# Why is the partition function called ''partition function''?

The partition function plays a central role in statistical mechanics.

But why is it called ''partition function''?

-

## 1 Answer

First, recall what a partition is. A partition of a set $X$ is a way to write $X$ as a disjoint union of subsets: $X=\coprod_i X_i$, $X_i\cap X_j=\emptyset$ for $i\neq j$. When the elements of the set $X$ are considered undistinguishable, what matters are the cardinals of the set only, and we have a partition of an integer number, $n=n_1+\ldots+n_k$. For numbers, the name "partition function" denotes the number of ways in which the number $n$ can be written like this. It is different than the "partition function" in statistical mechanics, but both refer to partitions.

In statistical mechanics, a partition describes how $n$ particles are distributed among $k$ energy levels. Probably the "partition function" is named so (indeed a bit uninspired), because it is a function associated to the way particles are partitioned among energy levels. An interesting explanation of this can be found in "The Partition Function: If That’s What It Is Why Don’t They Say So!". But I don't know a historical account of this.

-
It has to be said that Wikipedia (and specifically the German version) states that one should not confuse the partition function of physics (and probability theory) with that of number theory. – genneth Sep 17 '12 at 14:33
Just to make that comment a bit less obtuse: I thought of looking at the German because of the origin of the convention for $Z$. A direct translation (I think?) is 'sum over states'. – genneth Sep 17 '12 at 14:36
Right, it's Zustandsumme. – Luboš Motl Sep 18 '12 at 6:15
But there must have been a reason for calling it partition function rather than sum-over-states. – Arnold Neumaier Sep 18 '12 at 7:39
@ArnoldNeumaier: I don't have a reference, but the "partition of energy" is a common motif in early 20th century thermodynamics texts, for example, the "equipartition theorem", and I never had any thoughts that its this. I am not giving a quantum anachronistic interpretation, that's what I'm saying this answer does (just a little bit--- partitioning the particles into energy states is equivalent to partitioning the energy between particles, really, it's only humanistically different, in that it gives a different picture of what is partitioned). – Ron Maimon Sep 18 '12 at 23:45