# Conceptual explanation of the Single particle partition function

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.

• What exactly is 'configuration space' and could you expound in greater depth the ideal of an allowed volume in such configuration space? – Physkid May 5 '15 at 3:23
• Configuration space is an abstract space the components of which are the positions and momenta of every independent mass in the system. You'll note that this can be a very high dimensional space. But if you don't have this concept then you probably need a different formulation of the meaning of the partition function. Alas, I'm still working on my own conceptual foundation for statistical physics, so I don't have the right one to offer. – dmckee --- ex-moderator kitten May 5 '15 at 3:30