Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ζ 1, ζ 2,..., ζ N, then the partition function of the entire system is the product of the individual partition functions: Z = ∏ ∏ j = 1 N ζ ζ j .