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| In [[statistical mechanics]], the '''thermodynamic beta''' (or occasionally '''perk''') is the reciprocal of the [[thermodynamic temperature]] of a system. It can be calculated in the [[microcanonical ensemble]] from the formula
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| :<math>
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| \beta\triangleq\frac{1}{k_B}\left(\frac{\partial S}{\partial E}\right)_{V, N} = \frac1{k_B T} \,,</math>
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| where ''k''<sub>B</sub> 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, or in units where ''k''<sub>B</sub>=1 also has units reciprocal to that of temperature. Thermodynamic beta is essentially the connection between the [[information theory|information theoretic]]/[[statistical mechanics|statistical]] interpretation of a physical system through its entropy and the [[thermodynamics]] associated with its energy. It can be interpreted as the entropic response 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 where ''T'' has a singularity.<ref>{{Citation
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| | last = Kittel
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| | first = Charles
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| | last2 = Kroemer
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| | first2 = Herbert
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| | title = Thermal Physics
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| | place = United States of America
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| | publisher = W. H. Freeman and Company
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| | year = 1980
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| | edition = 2
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| | isbn = 978-0471490302
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| }}</ref>
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| ==Details==
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| === Statistical interpretation ===
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| 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''<sub>1</sub> and ''E''<sub>2</sub>. We assume ''E''<sub>1</sub> + ''E''<sub>2</sub> = some constant ''E''. The number of [[Microstate (statistical mechanics)|microstates]] of each system will be denoted by Ω<sub>1</sub> and Ω<sub>2</sub>. Under our assumptions Ω<sub>''i''</sub> depends only on ''E<sub>i</sub>''. Thus the number of microstates for the combined system is
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| :<math>\Omega = \Omega_1 (E_1) \Omega_2 (E_2) = \Omega_1 (E_1) \Omega_2 (E-E_1) . \,</math> | |
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| We will derive ''β'' from the [[fundamental assumption of statistical mechanics]]:
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| :''When the combined system reaches equilibrium, the number Ω is maximized.''
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| (In other words, the system naturally seeks the maximum number of microstates.) Therefore, at equilibrium,
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| :<math> | |
| \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.
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| </math>
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| But ''E''<sub>1</sub> + ''E''<sub>2</sub> = ''E'' implies
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| :<math>\frac{d E_2}{d E_1} = -1.</math>
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| So
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| :<math>\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) = 0
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| </math>
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| i.e.
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| :<math>
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| \frac{d}{d E_1} \ln \Omega_1 = \frac{d}{d E_2} \ln \Omega_2 \quad \mbox{at equilibrium.}
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| </math>
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| The above relation motivates a definition of ''β'':
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| :<math>\beta =\frac{d \ln \Omega}{ d E}.</math>
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| ===Connection with thermodynamic view===
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| On the other hand, when two systems are in equilibrium, they have the same [[thermodynamic temperature]] ''T''. Thus intuitively one would expect that ''β'' be related to ''T'' in some way. This link is provided by the fundamental assumption written as
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| :<math>S = k_B \ln \Omega, \,</math>
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| where ''k''<sub>B</sub> is the [[Boltzmann constant]]. So
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| :<math>d \ln \Omega = \frac{1}{k_B} d S .</math>
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| Substituting into the definition of ''β'' gives
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| :<math>\beta = \frac{1}{k_B} \frac{d S}{d E}.</math>
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| Comparing with the thermodynamic formula
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| :<math>\frac{d S}{d E} = \frac{1}{T} ,</math>
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| we have
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| :<math>\beta = \frac{1}{k_B T} = \frac{1}{\tau}</math>
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| where <math>\tau</math> is sometimes called the ''fundamental temperature'' of the system with units of energy.
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| ==See also==
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| * [[Boltzmann factor]]
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| * [[Boltzmann distribution]]
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| * [[Canonical ensemble]]
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| * [[Ising model]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Thermodynamic Beta}}
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| [[Category:Statistical mechanics]]
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| [[Category:Units of temperature]]
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