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The corresponding [[Gibbs measure]] then provides a probability distribution such that the expectation value of each <math>H_k</math> is a fixed value. More precisely, one has
:<math>\frac{\partial}{\partial \beta_k} \left(- \log Z \right) = \langle H_k\rangle = \mathrm{E}\left[H_k\right]</math>
with the angle brackets <math>\langle H_k \rangle</math> denoting the expected value of <math>H_k</math>, and <math>\mathrm{E}[\;]</math> being a common alternative notation. A precise definition of this expectation value is given below.
Although the value of <math>\beta</math> is commonly taken to be real, it need not be, in general; this is discussed in the section [[#Normalization|Normalization]] below. The values of <math>\beta</math> can be understood to be the coordinates of points in a space; this space is in fact a [[manifold]], as sketched below. The study of these spaces as manifolds constitutes the field of [[information geometry]].
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