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.
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.
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.
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.
Abstract This chapter analyses Landau’s Fermi liquid theory. A fundamental postulate of Landau’s Fermi liquid theory is that quasiparticle energies are a function of the momentum, p, and that the momentum, pF, corresponding to the Fermi energy continues to be given by a free particle expression. Thus, the quasiparticle states are in one-to-one correspondence with those of an ideal Fermi gas and each quasiparticle has a spin quantum number, σ ± 1/2. The remainder of the chapter deals with the quasiparticle interaction function; sum rules; and collisionless collective modes, which are defined as a collective mode as a time- (or frequency) and space- (or wave vector) dependent oscillation of the fluid involving a coordinated motion of its constituent particles. Sample problems are also provided at the end of the chapter.
The moderately dense quantum Boltzmann gas at equilibrium is shown to be equivalent to a gas of "dressed particles," or "quasiparticles," like those of Landau's theory of a Fermi liquid. The Landau quasiparticle interaction energy F(p) is related to the second virial coefficient by means of an inverse Laplace transform. Expressions are obtained for F(p), and consequently the "renormalized" single-quasiparticle energy ${\ensuremath{\epsilon}}_{p}$, in terms of both scattering phase shifts and the M\o{}ller operator. The theory is compared with the Hartree approximation, the results of Allen, Nicolis et al., the work of Baerwinkel and Grossmann, and with expressions from Fermi-fluid theory.
It is well known that the formula for the Fermi distribution is obtained from the formula for the Bose distribution if the argument of the polylogarithm, the activity a, the energy, and the number of particles change sign. The paper deals with the behavior of the Bose–Einstein distribution as a → 0; in particular, the neighborhood of the point a = 0 is studied in great detail, and the expansion of both the Bose distribution and the Fermi distribution in powers of the parameter a is used. During the transition from the Bose distribution to the Fermi distribution, the principal term of the distribution for the specific energy undergoes a jump as a → 0. In this paper,we find the value of the parameter a, close to zero, but not equal to zero, for which the Bose distribution (in the statistical sense) becomes zero. This allows us to find the point a , distinct from zero, at which a jump of the specific energy occurs. Using the value of the number of particles on the caustic, we can obtain the jump of the total energy of the Bose system to the Fermi system. Near the value a = 0, the author uses Gentile statistics, whichmakes it possible to study the transition fromthe Bose statistics to the the Fermi statistics in great detail. Here an important role is played by the self-consistent equation obtained by the author earlier.
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.