Thermodynamic beta - Alchetron, The Free Social Encyclopedia

In statistical mechanics, the thermodynamic beta (or occasionally perk) is the reciprocal of the thermodynamic temperature of a system. Also referred to as coldness, it can be calculated in the microcanonical ensemble from the formula

where k_B is the Boltzmann constant, S is the entropy, E is the energy, V is the volume, N is the particle number, and T is the absolute temperature. It has units reciprocal to that of energy; in units where k_B=1 it also has units reciprocal to that of temperature. Thermodynamic beta is essentially the connection between the information theoretic/statistical 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 by which the system will "perk up," i.e. randomize. 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.

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. Thus the number of microstates for the combined system is

Ω = Ω 1 ( E 1 ) Ω 2 ( E 2 ) = Ω 1 ( E 1 ) Ω 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,

d d E 1 Ω = Ω 2 ( E 2 ) d d E 1 Ω 1 ( E 1 ) + Ω 1 ( E 1 ) d d E 2 Ω 2 ( E 2 ) ⋅ d E 2 d E 1 = 0.

But E₁ + E₂ = E implies

d E 2 d E 1 = − 1.

So

Ω 2 ( E 2 ) d d E 1 Ω 1 ( E 1 ) − Ω 1 ( E 1 ) d d E 2 Ω 2 ( E 2 ) = 0

i.e.

d d E 1 ln ⁡ Ω 1 = d d E 2 ln ⁡ Ω 2 at equilibrium.

The above relation motivates a definition of β:

β = d ln ⁡ Ω d E .

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 B ln ⁡ Ω ,

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

d ln ⁡ Ω = 1 k B d S .

Substituting into the definition of β gives

β = 1 k B d S d E .

Comparing with the thermodynamic formula

d S d E = 1 T ,

we have

β = 1 k B T = 1 τ

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

References

Thermodynamic beta Wikipedia

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Contents

Statistical interpretation

Connection of statistical view with thermodynamic view

References