= Statistical mechanics
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<Statistical mechanics> connects <microstates> to <macrostates> through <probability distributions>. Its <microcanonical ensemble>, <canonical ensemble> and <grand canonical ensemble> specify different constraints on exchanges with the surroundings. An <ensemble average> describes repeated preparations under those constraints; it need not be the value in every individual preparation.
This makes the subject useful whenever tracking every microscopic detail would obscure the question being asked. To predict a <heat capacity>, we need how the <expected value> of <energy> changes with <temperature>, rather than a record of every unit's state. The <partition function> packages that calculation into one reusable object.
In his 1902 book, J. Willard Gibbs systematically developed the <canonical ensemble>. His preface contrasts following one system through time with studying a collection of systems with different microscopic conditions. Chapter IV defines the <canonical ensemble> through an <exponential function> of <energy> and establishes its normalization. This supplied a general strategy for predicting collective measurements by averaging over possible microscopic conditions, with the <partition function> organizing the calculation. https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics[Gibbs, Elementary Principles in Statistical Mechanics (1902), preface]{ref} https://en.wikisource.org/wiki/Elementary_Principles_in_Statistical_Mechanics/Chapter_IV[Gibbs (1902), Chapter IV, equations 90–93]{ref}
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