# Thermodynamic beta

SI temperature/coldness conversion scale: Temperatures in Kelvin scale are shown in blue (Celsius scale in green, Fahrenheit scale in red), coldness values in gigabyte per nanojoule are shown in black. Infinite temperature (coldness zero) is shown at the top of the diagram; positive values of coldness/temperature are on the right-hand side, negative values on the left-hand side.

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

${\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}$ (where T is the temperature and kB is Boltzmann constant).[1]

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,[2] based on earlier proposals for a "reciprocal temperature" function.[3][4]

Thermodynamic beta has units reciprocal to that of energy (in SI units, ${\displaystyle [\beta ]={\textrm {J}}^{-1}}$). In non-thermal units, it can also be measured in byte per joule, or more conveniently, gigabyte per nanojoule;[5] 1 K−1 is equivalent to about 13,062 gigabytes per nanojoule; at room temperature: T = 300K, β ≈ 44 GB/nJ39 eV−12.4×1020 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

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

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

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.[6]

## 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

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

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:

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

## 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

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

where kB is the Boltzmann constant and S is the classical thermodynamic entropy. So

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

Substituting into the definition of β gives

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

Comparing with the 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.