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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.

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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., $$ S-E/T=\log(Z)=-F/T $$

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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.

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  • $\begingroup$ What exactly is 'configuration space' and could you expound in greater depth the ideal of an allowed volume in such configuration space? $\endgroup$ – Physkid May 5 '15 at 3:23
  • $\begingroup$ 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. $\endgroup$ – dmckee --- ex-moderator kitten May 5 '15 at 3:30

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