Love wave - Revision history

en>Cydebot: Robot - Moving category Seismology and earthquake terminology to :Category:Seismology per CFD at Wikipedia:Categories for discussion/Log/2014 October 19.

2014-11-05T02:00:09Z

Robot - Moving category Seismology and earthquake terminology to Category:Seismology per CFD at Wikipedia:Categories for discussion/Log/2014 October 19.

← Older revision		Revision as of 04:00, 5 November 2014
Line 1:		Line 1:
	~~{{Unreferenced\|date=December 2009}}~~		Hello! My name is Melina. I smile that I can unify to the whole globe. I live in United States, in the TN region. I dream to head to the different nations, to obtain familiarized with interesting individuals.<br><br>Stop by my page: [http://www.whsa.co.uk/book-the-course/ Fifa 15 Coin generator]
	~~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~~
	~~:<math>~~
	~~\beta\triangleq\frac{1}{k_B}\left(\frac{\partial S}{\partial E}\right)_{V, N} = \frac1{k_B T} \,,</math>~~
	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~~
	~~\| last = Kittel~~
	~~\| first = Charles~~
	~~\| last2 = Kroemer~~
	~~\| first2 = Herbert~~
	~~\| title = Thermal Physics~~
	~~\| place = United States of America~~
	~~\| publisher = W. H. Freeman and Company~~
	~~\| year = 1980~~
	~~\| edition = 2~~
	~~\| isbn = 978-0471490302~~
	~~}}</ref>~~

	~~==Details==~~
	~~=== 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''<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~~

	:~~<math>\Omega = \Omega_1 (E_1) \Omega_2 (E_2) = \Omega_1 (E_1) \Omega_2 (E-E_1) . \,</math>~~

	~~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,~~

	:~~<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.~~
	~~</math>~~

	~~But ''E''<sub>1<~~/~~sub> + ''E''<sub>2<~~/~~sub> = ''E'' implies~~

	~~:<math>\frac{d E_2}{d E_1} = -1.</math>~~

	So

	~~:<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~~
	~~</math>~~

	i.e.

	~~:<math>~~
	~~\frac{d}{d E_1} \ln \Omega_1 = \frac{d}{d E_2} \ln \Omega_2 \quad \mbox{at equilibrium~~.}
	</~~math>~~

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

	~~:<math>\beta =\frac{d \ln \Omega}{ d E}.</math>~~

	~~===Connection with thermodynamic view===~~
	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

	~~:<math>S = k_B \ln \Omega, \,</math>~~

	~~where ''k''<sub>B</sub> is the [[Boltzmann constant]]. So~~

	~~:<math>d \ln \Omega = \frac{1}{k_B} d S .</math>~~

	~~Substituting into the definition of ''β'' gives~~

	~~:<math>\beta = \frac{1}{k_B} \frac{d S}{d E}.<~~/~~math>~~

	~~Comparing with the thermodynamic formula~~

	~~:<math>\frac{d S}{d E} = \frac{1}{T} ,</math>~~

	~~we have~~

	~~:<math>\beta = \frac{1}{k_B T} = \frac{1}{\tau}</math>~~

	~~where <math>\tau</math> is sometimes called the ''fundamental temperature'' of the system with units of energy.~~

	~~==See also==~~
	* [[Boltzmann factor]]
	* [[Boltzmann distribution]]
	* [[Canonical ensemble]]
	* [[Ising model]]

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Thermodynamic Beta}}~~
	~~[[Category:Statistical mechanics]]~~
	~~[[Category:Units of temperature]~~]

en>Salih: Reverted good faith edits by 2a02:810d:300:190:6138:5a62:90d7:fd7d (talk). (TW)

2014-01-22T03:40:16Z

Reverted good faith edits by 2a02:810d:300:190:6138:5a62:90d7:fd7d (talk). (TW)

← Older revision		Revision as of 05:40, 22 January 2014
Line 1:		Line 1:
	== ~~Ray Ban Folding Wayfarer Polarized~~ ==		{{Unreferenced\|date=December 2009}}
			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
			:<math>
			\beta\triangleq\frac{1}{k_B}\left(\frac{\partial S}{\partial E}\right)_{V, N} = \frac1{k_B T} \,,</math>
			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
			\| last = Kittel
			\| first = Charles
			\| last2 = Kroemer
			\| first2 = Herbert
			\| title = Thermal Physics
			\| place = United States of America
			\| publisher = W. H. Freeman and Company
			\| year = 1980
			\| edition = 2
			\| isbn = 978-0471490302
			}}</ref>

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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 ''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

			:<math>\Omega = \Omega_1 (E_1) \Omega_2 (E_2) = \Omega_1 (E_1) \Omega_2 (E-E_1) . \,</math>

			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,

			:<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.
			</math>

			But ''E''<sub>1</sub> + ''E''<sub>2</sub> = ''E'' implies

			:<math>\frac{d E_2}{d E_1} = -1.</math>

			So

			:<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
			</math>

			i.e.

			:<math>
			\frac{d}{d E_1} \ln \Omega_1 = \frac{d}{d E_2} \ln \Omega_2 \quad \mbox{at equilibrium.}
			</math>

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

			:<math>\beta =\frac{d \ln \Omega}{ d E}.</math>

			===Connection with thermodynamic view===
			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

			:<math>S = k_B \ln \Omega, \,</math>

			where ''k''<sub>B</sub> is the [[Boltzmann constant]]. So

			:<math>d \ln \Omega = \frac{1}{k_B} d S .</math>

			Substituting into the definition of ''β'' gives

			:<math>\beta = \frac{1}{k_B} \frac{d S}{d E}.</math>

			Comparing with the thermodynamic formula

			:<math>\frac{d S}{d E} = \frac{1}{T} ,</math>

			we have

			:<math>\beta = \frac{1}{k_B T} = \frac{1}{\tau}</math>

			where <math>\tau</math> is sometimes called the ''fundamental temperature'' of the system with units of energy.

			==See also==
			* [[Boltzmann factor]]
			* [[Boltzmann distribution]]
			* [[Canonical ensemble]]
			* [[Ising model]]

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Thermodynamic Beta}}
			[[Category:Statistical mechanics]]
			[[Category:Units of temperature]]

en>BattyBot: changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)

2012-07-17T02:46:26Z

changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)

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