Quantum distribution functions in systems with an arbitrary number of particles

TL;DR

Problem: describe entropy and distribution functions for non-interacting fermions and bosons with arbitrary, including small and non-integer, particle numbers in thermal equilibrium. Approach: derive exact finite-$N$ expressions using gamma and psi functions, yielding distribution functions with finite support bounded by $\theta_{j,\max}$, and show that in the large-$N$ limit these reduce to the Fermi-Dirac and Bose-Einstein forms $n_j^{(0)}=1/(e^{\theta_j}\pm 1)$ with $\theta_j=(\varepsilon_j-\mu)/T$. Key contributions: per-level entropies with finite-$N$ corrections, exact occupation relations $\psi[z_j(1-n_j)+1]-\psi(z_j n_j+1)=\theta_j$ for fermions and $\psi[z_j n_j+z_j]-\psi[z_j n_j+1]=\theta_j$ for bosons, explicit energy boundaries $\theta_{j,\max}$, and the recovery of FD/BE in the thermodynamic limit; the framework supports experimental tests in mesoscopic/nanostructured systems. Significance: provides a rigorous, finite-$N$ statistical-mechanics description for open quantum systems and guides interpretation of occupancy and entropy in quantum dots and related nanostructures.

Abstract

Expressions for the entropy and equations for the quantum distribution functions in systems of non-interacting fermions and bosons with an arbitrary, including small, number of particles are obtained in the paper

Figures (2)

• Figure 1: Distribution function of Fermi particles $n(\theta)$ over states in various approximations with multiplicities of level degeneracy: ( a) $z=2$, ( b) $z=10$. 1 -- distribution function (DF), calculated using the exact formula (15); 2 -- DF, calculated using approximate equation (9); 3 -- DF, calculated using formula (11); 4 -- conventional Fermi-Dirac DF (10).
• Figure 2: Distribution function of Bose particles $n(\theta)$ over states in various approximations with multiplicities of level degeneracy: ( a) $z=2$, ( b) $z=10$. 1 -- distribution function (DF), calculated using the exact formula (28); 2 -- DF, calculated using approximate equation (23); 3 -- conventional Bose-Einstein DF (24). The calculation using the approximate formula (25) gives a dependence that practically coincides with curve 1 (dotted line).