Thermodynamic beta

In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:$\beta = \frac$ (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Temperature", ''Archive for Rational Mechanics and Analysis'' 57:3, 281-29
abstract

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

Thermodynamic beta has units reciprocal to that of energy (in SI units, reciprocal joules, $$beta= \textrm^). In non-thermal units, it can also be measured in byte per joule, or more conveniently, gigabyte per nanojoule;P. Fraundorf (2003) "Heat capacity in bits", ''Amer. J. Phys.'' 71:11, 1142-1151. 1 K⁻¹ is equivalent to about 13,062 gigabytes per nanojoule; at room temperature: = 300K, β ≈ ≈ ≈ . The conversion factor is 1 GB/nJ = $8\ln2\times 10^$ 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
:$\beta = \frac1 \, =\frac\left(\frac\right)_$
(i.e., the partial derivative of the entropy with respect to the energy at constant volume and particle number ).

Advantages

Though completely equivalent in conceptual content to temperature, is generally considered a more fundamental quantity than temperature owing to the phenomenon of negative temperature, in which is continuous as it crosses zero whereas has a singularity.

In addition, has the advantage of being easier to understand causally: If a small amount of heat is added to a system, 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 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

:$\Omega = \Omega_1 (E_1) \Omega_2 (E_2) = \Omega_1 (E_1) \Omega_2 (E-E_1) . \,$

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,

:$\frac \Omega = \Omega_2 (E_2) \frac \Omega_1 (E_1) + \Omega_1 (E_1) \frac \Omega_2 (E_2) \cdot \frac = 0.$

But ''E''₁ + ''E''₂ = ''E'' implies

:$\frac = -1.$

So

:$\Omega_2 (E_2) \frac \Omega_1 (E_1) - \Omega_1 (E_1) \frac \Omega_2 (E_2) = 0$

i.e.

:$\frac \ln \Omega_1 = \frac \ln \Omega_2 \quad \mbox$

The above relation motivates a definition of ''β'':

:$\beta =\frac.$

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

:$S = k_ \ln \Omega,$

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

:$d \ln \Omega = \frac d S .$

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

:$\beta = \frac \frac.$

Comparing with thermodynamic formula

:$\frac = \frac ,$

we have

:$\beta = \frac = \frac$

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

See also

* Boltzmann distribution
* Canonical ensemble
* Ising model

References

{{DEFAULTSORT:Thermodynamic Beta
Statistical mechanics
Scalar physical quantities