Exact and approximate methods of calculating the sum of states for noninteracting classical and quantum particles occupying a finite number of modes

Abstract

We present exact expressions for the sum of states of noninteracting classical and quantum particles occupying a finite number of modes with arbitrary spacings. Exploiting a probabilistic analogy, we derive an analytic fourth-order approximation to the density of states, which captures its variance and kurtosis, and is superior to the previous, commonly used methods for all three particle statistics. Our approach employs a simple exact method of calculating the moments of the microcanonical density of states for quantum particles, which requires less computational effort than the commonly used saddle-point approximation. We test our methods numerically and discuss their applicability to various physical systems.

On the Merit of a Central Limit Theorem-based Approximation in Statistical Physics

“…Equation (31) shows the same behavior as (24). All comments made for (24) are thus valid also in this case.…”

“…and as a consequence The above Statement implies that if conditions 1, 2 and 3 are met for a particular system, then (24) and (31) actually give the same result and one is entitled to use the simpler one. We might have stated this Statement right after (31) in the section devoted to general results.…”

“…Particularly interesting, this model has a highly degenerate ground state, which might lead it to violate the third law of thermodynamics and thus to a non-zero value for ∂f AXY ∂T | T =0 . However one in generals expects the zeropoint entropy to depend only on the degree of degeneracy of the ground state, while the value predicted by GA depends on the energy of the ground state, which is clear from (24). Even in this case we then expect the exact solution to be very different from GA in the limit of low temperature.…”

“…in Ising model). The second term alone gives rise to another complementary error function which closely resembles the one analyzed in the previous subsection in order to obtain (24). Following the same steps we can then get…”

“…In this case one can apply both GA according to (24) (in x-or z-basis) and GGA according to (31) (in y-basis), and easily check that the three conditions of the Statement actually hold. Thus GA and GGA coincide in this case.…”