{'fraction_non_alphanumeric': 0.021566401816118047, 'fraction_numerical': 0.0, 'mean_word_length': 5.125, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Partition functions for many-particle systems.

The well-known cumulant-summation formula is applied to the low-temperature calculation of the quantum-mechanical partition functions for many particle systems. The motivation for this method is to be able to deal with many-particle states directly and avoid the usual starting point of independent single-particle states. The test case chosen for this method was the calculation of the canonical partition function and the grand-canonical partition function for the degenerate free-electron gas. The result is that the expected equivalence of the canonical and grand-canonical ensembles is not found at low temperatures. The role of few-particle fluctuations in the grand-canonical ensemble is shown to be extremely important to the thermal properties of the degenerate free-electron gas and in the difference between the ensembles.