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SEO Keyword summary for de.wikipedia.org/wiki/maxwell-boltzmann-verteilung
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be displaystyle
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Short Tail Keywords displaystyle der frac |
long Tail Keywords (2 words) kinetischen gastheorie der kinetischen herleitung mit mit hilfe idealen gas |
long Tail Keywords (3 words) herleitung mit hilfe der kinetischen gastheorie normalverteilung der komponenten frac 2pi frac hilfe der normalverteilung der komponenten der komponenten der geschwindigkeit |
de.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a de.wikipedia.org/wiki/maxwell-boltzmann-verteilung page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
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maxwellboltzmannverteilung ndash wikipedia
Meta description
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Content SEO
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Headings
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displaystyle xin infty sqrt frac operatorname erf leftfrac xsqrt arightsqrt xex asqrt sigma gamma approx leftasqrt rightgamma vvec etextkinfrac wepropto efrac etextkinkmathrm kmathrm vvmathord vxvyvz mathord pvmathrm vccdot cdot tmathrm int vec tilde pvec vleftfrac tright mathrm mvec vrightarrow viiint vvmathrm vtilde vmathrm iiint tiiint exp delta fnu xmv hat vsqrt tmsqrt rtm textdpvtextdv bar vfrac ldots vnn vint vpvtextdv tpi msqrt rtpi overline pvfrac nmoverline pvnkmathrm nkmathrm pvdv moverline breve ncdot vnright pvvtextdv tzquad quad vdvfrac tmdz mkmathrm tsqrt mpi textm vbar hsum nfrac muvec wfrac zbeta ebeta hvec xnvec mathbb nebeta pntextdtau textdtau hbar ntextd textd xntextd beta krm langle orangle nowtextdtau mathcal pxmathbb nrightarrow ymathbb pyyint ndelta yyxmathcal pxxtextdxlangle yyxrangle jin left nright beginalignedmathcal pjvec plangle prangle pwtextdtau pebeta htextdtau utextd pefrac msum npi textdp pnint nefrac pnendaligned ineq pfrac mpj pjint xvec fvec xtextd xfvec eax xint textdx leftint textdxright arightfrac pleftfrac rightfrac tfrac lambda pvint pdelta leftvfrac pmrighttextd pmmdelta pmv pvleftfrac mdelta pmvtextdp mefrac mmv theta pvsqrt trightfrac ttheta wikimedia foundation powered mediawiki
Mobile SEO de.wikipedia.org/wiki/maxwell-boltzmann-verteilung
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SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
man found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
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Navigation and internal links
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statistiken
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zipfmandelbrot
conwaymaxwellpoisson
discretephasetype
erweitert negativ binomial
parabolischfraktal
verallgemeinert invers gau
invers chiquadrat
scaleinvers chiquadrat
relativistisch breitwigner
shifted gompertz
truncated normal
exponential power
generalized hyperbolic
hyperbolicsecant
normalinvers gausch
dirichlet compound multinomial
gemischt multinomial
multivariat poisson
negativmultinomial
generalized dirichlet
multivariat student
normalskaliert invers gamma
poissondirichlet
matrix invers beta
matrix invers gamma
normalinverswishart
normalwishart
permanenter link
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versionsgeschichte
seiteninformationen
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wiki parameter
definitionsbereich
wahrscheinlichkeitsdichte
kumulierte verteilungsfunktion
erwartungswert
modus
varianz
quadratisch gemittelte geschwindigkeit
schiefe
wlbung
entropie
eulermascheronikonstante
statistischen physik
thermodynamik
kinetischen gastheorie
ideales gas
thermodynamisches gleichgewicht
james clerk maxwell
ludwig boltzmann
boltzmann
reale gase
kinetische energie
boltzmannfaktor
boltzmannkonstante
absolute temperatur
temperatur
kelvin
gasteilchen
geschwindigkeit
normal gau
integrieren
kugelschale
infinitesimaler
wasserstoff
helium
stickstoff
verteilungskurve
flssigen
festen
aggregatzustand
verdampfung
sublimation
kondensation
resublimation
phasenumwandlung
dynamisches gleichgewicht
phase
druck
sttigungsdampfdruck
phasendiagramm
zustandsgleichung
fundamentalgleichung
thermodynamisches potenzial
tripelpunkt
kritischer punkt
wahrscheinlichkeitsdichte
gesamtzahl
zustandsgleichung idealer gase
integriert
gleichverteilungssatzes
harmonischer mittelwert
hamiltonfunktion
kanonischen zustand
zustandssumme
deltadistribution
gaufunktion
thermischen debrogliewellenlnge
kugelkoordinaten
heavisidesprungfunktion
htheorem
benford
bernoulli
betabinomial
binomial
dirac
diskret uniform
hypergeometrisch
negativ hypergeometrisch
rademacher
verallgemeinert binomial
zipf
zweipunkt
gemischt poisson
geometrisch
logarithmisch
negativ binomial
poisson
verallgemeinert poisson
zeta
beta
cantor
dreieck
trapez
stetig uniform
beta prime
boseeinstein
chi
chiquadrat
erlang
exponential
extremwert
fermidirac
frchet
gamma
gammagamma
hartmanwatson
hotellings tquadrat
hyperexponentiale
invers normal
kolmogorowverteilung
lognormal
nichtzentriert chiquadrat
pareto
rayleigh
rosinrammler
weibull
cauchy
fishersz
fishertippett gumbel
landau
laplace
alphastabil
logistisch
studentschet
voigt
multinomial
multivariat hypergeometrisch
polyhypergeometrisch
dirichlet
multivariat normal
gleichverteilung auf der stiefelmannigfaltigkeit
matrixvonmisesfisherverteilung
wishart
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4faa98a21ac8133ab466999288849492be28b3d1 height: height attribute not set width: width attribute not set description: {\displaystyle v^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd height: height attribute not set width: width attribute not set description: {\displaystyle m} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d height: height attribute not set width: width attribute not set description: {\displaystyle [0,1]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/537ae16524e99d20d8be70b8c1385a7ae4ec8a1b height: height attribute not set width: width attribute not set description: {\displaystyle m_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3 height: height attribute not set width: width attribute not set description: {\displaystyle n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/47ae0f79cc7df81e1b9ef70e61757f6a5ce5c6ac height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {v}}={\sqrt {\frac {2k_{\mathrm {b} }t}{m}}}={\sqrt {\frac {2rt}{m}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7ff7d31693bf8cdb70e164deff544bd0b82c16 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {{\text{d}}p(v)}{{\text{d}}v}}=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1d09340f8f6c1979330c2f23e514e38f243a3b height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {v}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/533d2bd384193514fa66d22294b77adeb3b2ff18 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {v}}:={\frac {v_{1}+v_{2}+v_{3}+\ldots +v_{n}}{n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d5615ffa6233b0d09d5bafafb58a752c1e8de95f height: height attribute not set width: width attribute not set description: {\displaystyle v_{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bea5d6f8136c5a2d4e29747117628548c6db60e2 height: height attribute not set width: width attribute not set description: {\displaystyle n=1,2,3,\ldots ,n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ac408ac56a62abd5f1c69531d1e8903505f445e9 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {v}}=\int _{0}^{\infty }v\,p(v)\,{\text{d}}v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2908c86fdfec3374903b65d85dea44f103705b49 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {v}}={\sqrt {\frac {8k_{\mathrm {b} }t}{\pi m}}}={\sqrt {\frac {8rt}{\pi m}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d237b00f8856eef0c331e3906a3f40520976ec9 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\overline {v^{2}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7396488f55c32167bbec7429f7febeef95c17e29 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\overline {v^{2}}}}:={\sqrt {\frac {v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+\ldots +v_{n}^{2}}{n}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c9886421749ddfdc3055af6b472b97ea7c742913 height: height attribute not set width: width attribute not set description: {\displaystyle pv={\frac {1}{3}}nm{\overline {v^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ef64de358080f69b4d50fed46a622403d67064 height: height attribute not set width: width attribute not set description: {\displaystyle pv=nk_{\mathrm {b} }t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/95b6fa1d215e8d25ac97af04f9aea149ac5c9e88 height: height attribute not set width: width attribute not set description: {\displaystyle pv} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/29adb1787a6a64c6ed46b61407daec7390dfa2fd height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{3}}nm{\overline {v^{2}}}=nk_{\mathrm {b} }t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d80943460720b13201aad4d5d899f441468a254 height: height attribute not set width: width attribute not set description: {\displaystyle {\overline {v^{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/edd23806750efdd94dcc9ac353dcc47fdf367139 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\overline {v^{2}}}}={\sqrt {\frac {3k_{\mathrm {b} }t}{m}}}={\sqrt {\frac {3rt}{m}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd0980464ff734c5e88e1b218ad06b5d257ff70 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {2}}\approx 1{,}4} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21 height: height attribute not set width: width attribute not set description: {\displaystyle \infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5d49e4796010817ea731764f920bbcb1dcacada3 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\overline {v^{2}}}}={\sqrt {\int _{0}^{\infty }v^{2}\,p(v)\,dv}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9a30fca04341318fe926c428c2744550a5c14497 height: height attribute not set width: width attribute not set description: {\displaystyle {\overline {e_{\text{kin}}}}={\frac {1}{2}}m{\overline {v^{2}}}={\frac {3}{2}}k_{\mathrm {b} }t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ab951d779fcdeea3ec188bb5c73518c46b19de height: height attribute not set width: width attribute not set description: {\displaystyle n=3} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/08191db9bddb156391098a71d4700ab01912ef7b height: height attribute not set width: width attribute not set description: {\displaystyle {\breve {v}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/acd0b7ddb885f34a67237e1e9478c926d38a376f height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{\breve {v}}}:={\frac {1}{n}}\cdot \left({\frac {1}{v_{1}}}+{\frac {1}{v_{2}}}+{\frac {1}{v_{3}}}+\ldots +{\frac {1}{v_{n}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c0551c3264089a2a6eeba7b0977e79bd3d509959 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{\breve {v}}}=\int _{0}^{\infty }{\frac {p(v)}{v}}\,{\text{d}}v} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9150724d2938a2755657b46542e2611085f4c5e6 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {mv^{2}}{2k_{\mathrm {b} }t}}=z\quad } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/054f22efa8278b40f834a584af11106ff47c4785 height: height attribute not set width: width attribute not set description: {\displaystyle \quad v~dv={\frac {k_{\mathrm {b} }t}{m}}dz} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef9471a12f9e2d9327f20ff169f93ba85ac22c5 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{\breve {v}}}={\sqrt {\frac {2}{\pi }}}\cdot {\sqrt {\frac {m}{k_{\mathrm {b} }t}}}={\sqrt {\frac {2m}{\pi k_{\mathrm {b} }t}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e8eda97cb7ef2149b041464b336c80dc8cf05c56 height: height attribute not set width: width attribute not set description: {\displaystyle {\breve {v}}={\sqrt {\frac {\pi k_{\mathrm {b} }t}{2m}}}={\sqrt {\frac {\pi rt}{2{\text{m}}}}}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/maxwell-distr.png/220px-maxwell-distr.png height: 162 width: 220 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2790b6cafab4cc0f98a9ae8beb550947e60062f1 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {v}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/64883a97fcf614e35f5158db7afe85e90b0154e9 height: height attribute not set width: width attribute not set description: {\displaystyle {\hat {v}}<{\bar {v}}<{\sqrt {\overline {v^{2}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/647a3ea6445588a150f4139a263b5276a6a1c000 height: height attribute not set width: width attribute not set description: {\displaystyle {\bar {v}}={\sqrt {\frac {8}{3\pi }}}\cdot {\sqrt {\overline {v^{2}}}}\approx 0{,}921\cdot {\sqrt {\overline {v^{2}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8e50dc898b2f87ea36dddd650ce3219c77346f height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\sqrt {\pi }}{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4a293151cefb8e309fce3c55a81cf14e0dc25e8e height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {2}{3}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbcaeeb1536c34a673c94f59901a0fab054b708 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {2}{\sqrt {\pi }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8252273cd5c6a67190fe302c1ed49675c1dedf height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {8}{3\pi }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/97a23c0217f06a368bacaba22afe66b83374f3b3 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {4}{\pi }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2b64fed8bea2ac8743eb5e34fcc0a87a47317bea height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {3}{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e4780c326a5328c5188cf2eaa1517f2bf269e9e0 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {3\pi }{8}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/72e8ac1e0d7403564f2d2dd742b9b4d12ea7b7d0 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {6}{\pi }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89d7c88c1c93dce69a46052a8e276e231063de height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\pi }{4}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e87f39d296811cec9a68f75d149e12331fbd63ef height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {\pi }{6}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c23ef3378981f4ac807ae462aaf9304321726449 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {k_{\mathrm {b} }t}{m}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/77c7d598181c6fea69b3605a209758e5066b952c height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {8}{\pi }}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {3}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c6a52301c9e0601ac2a1e88232c13cf7e8f308 height: height attribute not set width: width attribute not set description: {\displaystyle {\sqrt {\frac {\pi }{2}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/995aa7049084304ca5592a99cc221515bef20cb0 height: height attribute not set width: width attribute not set description: {\displaystyle h=\sum _{i=1}^{n}{\frac {p_{i}^{2}}{2m}}+u({\vec {x}}_{1},\ldots ,{\vec {x}}_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025 height: height attribute not set width: width attribute not set description: {\displaystyle u} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2 height: height attribute not set width: width attribute not set description: {\displaystyle p_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aea5688a1c685024f0040a93221e8a7db7890c7c height: height attribute not set width: width attribute not set description: {\displaystyle w={\frac {1}{z(\beta )}}e^{-\beta h({\vec {x}}_{1},{\vec {p}}_{1},\,\ldots \,,{\vec {x}}_{n},{\vec {p}}_{n})}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bafff0fe0954fb055a672f76ccd9be0060687f70 height: height attribute not set width: width attribute not set description: {\displaystyle z(\beta )=\int _{\mathbb {r} ^{6n}}e^{-\beta h({\vec {x}}_{1},{\vec {p}}_{1},\,\ldots \,,{\vec {x}}_{n},{\vec {p}}_{n})}{\text{d}}\tau } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4bed56d9d797ab3e3736ffc10748cb54682e19 height: height attribute not set width: width attribute not set description: {\displaystyle {\text{d}}\tau ={\frac {1}{n!(2\pi \hbar )^{3n}}}\;{\text{d}}^{3}x_{1}{\text{d}}^{3}p_{1}\,\ldots \,{\text{d}}^{3}x_{n}{\text{d}}^{3}p_{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8 height: height attribute not set width: width attribute not set description: {\displaystyle \beta } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2030de8abb8307c014a848329a569911e25da071 height: height attribute not set width: width attribute not set description: {\displaystyle \beta ={\frac {1}{k_{\rm {b}}t}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/832148db32ac78dfb97dd251c0e1b6c477cd26b2 height: height attribute not set width: width attribute not set description: {\displaystyle \langle o\rangle =\int _{\mathbb {r} ^{6n}}o\,w\,{\text{d}}\tau } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab height: height attribute not set width: width attribute not set description: {\displaystyle x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2851b89ebafc8bd515866e6d3e83a8baab6f18fc height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}_{x}:\;\mathbb {r} ^{n}\rightarrow \mathbb {r} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6554bc466d3ea4aeffb97fccdb420601dc973054 height: height attribute not set width: width attribute not set description: {\displaystyle y:\;\mathbb {r} ^{n}\rightarrow \mathbb {r} ^{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b55b9dce5b24e1c772e99a8c9545dfe34c46dabc height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}_{y}(y)=\int _{\mathbb {r} ^{n}}\delta (y-y(x))\,{\mathcal {p}}_{x}(x)\,{\text{d}}x=\langle \delta (y-y(x))\rangle } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f height: height attribute not set width: width attribute not set description: {\displaystyle y} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b height: height attribute not set width: width attribute not set description: {\displaystyle {\vec {p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/62005a0ac26a2039868233b23adb168a6046b20f height: height attribute not set width: width attribute not set description: {\displaystyle j\in \left\{1,\ldots ,n\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fde68acce6dc02569eb61a407d3cb75658800eb7 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\mathcal {p}}_{{\vec {p}}_{j}}({\vec {p}})&=\langle \delta ({\vec {p}}_{j}-{\vec {p}})\rangle =\int _{\mathbb {r} ^{6n}}\delta ({\vec {p}}_{j}-{\vec {p}})\,w\,{\text{d}}\tau ={\frac {1}{z(\beta )}}\int _{\mathbb {r} ^{6n}}\delta ({\vec {p}}_{j}-{\vec {p}})\,e^{-\beta h}\,{\text{d}}\tau \\&={\frac {\int _{\mathbb {r} ^{6n}}{\delta }({\vec {p}}_{j}-{\vec {p}})\,e^{-\beta h}\,{\text{d}}\tau }{\int _{\mathbb {r} ^{6n}}{e^{-\beta h}}\,{\text{d}}\tau }}\\&={\frac {\int _{\mathbb {r} ^{3n}}{e^{-\beta u}\,{\text{d}}^{3}x_{1}\,\ldots \,{\text{d}}^{3}x_{n}\;}\ \int _{\mathbb {r} ^{3n}}{\delta ({\vec {p}}_{j}-{\vec {p}})\,e^{-{\frac {\beta }{2m}}\sum _{i=1}^{n}{p_{i}^{2}}}\,{\text{d}}p_{1}\,\ldots \,{\text{d}}^{3}p_{n}\;}}{\int _{\mathbb {r} ^{3n}}{e^{-\beta u}\,{\text{d}}^{3}x_{1}\,\ldots \,{\text{d}}^{3}x_{n}\;}\ \int _{\mathbb {r} ^{3n}}{e^{-{\frac {\beta }{2m}}\sum _{i=1}^{n}{p_{i}^{2}}}\,{\text{d}}^{3}p_{1}\,\ldots \,{\text{d}}^{3}p_{n}\;}}}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be height: height attribute not set width: width attribute not set description: {\displaystyle i\neq j} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/499e0821b28c43e9bc2a6360b937de535057bc62 height: height attribute not set width: width attribute not set description: {\displaystyle p_{j}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9c27a78f41e1267b371321e19f7a50cf02f277bf height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}_{{\vec {p}}_{j}}({\vec {p}})={\frac {\int _{\mathbb {r} ^{3}}\delta ({\vec {p}}_{j}-{\vec {p}})\,e^{-{\frac {\beta }{2m}}p_{j}^{2}}\,{\text{d}}^{3}p_{j}}{\int _{\mathbb {r} ^{3}}e^{-{\frac {\beta }{2m}}p_{j}^{2}}{\text{d}}^{3}p_{j}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92a921350c3a96dbcbccef6ea4084bb1c54af height: height attribute not set width: width attribute not set description: {\displaystyle \int _{\mathbb {r} ^{3}}{\delta }({\vec {x}}-{\vec {x}}_{0})f({\vec {x}})\,{\text{d}}^{3}x=f({\vec {x}}_{0})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c44d18c50159346e693f9dd9476acdca0133a248 height: height attribute not set width: width attribute not set description: {\displaystyle x^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0f260c9a311f3c25bf8db0d7b999c5ca26932e01 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{\mathbb {r} ^{3}}{e^{-ax^{2}}{\text{d}}^{3}x}=\int _{\mathbb {r} ^{3}}{e^{-ax_{1}^{2}}e^{-ax_{2}^{2}}e^{-ax_{3}^{2}}{\text{d}}x_{1}{\text{d}}x_{2}{\text{d}}x_{3}}=\left(\int _{-\infty }^{\infty }{e^{-ax^{2}}{\text{d}}x}\right)^{3}\,=\left({\frac {\pi }{a}}\right)^{\frac {3}{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3305ef561c746aa29a646218a57510dee4f32923 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}_{{\vec {p}}_{j}}({\vec {p}})=\left({\frac {\beta }{2m\pi }}\right)^{\frac {3}{2}}e^{-{\frac {\beta }{2m}}p^{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/86d8dea1f980e78303926f264cfea49590cc83fc height: height attribute not set width: width attribute not set description: {\displaystyle ({\tfrac {\beta }{2m\pi }})^{3/2}=({\tfrac {\lambda }{h}})^{3}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a height: height attribute not set width: width attribute not set description: {\displaystyle \lambda } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8798c5a5eb46b85c191995604acbe103195fdfd8 height: height attribute not set width: width attribute not set description: {\displaystyle v=|{\vec {p}}|/m} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a35f9835048b907471303728bc721cec349c4d height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}(v)=\int _{\mathbb {r} ^{3}}{\mathcal {p}}_{{\vec {p}}_{j}}({\vec {p}})\,\delta \left(v-{\frac {|{\vec {p}}|}{m}}\right)\,{\text{d}}^{3}p=\left({\frac {\beta }{2m\pi }}\right)^{\frac {3}{2}}\int _{\mathbb {r} ^{3}}e^{-{\frac {\beta }{2m}}p^{2}}\,\delta \left(v-{\frac {|{\vec {p}}|}{m}}\right)\,{\text{d}}^{3}p} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a7311690da232979835503ba668f5d086dfd1799 height: height attribute not set width: width attribute not set description: {\displaystyle \delta (v-|{\vec {p}}|/m)=m\,\delta (p-mv)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4992a47c80eb76978faa7d14947ea2293634a6 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}(v)=\left({\frac {\beta }{2m\pi }}\right)^{\frac {3}{2}}4\pi \int _{0}^{\infty }p^{2}e^{-{\frac {\beta }{2m}}p^{2}}m\,\delta (p-mv)\,{\text{d}}p} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1da3683bc73a230df1b579c4b8545a865365c113 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}(v)=\left({\frac {\beta }{2m\pi }}\right)^{\frac {3}{2}}4\pi (mv)^{2}m\,e^{-{\frac {\beta }{2m}}(mv)^{2}}\theta (v)=\left({\frac {\beta m}{2\pi }}\right)^{\frac {3}{2}}4\pi v^{2}e^{-\beta {\frac {m}{2}}v^{2}}\theta (v)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c5eb61492428f2e886c693e256c68e2f927e88d5 height: height attribute not set width: width attribute not set description: {\displaystyle \theta (v)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/14611a8c666763d4d172107be385d4da7623b1ca height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {p}}(v)={\sqrt {\frac {2}{\pi }}}\left({\frac {m}{k_{\mathrm {b} }t}}\right)^{\frac {3}{2}}v^{2}\,e^{-{\frac {mv^{2}}{2k_{\mathrm {b} }t}}}\,\theta (v)} |
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