Obtaining the thermodynamic relations for the Gibbs ensemble using the maximum entropy method

As a generating functional of the Gibbs ensemble, we use the Laplace transform of the complex (or generalized) Poisson measure. We use the maximum entropy principle to determine the form of the generating function of this distribution. We consider the cases where only the mathematical expectation is known and where the mathematical expectation and the second moment are known. In the latter case, the equation of state has a transcendental form. In the both cases, if there is no interaction, then the obtained relations lead to expressions for an ideal gas.