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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.

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1 Details
- 1.1 Statistical interpretation
- 1.2 Connection with thermodynamic view
2 See also

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 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 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 E₁ + E₂ = E implies

$\frac{d E_2}{d E_1} = -1.$

$\Omega_2 (E_2) \frac{d}{d E_1} \Omega_1 (E_1) - \Omega_1 (E_1) \frac{d}{d E_2} \Omega_2 (E_2) = 0$

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:

$\beta \equiv \frac{d \ln \Omega}{ d E}.$

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

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