Thermodynamic beta
# Thermodynamic beta

In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:

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

(where T is the temperature and $k_B$ is Boltzmann constant).

It was originally introduced in 1971 (as Kältefunktion "coldness function") by Ingo Müller [de], one of the proponents of the rational thermodynamics school of thought, based on earlier proposals for a "reciprocal temperature" function.

Thermodynamic beta has units reciprocal to that of energy (in SI units, reciprocal joules, $[\beta ]={\textrm {J}}^{-1}$). In non-thermal units, it can also be measured in byte per joule, or more conveniently, gigabyte per nanojoule; 1 $K^{−1}$ is equivalent to about 13,062 gigabytes per nanojoule; at room temperature: T = 300K, β ≈ 44 GB/nJ ≈ 39 $eV^{−1}$ ≈ $2.4×10^{20}$ $J^{−1}.$ The conversion factor is 1 GB/nJ = $8\ln 2\times 10^{18}$ $J^{−1}.$

## Description

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

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

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

### Advantages

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.

In addition, β has the advantage of being easier to understand causally: If a small amount of heat is added to a system, β is the increase in entropy divided by the increase in heat. Temperature is difficult to interpret in the same sense, as it is not possible to "Add entropy" to a system except indirectly, by modifying other quantities such as temperature, volume, or number of particles.

## 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 $E_1$ and $E_2.$ We assume $E_1$ + $E_2$ = some constant E. The number of microstates of each system will be denoted by $Ω_1$ and $Ω_2.$ Under our assumptions $Ω_i$ depends only on $E_i.$ We also assume that any microstate of system 1 consistent with $E_1$ can coexist with any microstate of system 2 consistent with $E_2.$ 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 $E_1$ + $E_2$ = 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

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

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

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

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

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

Comparing with thermodynamic formula

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

we have

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

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

