When can we treat identical particles as distinguishable? An unfamiliar classical limit

Abstract

Consider a system of N identical particles having an internal degree of freedom α whose energy enters additively. In statistical mechanics the question arises, ‘‘When can we compute the equilibrium properties related to α as if the system were a collection of N noninteracting distinguishable particles?’’ Most texts leave the student very unclear about this. Consider the case α=intrinsic spin. Then for a paramagnetic solid the answer is ‘‘for any density, temperature, or B field.’’ For an ideal gas we find the general answer ‘‘if and only if 〈nr〉 ≪1, all single particle states r, that is, degeneracy effects are negligible.’’ It is well known that this criterion is necessary for the so-called ‘‘classical limit’’ of low densities or high temperatures. However, it is not equivalent to it; this generalized classical regime can hold for arbitrary densities or temperatures, and lead to unfamiliar magnetic properties, if the B field is strong enough. In other words, very strong B fields can inhibit degeneracy even for high densities or low temperatures. Neutron stars are briefly considered.