Thermodynamic beta

Via the statistical definition of temperature as a function of entropy, the coldness function can be calculated in the microcanonical ensemble from the formula (i.e., the partial derivative of the entropy S with respect to the energy E at constant volume V and particle number N).

[4] In addition, β has the advantage of being easier to understand causally: If a small amount of heat is added to a system, β is the increase in entropy divided by the increase in heat.

From the statistical point of view, β is a numerical quantity relating two macroscopic systems in equilibrium.

Consider two systems, 1 and 2, in thermal contact, with respective energies E1 and E2.

Thus, the number of microstates for the combined system is We will derive β from the fundamental assumption of statistical mechanics: (In other words, the system naturally seeks the maximum number of microstates.)

This link is provided by Boltzmann's fundamental assumption written as where kB is the Boltzmann constant, S is the classical thermodynamic entropy, and Ω is the number of microstates.

is called the fundamental temperature of the system, and has units of energy.

The thermodynamic beta was originally introduced in 1971 (as Kältefunktion "coldness function") by Ingo Müller [de], one of the proponents of the rational thermodynamics school of thought,[5][6] based on earlier proposals for a "reciprocal temperature" function.