For a gas of N identical classical noninteracting particles in three dimensions, the partition function is Z = 1 N ! h 3 N ∫ ∫ exp ⁡ ⁡ ( − − β β ∑ ∑ i = 1 N H ( q i , p i ) ) d 3 q 1 ⋯ ⋯ d 3 q N d 3 p 1 ⋯ ⋯ d 3 p N = Z single N N ! where