The Single particle partition function is defined mathematically as $$\text{Z=$\sum $}g_ie^{\left(\frac{-E_i}{K_BT}\right)}$$
But what is the physical interpretation of the partition function and it's significance to Thermodynamics? I'm seeking a simple yet understandable intuition.
But what is the physical interpretation of the partition function and it's significance to Thermodynamics? I'm seeking a simple yet understandable intuition.
The partition function has one simple physical interpretation in terms of Thermodynamic functions: Its natural log is proportional to the Free Energy (the proportionality constant is the negative inverse temperature).
The Free Energy, which is well-known from Thermodynamics, is given by $$ F=E-TS\;, $$ where E is the Thermodynamics Energy, T is the temperature, and S is the Entropy.
From a Statistical Mechanics perspective, remember that the probability to be in a state $n$ is given by $$ p_n\equiv \frac{e^{-E_n/T}}{Z}\;, $$ where $Z$ is your partition function, T is the temperature, and $E_n$ is the energy of state $n$.
The statistical definition of Entropy is $$ S=-\sum_n p_n \log(p_n) $$ $$ =-\sum_n p_n (-E_n/T-\log(Z)) $$ $$ =E/T+\log(Z) $$
I.e., $$ S-E/T=\log(Z)=-F/T $$
Partition functions are a measure of the allowed volume in (microscopic-)configuration space for the system, and as such they are the normalizing function for probabilities expressed as volumes in configuration space (and assuming the applicability of the ergodic hypothesis).
I know that this is very abstract, but it is also very general.
