# Thermodynamic beta

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

*k*

_{B}is Boltzmann constant).

^{[1]}

It was originally introduced in 1971 (as *Kältefunktion* "coldness function") by Ingo Müller , one of the proponents of the rational thermodynamics school of thought,^{[2]}^{[3]} based on earlier proposals for a "reciprocal temperature" function.^{[4]}^{[5]}

Thermodynamic beta has units reciprocal to that of energy (in SI units, reciprocal joules, ). In non-thermal units, it can also be measured in byte per joule, or more conveniently, gigabyte per nanojoule;^{[6]} 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 = J^{−1}.

## Description[edit]

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

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

### Advantages[edit]

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.^{[7]}

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[edit]

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*can coexist with any microstate of system 2 consistent with

_{1}*E*. Thus, the number of microstates for the combined system is

_{2}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,

But *E*_{1} + *E*_{2} = *E* implies

So

i.e.

The above relation motivates a definition of *β*:

## Connection of statistical view with thermodynamic view[edit]

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

where *k*_{B} is the Boltzmann constant, *S* is the classical thermodynamic entropy, and Ω is the number of microstates. So

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle d \ln \Omega = \frac{1}{k_{\rm B}} d S .}**

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

Comparing with thermodynamic formula

we have

where is called the *fundamental temperature* of the system, and has units of energy.

## See also[edit]

## References[edit]

**^**J. Meixner (1975) "Coldness and Temperature",*Archive for Rational Mechanics and Analysis***57**:3, 281-290 abstract.**^**Müller, Ingo (1971). "Die Kältefunktion, eine universelle Funktion in der Thermodynamik wärmeleitender Flüssigkeiten" [The cold function, a universal function in the thermodynamics of heat-conducting liquids].*Archive for Rational Mechanics and Analysis*.**40**: 1–36. doi:10.1007/BF00281528.**^**Müller, Ingo (1971). "The Coldness, a Universal Function in Thermoelastic Bodies".*Archive for Rational Mechanics and Analysis*.**41**: 319–332. doi:10.1007/BF00281870.**^**Day, W.A. and Gurtin, Morton E. (1969) "On the symmetry of the conductivity tensor and other restrictions in the nonlinear theory of heat conduction",*Archive for Rational Mechanics and Analysis***33**:1, 26-32 (Springer-Verlag) abstract.**^**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).**^**P. Fraundorf (2003) "Heat capacity in bits",*Amer. J. Phys.***71**:11, 1142-1151.**^**Kittel, Charles; Kroemer, Herbert (1980),*Thermal Physics*(2 ed.), United States of America: W. H. Freeman and Company, ISBN 978-0471490302