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In statistical mechanics, the thermodynamic beta is a numerical quantity related to the thermodynamic temperature of a system. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and thermodynamics.
Additional recommended knowledge
From the statistical point of view, β is a numerical quantity relating two macroscopic systems in equilibrium. The exact formulation is as follows. Consider two systems, 1 and 2, in thermal contact, with respective energies E1 and E2. We assume E1 + E2 = some constant E. The number of microstates of each system will be denoted by Ω1 and Ω2. Under our assumptions Ωi depends only on Ei. Thus the number of microstates for the combined system is
We will derive β from the following fundamental assumption:
(In other words, the system naturally seeks the maximum number of microstates.) Therefore, at equilibrium,
But E1 + E2 = E implies
The above relation motivates the definition of β:
Connection with thermodynamic view
On the other hand, when two systems are in equilibrium, they have the same thermodynamic temperature T. Thus intuitively one would expect that β be related to T in some way. This link is provided by the formula
where k is the Boltzmann constant. So
Substituting into the definition of β gives
Comparing with the thermodynamic formula
where τ is sometimes called the fundamental temperature of the system with units of energy.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Thermodynamic_beta". A list of authors is available in Wikipedia.|