Thermodynamic beta

File:ColdnessScale.png

SI temperature/coldness (1/kT) scale.

In statistical mechanics, the thermodynamic beta (or occasionally perk) is the reciprocal of the thermodynamic temperature of a system. Also referred to as coldness, 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 k_B 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; in units where k_B=1 it 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 express the response of entropy 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 whereas T has a singularity.

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 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 of statistical view with thermodynamic view

When two systems are in equilibrium, they have the same thermodynamic temperature T. Thus intuitively, one would expect β (as defined via microstates) to be related to T in some way. This link is provided by Boltzmann's fundamental assumption written as

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

where k_B is the Boltzmann constant and S is the classical thermodynamic entropy. 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 called the fundamental temperature of the system, and has units of energy.

See also

• Boltzmann factor
• Boltzmann distribution
• Canonical ensemble
• Ising model