# 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. Commented Sep 17, 2012 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'. Commented Sep 17, 2012 at 14:36
• Right, it's Zustandsumme. Commented Sep 18, 2012 at 6:15
• This is not exactly precisely accurate (but close)--- the name derives from how energy is partitioned between particles, which is subtly different in point of view than how particles are partitioned in energy. It's classical while the former is quantum. Commented Sep 18, 2012 at 13:20
• @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). Commented Sep 18, 2012 at 23:45