In statistical mechanics, the entropy  of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of , the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy ):
This equation, which relates the microscopic details, or microstates, of the system (via ) to its macroscopic state (via the entropy ), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.
The constant of proportionality  serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius:
One could choose instead a rescaled dimensionless entropy in microscopic terms such that
This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy.
The characteristic energy  is thus the energy required to increase the rescaled entropy by one nat.