# Who introduced the partition function?

Is it by Gibbs? Or Boltzmann? I do not expect Maxwell. The point is, how was the textbook introduction of partition function today developed.

• The notation Z is due to Planck (1921), as the first letter of Zustandssumme (sum over states). – KCd Feb 12 '17 at 4:25
• Marginally related: physics.stackexchange.com/questions/37617/… – Mars Jul 10 '17 at 16:14
• I immediately thought of partition theory from number theory, which was worked on by Ramanujan, and Hardy and Littlewood later found an exact formula. You might want to add some clarifying text to your question. – Spencer Jun 8 at 23:48

## 1 Answer

From the reading I have done, it is mainly Gibbs. Boltzmann introduced the idea of a factor having the form of exp(-alpha*E) in his re-derivation of Maxwell's distribution for the speeds of gases in 1868. Later the constant alpha was determined to be 1/kT, with k Boltzmann's constant -- I believe this determination was also carried out by Boltzmann. But Boltzmann's definition of entropy was contained in the H-theorem (1872) and later the formula S = k log (Omega) (1878), which deal with an average over single-particle trajectories in phase space, and a constant-energy average, respectively. It was Gibbs who came up with the idea of the canonical and grand canonical ensembles, where the partition function is the essential quantity. Because of this, I've seen the Boltzmann factor referred to as the Gibbs factor in a couple of places. So while I can't point to one website or source that attributes the partition function to Gibbs, I think all this evidence points very strongly to Gibbs having developed it. A good source on Gibbs' work in this area is Lynde Wheeler's biography, Chapter 10.