Based on the canonical ensemble, we suggested the simple scheme for taking into account Gaussian fluctuations in a finite system of ideal boson gas. Within framework of scheme we investigated the influence of fluctuations on the particle distribution in Bose -gas for two cases - with taking into account the number of particles in the ground state and without this assumption. The temperature and fluctuation parameter dependences of the modified Bose-Einstein distribution have been determined. Also the dependence of the condensation temperature on the fluctuation distribution parameter has been obtained.
The statistical problem of the distribution of
s quanta of equal energy
ε0 and total energy
E among
N distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by
p(κ)=(N−1N+s−1)(N−2N+s−κ−2);κ=0,1,2,⋯,s In this context, the special case (
N=4,
s=4) presented indicates that the alternative concept of most probable state is not valid for finite values of
s and
N. In the present article we derive alternatively
p(κ) by distributing
s quanta over
N particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $;
sεo=E∼ fixed, where the energy of the particles becomes continuous,
p(κ) transforms to the Boltzmann law
P(ε)=⟨ε⟩1e−⟨ε⟩ε;0≤ε<+∞ where
⟨ε⟩=E/N. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.
A simple method has been introduced to derive the all order quantum corrected Bose-Einstein distribution as the solution of the Wigner equation. The process is a perturbative one where the Bose-Einstein distribution has been taken as the unperturbed solution. This solution has been applied to calculate the number density of the bosons at finite temperature. The study may be important to investigate the properties of bosons and bose condensates at finite temperature. This process can also be applied to obtain the quantum corrected Fermi distribution.
If the N bosons that compose an ideal Bose-Einstein gas with energy E and volume V are each assumed to have the average energy of the system E/N, the entropy is easily expressed in terms of the number of bosons N and the number of single-particle microstates n they can occupy. Because the entropy derived is a function of only N and n, and the latter is a function of the extensive variables, E, V, and N, this entropy describes all that can be known of the thermodynamics of this system. In particular, the entropy very simply recovers the Sakur-Tetrode entropy in the classical limit and at sufficiently low temperature describes an unstable system. A thermodynamic stability analysis recovers the Bose-Einstein condensate and a two-phase region. Apart from numerical factors of order one, results are identical with those derived via standard, probabilistic methods.
A new quantum mechanical distribution function
nI(ε), is derived for the condition
n≥g, where in contrast to the exclusion principle
n≤g for fermions, each energy state must be populated by at least one particle. Although the particles share many features with bosons, the anomalous behavior of
nI(ε) precludes Bose-Einstein condensation (BEC) due to the required occupancy of the excited states, which creates a permanently pressurized background at
T=0, similar to the degeneracy pressure of fermions. An exhaustive classification scheme is presented for both distinguishable and indistinguishable, particles and energy levels based on Richard Stanley's twelvefold way in combinatorics.