Landau functions for noninteracting bosons

Abstract

We discuss the statistics of Bose-Einstein condensation (BEC) in a canonical ensemble of N noninteracting bosons in terms of a Landau function L BEC N (q) defined by the logarithm of the probability distribution of the order parameter q for BEC. We also discuss the corresponding Landau function for spontaneous symmetry breaking (SSB), which for finite N should be distinguished from L BEC N (q). Only for N → ∞ BEC and SSB can be described by the same Landau function which depends on the dimensionality and on the form of the external potential in a surprisingly complex manner. For bosons confined by a three-dimensional harmonic trap the Landau function exhibits the usual behavior expected for continuous phase transitions.The concept of a Landau function L(p) has been extremely useful to understand the nature of phase transitions [1]. For example, in the vicinity of a second order (continuous) thermal phase transition L(p) smoothly develops minima at non-zero values of the relevant order parameter p when the temperature T drops below the critical temperature T c , see Fig. 1 (a). On the other hand, in the case of a first order phase transition the location of the minima of L(p) changes discontinuously at T = T c , as shown in Fig. 1(b). While for infinite systems and in the absence of spontaneous symmetry breaking (SSB) L(p) can be identified with the Gibbs potential, this identification is in general not valid for finite systems or in the symmetry broken phase. In the latter case the Gibbs potential, which by construction is a convex function of p, exhibits a plateau as shown in Fig. 1. Microscopically, L(p) is defined in terms of the logarithm of the probability distribution of the order parameter in a given statistical ensemble [1]. From this probabilistic definition it is clear that L(p) does not have a strictly thermodynamic interpretation.For non-interacting systems the Landau function is in (b) (a) rag replacements T > Tc T = Tc T < Tc p L FIG. 1: Typical evolution of the Landau function L as a function of the relevant order parameter p close to (a) second order and (b) first order phase transitions. The dashed line denotes the plateau of the corresponding Gibbs potential for T < Tc.most cases not very interesting, because the correlations responsible for phase transitions are usually due to interactions between the microscopic degrees of freedom. An important exception are non-interacting bosons, where at low temperatures the correlations imposed by quantum statistics are sufficient to give rise to a phase transition, the Bose-Einstein condensation (BEC) [2,3]. In the condensed state the fraction of particles that occupy the single-particle state with the lowest energy is of the order of unity. More precisely, the order parameter for BEC can be chosen to be the expectation value of the operatorq = b † 0 b 0 /N , where b 0 annihilates a boson in the single-particle ground state and N is the total number of particles in the system.Conceptually, BEC should be distinguished from SSB, which manifests itself via a...

Analytical theory of mesoscopic Bose-Einstein condensation in an ideal gas

“…In particular, we rigorously prove that the cut-off distribution is the exact solution to a well-known recursion relation. Thus, in Sections IV and V we find analytically the Landau function [29,47], that is the logarithm of the probability distribution of the order parameter, which plays a part of an effective fluctuation Hamiltonian and, due to an absence of interatomic interaction in the ideal gas, is the actual Hamiltonian for the mesoscopic BEC in the ideal gas in the canonical ensemble. On this basis we find the universal scaling and structure of the order parameter (Sec.…”

“…However, a clear and full physical picture of the statistics and dynamics of BEC in the mesoscopic systems is absent until now not only in a general case of an interacting gas, but even in the case of an ideal gas (for a review, see e.g. [2,13,25,27,[29][30][31][32][33][34][35][36] and references therein). In particular, one of the most interesting in the statistical physics of BEC results, namely, a formula for the anomalously large variance of the ground-state occupation, (n 0 −n 0 ) 2 ∝ N 4/3 (T /T c ) 2 , found both for the ideal gas [27,[37][38][39][40][41][42] and for the weakly interacting gas [43][44][45][46], is valid only far enough from the critical point, where fluctuations of the order parameter are already relatively small, (n 0 −n 0 ) 2 ≪n 2 0 .…”

“…1) is strongly asymmetric and peculiar for all N < N c and N ∼ N c , including the critical region. In terms of the Landau function [29,47], − ln ρ n , this result means that, even without any interatomic interaction in the gas, the constraint nonlinearity, that originates from many-body Fock space cut off in the canonical ensemble, produces the infinite potential wall at n = N in the effective fluctuation Hamiltonian and makes it highly asymmetric. We find that the outlined constraint-cut-off mechanism is responsible for all unusual critical phenomena of the BEC phase transition in the ideal gas and, in particular, makes the BEC statistics strongly non-Gaussian.…”

“…6 and is truly universal since it contains no free or any physical parameters of the system at all and involve only pure mathematical numbers π and s m defined in Eq. (29).…”

“…The essence of the infrared universality is in the fast convergence of the sums s m in Eq. (29), which determine the generating cumulantsκ (∞) m in Eq. (13) in the thermodynamic limit, to the value g 1 = 6, that is equal to the contribution from the lowest 6-fold degenerate energy level ǫ 1 , with increasing the order m, i.e.…”