Thermodynamic beta

 ( Statistical Mechanics ) 

In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:$$\backslash beta\; \backslash equiv\; \backslash frac\{1\}\{k\_\{\backslash rm\; B\}T\}$$ (where is the temperature and is Boltzmann constant).

Thermodynamic beta has units reciprocal to that of energy (in SI units, reciprocal joules, $\backslash beta\; =\; \backslash textrm\{J\}^\{-1\}$). 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: = 300K, β ≈ ≈ ≈ . The conversion factor is 1 GB/nJ = $8\backslash ln2\backslash times\; 10^\{18\}$ J⁻¹.

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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

$\backslash beta\; =\; \backslash frac1\{k\_\{\backslash rm\; B\}\; T\}\; \backslash ,\; =\backslash frac\{1\}\{k\_\{\backslash rm\; B\}\}\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; E\}\backslash right)\_\{V,\; N\}$

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

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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.

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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

$\backslash Omega\; =\; \backslash Omega\_1\; (E\_1)\; \backslash Omega\_2\; (E\_2)\; =\; \backslash Omega\_1\; (E\_1)\; \backslash Omega\_2\; (E-E\_1)\; .\; \backslash ,$

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}{d E_1} \Omega = \Omega_2 (E_2) \frac{d}{d E_1} \Omega_1 (E_1) + \Omega_1 (E_1) \frac{d}{d E_2} \Omega_2 (E_2) \cdot \frac{d E_2}{d E_1} = 0.

But E₁ + E₂ = E implies

$\backslash frac\{d\; E\_2\}\{d\; E\_1\}\; =\; -1.$

So

$\backslash Omega\_2\; (E\_2)\; \backslash frac\{d\}\{d\; E\_1\}\; \backslash Omega\_1\; (E\_1)\; -\; \backslash Omega\_1\; (E\_1)\; \backslash frac\{d\}\{d\; E\_2\}\; \backslash Omega\_2\; (E\_2)\; =\; 0$

i.e.

$\backslash frac\{d\}\{d\; E\_1\}\; \backslash ln\; \backslash Omega\_1\; =\; \backslash frac\{d\}\{d\; E\_2\}\; \backslash ln\; \backslash Omega\_2\; \backslash quad\; \backslash mbox\{at\; equilibrium.\}$

The above relation motivates a definition of β:

$\backslash beta\; =\backslash frac\{d\; \backslash ln\; \backslash Omega\}\{\; d\; E\}.$

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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

$S\; =\; k\_\{\backslash rm\; B\}\; \backslash ln\; \backslash Omega,$

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

$d\; \backslash ln\; \backslash Omega\; =\; \backslash frac\{1\}\{k\_\{\backslash rm\; B\}\}\; d\; S\; .$

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

$\backslash beta\; =\; \backslash frac\{1\}\{k\_\{\backslash rm\; B\}\}\; \backslash frac\{d\; S\}\{d\; E\}.$

Comparing with thermodynamic formula

$\backslash frac\{d\; S\}\{d\; E\}\; =\; \backslash frac\{1\}\{T\}\; ,$

we have

$\backslash beta\; =\; \backslash frac\{1\}\{k\_\{\backslash rm\; B\}\; T\}\; =\; \backslash frac\{1\}\{\backslash tau\}$

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

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

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See also

• Boltzmann distribution
• Canonical ensemble
• Ising model

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