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In statistical mechanics, is the partition function a mathematical cumulative distribution function as in probability mathematics ?

Why is the partition function the fundamental objects for statistical mechanisms ?

Why isn't it the probability density function ?

Mathieu Krisztian
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  • @By Symmetry : no, I ask explicitely if the partition function is the cumulative distribution function $F(x)=P(X\le x)$ – Mathieu Krisztian Dec 25 '21 at 08:09
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    @MathieuKrisztian Then the answer is obviously no: it is not even bounded by $1$ in general! But, as explained on the linked page, it is related to a classical quantity in probability theory: the moment generating function (and its logarithm is the generating function of the cumulant). – Yvan Velenik Dec 25 '21 at 09:51
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    Note also that it is not true that "the partition function the fundamental objects for statistical mechanisms". The fundamental object is the relevant probability measure (which depends on the chosen ensemble). However, partition functions, being generating functions, provide useful information on the system (for some specific observables). Moreover, they are the quantities directly related to thermodynamic potentials (the relevant thermodynamic potential being, roughly speaking, the rate of exponential growth of the partition function). – Yvan Velenik Dec 25 '21 at 11:31

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