Partition function (statistical mechanics): Difference between revisions

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To make it into a dimensionless quantity, we must divide it by ''h'', which is some quantity with units of [[action (physics)|action]] (usually taken to be the [[Planck constant]]).

For generalized cases, the partition function of <math> N </math> particles in <math> d </math>-dimensions is given by

<math display="block"> Z = \frac{1}{h^{Nd}} \int \prod_{i=1}^{N} e^{-\beta \mathcal{H}(q_i\textbf{q}_i, p_i\textbf{p}_i)} \, d^d q_i\textbf{q}_i \, d^d p_i\textbf{p}_i, </math>

==== Classical continuous system (multiple identical particles) ====

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For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.

Consider a system ''S'' embedded into a [[heat bath]] ''B''. Let the total [[energy]] of both systems be ''E''. Let ''p_i'' denote the [[probability]] that the system ''S'' is in a particular [[Microstate (statistical mechanics)|microstate]], ''i'', with energy ''E_i''. According to the [[Statistical mechanics#Fundamental postulate|fundamental postulate of statistical mechanics]] (which states that all attainable microstates of a system are equally probable), the probability ''p_i'' will be inversely proportional to the number of microstates of the total [[Closed system (thermodynamics)|closed system]] (''S'', ''B'') in which ''S'' is in microstate ''i'' with energy ''E_i''. Equivalently, ''p_i'' will be proportional to the number of microstates of the heat bath ''B'' with energy {{nowrap|''E'' − ''E_i''}}. We then normalize this by dividing by the total number of microstates in which the constraints we have imposed on the entire system; both S and the heat bath; hold. In this case the only constraint is that the total energy of both systems is ''E'', so:

<math display="block">p_i = \frac{\Omega_B(E - E_i)}{\Omega_{(S,B)}(E)}.</math>