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Does anyone know why is the function $Z = \sum e^{-\beta E_i}$ called "partition function"?

For example, does it have a connection to the mathematical term "partition of $A$" which is a representation of the set $A$ as a disjoint union of it's subsets (and defines an equivalence relation over $A$)?

EDIT: The explanation below and the explanation here indeed almost give me a full answer. I just want to be sure: We are divding the whole quantity be it's energy states and not by it's particles. This means that a class in a partition can have lots of particles and can be related to one energy state exactly. And we have to know the distribution function of the particles in the system. Am I right? or mayby every particle has it's own distribution?

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    $\begingroup$ This is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. From Wikiepdia. I have to say that this "explanation" does not satisfy me completely, though... $\endgroup$
    – valerio
    Dec 2, 2016 at 11:25
  • $\begingroup$ @valerio92 So, each probability defines a different sate of the sysrem, and under this state the particles are arranged in sets where each set contains all particles with a specific energy $E_i$? $\endgroup$
    – user135172
    Dec 2, 2016 at 13:46

1 Answer 1

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It is appropriate to call $Z$,

$$Z=\sum_{i \in \, \mathrm{states}}\exp \left( -\beta E_i\right)$$

the partition function as it describes how probabilities are distributed amongst all the states with energies $E_0, E_1$, and so forth. To see this, note that the expected value of a property $Q$ is, $$\langle Q \rangle = \frac{1}{Z}\sum_{i \in \, \mathrm{states}}Q_i\exp \left( -\beta E_i\right).$$

This is analogous to the fact that in probability, for a discrete variable $X$ which can take values $\{x_i\}$ the expected value is,

$$\langle X \rangle = \sum_{i} x_i p_i$$

for a normalised distribution $\sum_i p_i = 1$, with probabilities $p_i$ for each value $x_i$. Thus the partition function provides an appropriate weight for each state.

It is denoted by $Z$ after the German word, Zustandssumme which roughly translates to a sum over states, which is what we are instructed to do,$\sum_{i \in \, \mathrm{states}}$, to obtain it.

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  • $\begingroup$ So, the partition is a partition of a distribution into all of it's energy states? $\endgroup$
    – user135172
    Dec 2, 2016 at 12:36
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    $\begingroup$ @user135172 The partition function provides a partition of the respective probabilities of the states of a system, but it is also more than that. For example, the partition function in a quantum field theory can be seen to be the generator of its correlation functions. It is quite a multi-faceted object. $\endgroup$
    – JamalS
    Dec 2, 2016 at 12:57

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