variation of the energy with respect to np. The Landau f-functions (the second variations of the energy) are closely related to the excitation energy and play important roles in the phase transition. The excitation-energy dependence of Landau f-functions are obtained for the dilute gas.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
Low-lying excitations in a liquid or a solid are considered to have a long lifetime and to obey the Bose or Fermi statistics. They will be called "quasiparticles" in general. The low-lying energy levels of the system are considered to be characterized by a set of occupation numbers of the quasiparticles. The expressions for the grand partition function and the distribution function (the average occupation number) of the quasiparticles have the same structure as that for the classical lattice gas, under a nonuniform external field and with two-, three-,..., body interactions. Therefore the virial expansion formula for the classical lattice gas can be applied to the system of quasiparticles. The result is simplified by assuming that the energy correction due to the simultaneous excitations of two, three,..., nnn quasiparticles are of $O({V}^{\ensuremath{-}1}),O({V}^{\ensuremath{-}2}),\ensuremath{\cdots},O({V}^{\ensuremath{-}n+1})$, respectively, where VVV is the volume of the system. The resulting expression for the distribution function of the quasiparticles is either a Fermi or a Bose distribution function with an effective energy. The expression for the entropy in terms of the distribution function is the same as that for an ideal gas. This shows that Landau's formula for the distribution function of quasiparticles for a Fermi liquid is valid under conditions which are more general and more practical than those given by Landau. Finally the situation in the Husimi-Temperley model is discussed in comparison with that in the Ising model from the point of view of the present formulation.
We obtain thermodynamical functions for a dilute Bose gas of hard spheres by using the efffective Hamiltonian which is obtained by a quasi-linear transformation proposed previously and may be recognized as an improvement of the model Hamiltonian of Lee and Yang which is not correct about the terms of second order in the diameter of hard spheres when it is applied to the problem of the phase transition. The Bose condensation occurs at a temperature below which the degenerate phase of the quasi-particles appears. We evaluate the amount of the gap of the specific heat and that of the compressibility at the transition point.
Bose system of zero spin particles is considered in the presence of the Bose–Einstein condensate in the vicinity of the phase transition point. The system is investigated in the framework of the Bogolyubov model with the separated condensate. In this model an effective Hamiltonian of the system is introduced by replacing condensate creation and annihilation operators in system Hamiltonian by n01/2 where n0 is occupation number of the condensate state. According to Bogolyubov, the grand canonical thermodynamic potential related to the effective Hamiltonian is considered as nonequilibrium thermodynamic potential. In the present paper this potential is investigated as a function of the small variable n0. With the help of the thermodynamic perturbation theory it is shown that it is expanded in a series over integer powers of n0. This corresponds to the basic idea of the Landau theory of the phase transitions of the second kind. Coefficients at terms of the first and second orders in n0 in the expansion are calculated for Bose gas in the main approximation in small interaction. Calculation of the coefficients at terms of the third and fourth orders needs accounting contributions of the thermodynamic perturbation theory at least of the 4th order and will be done elsewhere. It is established that the results obtained for Bose gas do not fit into the Landau theory of phase transitions of the second kind. Some comments that discuss the situation are given.
In Landau's phenomenological Fermi-liquid theory (FLT), most physical quantities are derived from the knowledge of the energy variation δE[δn] corresponding to a change δn of the quasi-particle (QP) distribution function n≡{nkσ}. We show that the internal energy E[n] (or, more precisely, the thermodynamic potential Φ[n]), expressed as a function of the QP distribution n, can be interpreted as an effective potential (in the sense of field theory), which is obtained from the free energy by a Legendre transformation. This allows to obtain explicitly δΦ (or δE) starting from a microscopic Hamiltonian and to relate the Landau f function to the forward-scattering two-particle vertex without considering the collective modes as in the standard diagrammatic derivation of FLT. Out-of-equilibrium properties are obtained by extending the definition of the effective potential to space- and time-dependent configurations. Φ[n] is then a functional of the Wigner distribution function n≡{nkσ (r,t)}. It contains information about both the static and dynamic properties of the Fermi liquid. In particular, it yields the quantum Boltzmann equation satisfied by nkσ (r, t). Finally, we show how δΦ[δn] can be derived (in the static case) using a finite-temperature renormalization-group approach. In agreement with previous results based on this technique, we find that the Landau f function is defined by the fixed-point value of the Ω-limit of the forward-scattering two-particle vertex.