Thermodynamic beta

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.

Details

Statistical interpretation

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 "E"₁ and "E"₂. We assume "E"₁ + "E"₂ = some constant "E". The number of microstates of each system will be denoted by Ω₁ and Ω₂. Under our assumptions Ω_"i" depends only on "E_i". Thus the number of microstates for the combined system is

:$Omega\; =\; Omega\_1\; (E\_1)\; Omega\_2\; (E\_2)\; =\; Omega\_1\; (E\_1)\; Omega\_2\; (E-E\_1)\; .\; ,$

We will derive "β" from the following fundamental assumption:

:"When the combined system reaches equilibrium, the number Ω is maximized."

(In other words, the system naturally seeks the maximum number of microstates.) Therefore, at equilibrium,

:$frac\{d\}\{d\; E\_1\}\; Omega\; =\; Omega\_2\; (E\_2)\; frac\{d\}\{d\; E\_1\}\; Omega\_1\; (E\_1)\; +\; Omega\_1\; (E\_1)\; frac\{d\}\{d\; E\_2\}\; Omega\_2\; (E\_2)\; cdot\; frac\{d\; E\_2\}\{d\; E\_1\}\; =\; 0.$

But "E"₁ + "E"₂ = "E" implies

:$frac\{d\; E\_2\}\{d\; E\_1\}\; =\; -1.$

So

:$Omega\_2\; (E\_2)\; frac\{d\}\{d\; E\_1\}\; Omega\_1\; (E\_1)\; -\; Omega\_1\; (E\_1)\; frac\{d\}\{d\; E\_2\}\; Omega\_2\; (E\_2)\; =\; 0$

i.e.

:$frac\{d\}\{d\; E\_1\}\; ln\; Omega\_1\; =\; frac\{d\}\{d\; E\_2\}\; ln\; Omega\_2\; quad\; mbox\{at\; equilibrium.\}$

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

:$S\; =\; k\; ln\; Omega,\; ,$

where "k" is the Boltzmann constant. So

:$d\; ln\; Omega\; =\; frac\{1\}\{k\}\; d\; S\; .$

Substituting into the definition of "β" gives

:

Comparing with the thermodynamic formula

:$frac\{d\; S\}\{d\; E\}\; =\; frac\{1\}\{T\}\; ,$

we have

:

where $au$ is sometimes called the "fundamental temperature" of the system with units of energy.

ee also

* Boltzmann distribution
* Canonical ensemble