Thermodynamic beta

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.^[1] Also referred to as coldness,^[2] it can be calculated in the microcanonical ensemble from the formula

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

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.^[3]

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,

${\frac {d}{dE_{1}}}\Omega =\Omega _{2}(E_{2}){\frac {d}{dE_{1}}}\Omega _{1}(E_{1})+\Omega _{1}(E_{1}){\frac {d}{dE_{2}}}\Omega _{2}(E_{2})\cdot {\frac {dE_{2}}{dE_{1}}}=0.$

But E₁ + E₂ = E implies

${\frac {dE_{2}}{dE_{1}}}=-1.$

So

$\Omega _{2}(E_{2}){\frac {d}{dE_{1}}}\Omega _{1}(E_{1})-\Omega _{1}(E_{1}){\frac {d}{dE_{2}}}\Omega _{2}(E_{2})=0$

i.e.

${\frac {d}{dE_{1}}}\ln \Omega _{1}={\frac {d}{dE_{2}}}\ln \Omega _{2}\quad {\mbox{at equilibrium.}}$

The above relation motivates a definition of β:

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

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

$d\ln \Omega ={\frac {1}{k_{B}}}dS.$

Substituting into the definition of β gives

$\beta ={\frac {1}{k_{B}}}{\frac {dS}{dE}}.$

Comparing with the thermodynamic formula

${\frac {dS}{dE}}={\frac {1}{T}},$

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

References

1. J. Castle, W. Emmenish, R. Henkes, R. Miller, and J. Rayne (1965) Science by Degrees: Temperature from Zero to Zero (Westinghouse Search Book Series, Walker and Company, New York) snippets.
2. Claude Garrod (1995) Stat Mech and Thermodynamics (Oxford U. Press).
3. Kittel, Charles; Kroemer, Herbert (1980), Thermal Physics (2 ed.), United States of America: W. H. Freeman and Company, ISBN 978-0471490302