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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 k_B is Boltzmann constant).

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

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 small amount of energy is added to the system, 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).

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 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:

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

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 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.

History

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The thermodynamic beta 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.

See also

• Boltzmann distribution
• Canonical ensemble
• Ising model

References

1. ^ Day, W. A.; Gurtin, Morton E. (1969-01-01). "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. Bibcode:1969ArRMA..33...26D. doi:10.1007/BF00248154. ISSN 1432-0673.
2. Meixner, J. (1975-09-01). "Coldness and temperature". Archive for Rational Mechanics and Analysis. 57 (3): 281–290. Bibcode:1975ArRMA..57..281M. doi:10.1007/BF00280159. ISSN 1432-0673.
3. Fraundorf, P. (2003-11-01). "Heat capacity in bits". American Journal of Physics. 71 (11): 1142–1151. Bibcode:2003AmJPh..71.1142F. doi:10.1119/1.1593658. ISSN 0002-9505.
4. Kittel, Charles; Kroemer, Herbert (1980), Thermal Physics (2 ed.), United States of America: W. H. Freeman and Company, ISBN 978-0471490302
5. 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.
6. Müller, Ingo (1971). "The Coldness, a Universal Function in Thermoelastic Bodies". Archive for Rational Mechanics and Analysis. 41 (5): 319–332. Bibcode:1971ArRMA..41..319M. doi:10.1007/BF00281870.
7. Castle, J.; Emmenish, W.; Henkes, R.; Miller, R.; Rayne, J. (1965). Science by Degrees: Temperature from Zero to Zero. New York: Walker and Company.

• Statistical mechanics
• Scalar physical quantities