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For the general case, one considers a set of functions <math>\{H_k(x_1,\cdots)\}</math> that each depend on the random variables <math>X_i</math>. These functions are chosen because one wants to hold their expectation values constant, for one reason or another. To constrain the expectation values in this way, one applies the method of [[Lagrange multiplier]]s. In the general case, [[maximum entropy method]]s illustrate the manner in which this is done.
Some specific examples are in order. In basic thermodynamics problems, when using the [[canonical ensemble]], the use of just one parameter <math>\beta</math> reflects the fact that there is only one expectation value that must be held constant: the [[Thermodynamic free energy|free energy]] (due to [[conservation of energy]]). For chemistry problems involving chemical reactions, the [[grand canonical ensemble]] provides the appropriate foundation, and there are two Lagrange multipliers. One is to hold the energy constant, and another, the [[fugacity]], is to hold the particle count constant (as chemical reactions involve the recombination of a fixed number of atoms).
For the general case, one has
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