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In [[physics]], the '''Maxwell–Jüttner distribution''' is the distribution of speeds of particles in a hypothetical gas of relativistic particles. Similar to [[Maxwell's distribution]], the Maxwell–Jüttner distribution considers a classical ideal gas where the particles are dilute and do not significantly interact with each other. The distinction from Maxwell's case is that effects of [[special relativity]] are taken into account. In the limit of low temperatures ''T'' much less than ''mc''<sup>2</sup>/''k'' (where ''m'' is the mass of the kind of particle making up the gas, ''c'' is the [[speed of light]] and ''k'' is [[Boltzmann's constant]]), this distribution becomes identical to the Maxwell–Boltzmann distribution.
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The distribution can be attributed to [[Ferencz Jüttner]], who derived it in 1911.<ref>{{cite doi|10.1002/andp.19113390503}}</ref> It has become known as the Maxwell–Jüttner distribution by analogy to the name Maxwell-Boltzmann distribution that is commonly used to refer to Maxwell's distribution.
The breach held up from June 16 to Sept. 5, 2014, according to the location. Jimmy Johns has not said what number of cards were affected.<br><br>Hackers may have stolen credit and debit card numbers and perhaps the cardholder's name, verification code and/or the card's expiration date, Jimmy Johns said in a news release sunday.<br><br>Shares of Orion Energy bodies Inc. Jumped Thursday after organization said it has won a $6.2 million contract to retrofit the factory of an undisclosed automobile manufacturer with LED lighting.<br><br>The Manitowoc company said the auto plant will replace 465 watt lights with 173 watt light emitting diode furnishings.<br><br>the company previously announced a $2.4 million order to retrofit an guaranteeing assembly plant in Canada.<br><br>Are pleased to announce our continuing partnership with this customer to provide facility upgrdades for another of their making plants, claimed John Scribante, chief executive officer, In a statement.<br><br>The energy savings from the new contract need to reduce energy use by 327.5 million kw hours, And cut greenhouse gas emissions by more than 200,000 a lot, in Orion.<br><br>Orion has invested in what it sees as a market shift toward greater acceptance and installation of LED lighting.<br><br>group bought Harris LED of Florida last year and LED lighting sales totaled about $2.5 million, Or 21% of Orion lighting sales in the most up-to-date quarter. which often up from $1.1 million, Or 7% of all lighting effects sales, through the April June quarter of 2013.<br><br>Believe LED products will be the new driver of our company sales in in the future, supposed Scott Jensen, Chief fiscal officer, During an investor business call last month.<br><br>you can actually shares were trading around $6 a share, Up all over 20%, Thursday mid-day. The shares have traded between say $3.39 or $8.11 within the last year.<br><br>Duke ATC power line would link wy wind farm, Utah energy remembrance project<br><br>Wind tower orders fixed for Manitowoc plant<br><br>Orion imparts $6.2 million LED lamps contract, explains to you jump<br><br>PSC kits Oct. 8 public hearing on We systems rate case<br><br>ZBB Energy prices stock promising at $1.12 a have<br><br>UW Madison awarded $4 million for nuclear power R ZBB announces investment in Chinese three way partnership, Stock giving away<br><br>MGE Energy boosts dividend to shareholders by 4%<br><br>Eaton awarded nearly $5 million for fuel saving studies<br><br>Madison utility potential clients will receive $6.7 million reimburse<br><br>With new reports this week demonstrating that the dangers of climate change are mounting, The City of Milwaukee and Wisconsin's largest company have launched an initiative to cut carbon emissions by making buildings more cost-effective.<br><br>Johnson control buttons Inc. And Milwaukee are partners in a global energy efficiency step, [http://en.wiktionary.org/wiki/Sponsored Sponsored] by the un, That aims to tackle greenhouse gas emissions through energy saving projects like those taking place in downtown Milwaukee and the big apple.<br><br>In case you cherished this post as well as you desire to obtain details with regards to [http://www.pelicola.net/ ゴルフ] generously pay a visit to the webpage.
 
==The distribution function ==
[[Image:mj.png|thumbnail | right | 400px | Maxwell–Jüttner distribution over [[Lorentz factor]] (relativistic Maxwellian), for a gas at different temperatures. Speed is represented in terms of the [[Lorentz factor]].]]
 
As the gas becomes hotter and ''kT'' approaches or exceeds ''mc''<sup>2</sup>, the probability distribution for <math>\gamma=1/\sqrt{1-v^2/c^2}</math> in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:<ref name="Synge">
{{cite book
|last=Synge |first=J.L
|year=1957
|title=The Relativistic Gas
|series=Series in physics
|publisher=[[North-Holland]]
|lccn=57003567
}}</ref>
 
:<math> f(\gamma) = \frac {\gamma^2 \beta }{\theta K_2(1/\theta)}
\exp
\left(
- \frac {\gamma}{\theta}
\right)
</math>
 
where <math>\beta = \frac {v}{c}=\sqrt{1-1/\gamma^2},</math> <math>\theta=\frac{kT}{mc^2},</math> and <math>K_2</math> is the modified [[Bessel function]] of the second kind.
 
Alternatively, this can be written in terms of the momentum as
 
:<math> f(p) = \frac{1}{4 \pi m^3 c^3 \theta K_2(1/\theta)} \exp\left( -\frac{\gamma(p)}{\theta}\right)
</math>
 
where <math>\gamma(p) = \sqrt{1+\left(\frac{p}{mc}\right)^2}</math>. The Maxwell–Jüttner equation is covariant, but not [[Manifest covariance|manifestly]] so, and the temperature of the gas does not vary with the gross speed of the gas.<ref>{{cite journal
|last1=Chacon-Acosta|first1=Guillermo
|last2=Dagdug|first2=Leonardo
|last3=Morales-Tecotl|first3=Hugo A.
|year=2009 |title=On the Manifestly Covariant Jüttner Distribution and Equipartition Theorem |journal=ArXiv:0910.1625v1 |arxiv=0910.1625|bibcode = 2010PhRvE..81b1126C |doi = 10.1103/PhysRevE.81.021126 }}</ref>
 
==Limitations==
 
Some limitations of the Maxwell–Jüttner distributions are shared with the classical ideal gas: neglect of interactions, and neglect of quantum effects. An additional limitation (not important in the classical ideal gas) is that the Maxwell–Jüttner distribution neglects antiparticles.
 
If particle-antiparticle creation is allowed, then once the thermal energy ''kT'' is a significant fraction of ''mc''<sup>2</sup>, particle-antiparticle creation will occur and begin to increase the number of particles while generating antiparticles (the number of particles is not conserved, but instead the conserved quantity is the difference between particle number and antiparticle number). The resulting thermal distribution will depend on the [[chemical potential]] relating to the conserved particle-antiparticle number difference. A further consequence of this is that it becomes necessary to incorporate statistical mechanics for indistinguishable particles, because the occupation probabilities for low kinetic energy states becomes of order unity. For [[fermion]]s it is necessary to use [[Fermi–Dirac statistics]] and the result is analogous to the thermal generation of electron-[[electron hole|hole]] pairs in [[semiconductors]]. For [[boson]]ic particles, it is necessary to use the [[Bose–Einstein statistics]].<ref>See first few paragraphs in [http://arxiv.org/abs/hep-th/9604039v1] for extended discussion.</ref>
 
==References==
<references/>
 
{{ProbDistributions|continuous-semi-infinite}}
 
{{DEFAULTSORT:Maxwell-Juttner distribution}}
[[Category:Concepts in physics|Concepts in physics]]
[[Category:Special relativity]]
[[Category:Continuous distributions]]
[[Category:Gases]]
[[Category:Maxwell–Boltzmann statistics| ]]
[[Category:Particle distributions]]
[[Category:Probability distributions]]

Latest revision as of 02:24, 3 November 2014

Jimmy Johns confirmed Wednesday that it had a across the united states data breach at 216 locations this summer, Including five sandwich diners in Wisconsin Milwaukee, Sheboygan, Oshkosh, Portage and riv Falls.

The breach held up from June 16 to Sept. 5, 2014, according to the location. Jimmy Johns has not said what number of cards were affected.

Hackers may have stolen credit and debit card numbers and perhaps the cardholder's name, verification code and/or the card's expiration date, Jimmy Johns said in a news release sunday.

Shares of Orion Energy bodies Inc. Jumped Thursday after organization said it has won a $6.2 million contract to retrofit the factory of an undisclosed automobile manufacturer with LED lighting.

The Manitowoc company said the auto plant will replace 465 watt lights with 173 watt light emitting diode furnishings.

the company previously announced a $2.4 million order to retrofit an guaranteeing assembly plant in Canada.

Are pleased to announce our continuing partnership with this customer to provide facility upgrdades for another of their making plants, claimed John Scribante, chief executive officer, In a statement.

The energy savings from the new contract need to reduce energy use by 327.5 million kw hours, And cut greenhouse gas emissions by more than 200,000 a lot, in Orion.

Orion has invested in what it sees as a market shift toward greater acceptance and installation of LED lighting.

group bought Harris LED of Florida last year and LED lighting sales totaled about $2.5 million, Or 21% of Orion lighting sales in the most up-to-date quarter. which often up from $1.1 million, Or 7% of all lighting effects sales, through the April June quarter of 2013.

Believe LED products will be the new driver of our company sales in in the future, supposed Scott Jensen, Chief fiscal officer, During an investor business call last month.

you can actually shares were trading around $6 a share, Up all over 20%, Thursday mid-day. The shares have traded between say $3.39 or $8.11 within the last year.

Duke ATC power line would link wy wind farm, Utah energy remembrance project

Wind tower orders fixed for Manitowoc plant

Orion imparts $6.2 million LED lamps contract, explains to you jump

PSC kits Oct. 8 public hearing on We systems rate case

ZBB Energy prices stock promising at $1.12 a have

UW Madison awarded $4 million for nuclear power R ZBB announces investment in Chinese three way partnership, Stock giving away

MGE Energy boosts dividend to shareholders by 4%

Eaton awarded nearly $5 million for fuel saving studies

Madison utility potential clients will receive $6.7 million reimburse

With new reports this week demonstrating that the dangers of climate change are mounting, The City of Milwaukee and Wisconsin's largest company have launched an initiative to cut carbon emissions by making buildings more cost-effective.

Johnson control buttons Inc. And Milwaukee are partners in a global energy efficiency step, Sponsored by the un, That aims to tackle greenhouse gas emissions through energy saving projects like those taking place in downtown Milwaukee and the big apple.

In case you cherished this post as well as you desire to obtain details with regards to ゴルフ generously pay a visit to the webpage.

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