Instatistical thermodynamics,thermodynamic beta, also known ascoldness,[1] is the reciprocal of thethermodynamic temperature of a system: (whereT is the temperature andkB isBoltzmann constant).[2]
Thermodynamic beta has units reciprocal to that of energy (inSI units, reciprocaljoules,). In non-thermal units, it can also be measured inbyte per joule, or more conveniently, gigabyte per nanojoule;[3] 1 K−1 is equivalent to about 13,062 gigabytes per nanojoule; at room temperature:T = 300K, β ≈44 GB/nJ ≈39 eV−1 ≈2.4×1020 J−1. The conversion factor is 1 GB/nJ = J−1.
Thermodynamic beta is essentially the connection between theinformation theory andstatistical mechanics interpretation of a physical system through itsentropy and thethermodynamics associated with itsenergy. It expresses the response of entropy to an increase in energy. If a small amount of energy is added to the system, thenβ 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 themicrocanonical ensemble from the formula
(i.e., thepartial derivative of the entropyS with respect to the energyE at constant volumeV and particle numberN).
Though completely equivalent in conceptual content to temperature,β is generally considered a more fundamental quantity than temperature owing to the phenomenon ofnegative temperature, in whichβ is continuous as it crosses zero whereasT has a singularity.[4]
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.
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 energiesE1 andE2. We assumeE1 +E2 = some constantE. The number ofmicrostates of each system will be denoted by Ω1 and Ω2. Under our assumptions Ωi depends only onEi. We also assume that any microstate of system 1 consistent withE1 can coexist with any microstate of system 2 consistent withE2. Thus, the number of microstates for the combined system is
We will deriveβ from thefundamental assumption of statistical mechanics:
(In other words, the system naturally seeks the maximum number of microstates.) Therefore, at equilibrium,
ButE1 +E2 =E implies
So
i.e.
The above relation motivates a definition ofβ:
When two systems are in equilibrium, they have the samethermodynamic temperatureT. Thus intuitively, one would expectβ (as defined via microstates) to be related toT in some way. This link is provided by Boltzmann's fundamental assumption written as
wherekB is theBoltzmann constant,S is the classical thermodynamic entropy, and Ω is the number of microstates. So
Substituting into the definition ofβ from the statistical definition above gives
Comparing with thermodynamic formula
we have
where is called thefundamental temperature of the system, and has units of energy.
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The thermodynamic beta was originally introduced in 1971 (asKältefunktion "coldness function") byIngo Müller [de], one of the proponents of therational thermodynamics school of thought,[5][6] based on earlier proposals for a "reciprocal temperature" function.[1][7][non-primary source needed]