# Thermodynamic beta

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In statistical mechanics, the thermodynamic beta (or occasionally perk) is the reciprocal of the thermodynamic temperature of a system. It can be calculated in the microcanonical ensemble from the formula

$\beta\triangleq\frac{1}{k_B}\left(\frac{\partial S}{\partial E}\right)_{V, N} = \frac1{k_B T} \,,$

where kB is the Boltzmann constant, S is the entropy, E is the energy, V is the volume, N is the particle number, and T is the absolute temperature. It has units reciprocal to that of energy, or in units where kB=1 also has units reciprocal to that of temperature. Thermodynamic beta is essentially the connection between the information theoretic/statistical interpretation of a physical system through its entropy and the thermodynamics associated with its energy. It can be interpreted as the entropic response to an increase in energy. If a system is challenged with a small amount of energy, then β describes the amount by which the system will "perk up," i.e. randomize. Though completely equivalent in conceptual content to temperature, β is generally considered a more fundamental quantity than temperature owing to the phenomenon of negative temperature, in which β is continuous as it crosses zero where T has a singularity.[1]

## 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 Ωi depends only on Ei. 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 fundamental assumption of statistical mechanics:

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

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

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 a definition of β:

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

### 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 fundamental assumption written as

$S = k_B \ln \Omega, \,$

where kB is the Boltzmann constant. So

$d \ln \Omega = \frac{1}{k_B} d S .$

Substituting into the definition of β gives

$\beta = \frac{1}{k_B} \frac{d S}{d E}.$

Comparing with the thermodynamic formula

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

we have

$\beta = \frac{1}{k_B T} = \frac{1}{\tau}$

where $\tau$ is sometimes called the fundamental temperature of the system with units of energy.

## References

1. Kittel, Charles; Kroemer, Herbert (1980), Thermal Physics (2 ed.), United States of America: W. H. Freeman and Company, ISBN 978-0471490302
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