Thermodynamic beta

Thermodynamic beta is essentially the connection between the information theory and statistical mechanics interpretation of a physical system through its entropy and the thermodynamics associated with its energy. It expresses 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 the system will randomize.

Via the statistical definition of temperature as a function of entropy, the coldness function can be calculated in the microcanonical ensemble from the formula

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

(i.e., the partial derivative of the entropy S with respect to the energy E at constant volume V and particle number N).

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. We also assume that any microstate of system 1 consistent with E₁ can coexist with any microstate of system 2 consistent with E₂. Thus, the number of microstates for the combined system is

${\displaystyle \Omega =\Omega _{1}(E_{1})\Omega _{2}(E_{2})=\Omega _{1}(E_{1})\Omega _{2}(E-E_{1}).\,}$

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,

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

But E₁ + E₂ = E implies

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

So

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

i.e.

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

The above relation motivates a definition of β:

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

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

${\displaystyle S=k_{\rm {B}}\ln \Omega ,}$

where k_B is the Boltzmann constant, S is the classical thermodynamic entropy, and Ω is the number of microstates. So

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

Substituting into the definition of β from the statistical definition above gives

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

Comparing with thermodynamic formula

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

we have

${\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}={\frac {1}{\tau }}}$

where ${\displaystyle \tau }$ is called the fundamental temperature of the system, and has units of energy.

• Boltzmann distribution
• Canonical ensemble
• Ising model