Partitions and the Fermi-Dirac Distribution

For systems of electrons that can exchange energy and particles with a large medium, the celebrated Fermi Dirac (FD) distribution provides the probability for an electron to occupy a given energy level after the total energy of the system has been increased. This powerful tool derives from the laws of statistical mechanics (see, e.g., Landau and Lifchitz [3]). But this result does not apply to small isolated systems. Arnaud et al. [2] propose to compute the exact distribution in such systems by direct enumeration. This is done in the present paper, with the help of certain partitions of integers. The systems considered are isolated and one-dimensional with evenly spaced energy levels (the spacing is conveniently assumed to be unity). Two electrons cannot occupy the same level according to the Pauli exclusion principle, the electron spin being presently ignored. Figure 1 shows the initial (zero temperature) configuration of the system (n=0), and the possible configurations (microstates) after an increase of the energy by n=3 and n=6 units. The total number N of electrons of the system is assumed to be larger than n, and the energy levels are indexed by the integer k, with k=0 labeling the electron on top of the initial configuration. The number of microstates for a given integral value n of added energy is the number p(n) of partitions of n (for references to the theory of partitions, see, e.g., Andrews [1]). In the physical model, the microstates are assumed to be equally likely. The desired probability follows from the computation of the number m(n, k) of microstates that exhibit an electron at doi:10.1006 jcta.2000.3059, available online at http: www.idealibrary.com on