thermodynamic potential of the microcanonical ensemble is entropy. There are at least three possible definitions, each given in terms of the phase volume function , which counts the total number of states with energy less than (see the Precise expressions section for the mathematical definition of ): In the microcanonical ensemble, the temperature is a derived quantity rather than an external control parameter. It is defined as the derivative of the chosen entropy with respect to energy. For example, one can define the "temperatures" and as follows: Like entropy, there are multiple ways to understand temperature in the microcanonical ensemble. More generally, the correspondence between these ensemble-based definitions and their thermodynamic counterparts is not perfect, particularly for finite systems. The microcanonical pressure and chemical potential are given by: Phase transitions Under their strict definition, phase transitions correspond to nonanalytic behavior in the thermodynamic potential or its derivatives. Using this definition, phase transitions in the microcanonical ensemble can occur in systems of any size. This contrasts with the canonical and grand canonical ensembles, for which phase transitions can occur only in the thermodynamic limit—i.e., in systems with infinitely many degrees of freedom. Roughly speaking, the reservoirs defining the canonical or grand canonical ensembles introduce fluctuations that "smooth out" any nonanalytic behavior in the free energy of finite systems. This smoothing effect is usually negligible in macroscopic systems, which are sufficiently large that the free energy can approximate nonanalytic behavior exceedingly well. However, the technical difference in ensembles may be important in the theoretical analysis of small systems. Information entropy For a given mechanical system (fixed , ) and a given range of energy, the uniform distribution of probability over microstates (as in the microcanonical ensemble) maximizes the ensemble average . Thermodynamic analogies Early work in statistical mechanics by Ludwig Boltzmann led to his eponymous entropy equation for a system of a given total energy, , where is the number of distinct states accessible by the system at that energy. Boltzmann did not elaborate too deeply on what exactly constitutes the set of distinct states of a system, besides the special case of an ideal gas. This topic was investigated to completion by Josiah Willard Gibbs who developed the generalized statistical mechanics for arbitrary mechanical systems, and defined the microcanonical ensemble described in this article. Gibbs investigated carefully the analogies between the microcanonical ensemble and thermodynamics, especially how they break down in