Revision of the fractional exclusion statistics

I discuss the concept of fractional exclusion statistics (FES) and I show that in order to preserve the thermodynamic consistency of the formalism, the exclusion statistics parameters should change if the species of particles in the system are divided into subspecies. Using a simple and intuitive model I deduce the general equations that have to be obeyed by the exlcusion statistics parameters in any FES system. Introduction. – In Ref. [1] Haldane introduced the fruitful concept of fractional exclusion statistics (FES). Although many authors analyzed the physical properties of FES systems and the microscopic reasons for the manifestation of this type of statistics (see [2–15] and references therein, just as examples), there are important properties that have been overlooked. In Ref. [16] I proved that if the mutual exclusion statistics parameters (see below the definitions) are defined in the typical way (e.g. like in [1,2]), then the thermodynamics of the system is inconsistent. To restore the thermodynamics, I conjectured in the same paper that any of the mutual exclusion statistics parameters should be proportional to the dimension of the space on which it acts. In another paper [17] I showed that FES is manifesting in general in systems of interacting particles and the calculated exclusion statistics parameters have indeed the properties conjectured in [16]. In this letter I analyze the basic properties of the mutual exclusion statistics parameters based on simple, general arguments and I show that the conjectures introduced in [16] are, simply, necessary conditions for the logical consistency of the formalism. This is not surprising, since the inconsistency of the thermodynamics proved in Ref. [16] could have been only a consequence of an unconsistent undelying physical model. A simple model. – Let us assume that we have a system formed of only two species of particles, 0 and 1, like in Fig. 1. We denote the exclusion statistics parameters of this system by α̃00, α̃01, α̃10 and α̃11, and we start in the standard way [1, 2] by writing the total number of configurations corresponding to N0 particles of species 0 and N1 particles of species 1 as