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| {{Hide in print|1={{Probability distribution|
| | == ヤンの三種類に達するの「 '3'色 'それは == |
| name =Maxwell–Boltzmann|
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| type =density|
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| pdf_image = [[Image:Maxwell-Boltzmann distribution pdf.svg|325px]]|
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| cdf_image =[[Image:Maxwell-Boltzmann distribution cdf.svg|325px]]|
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| parameters =<math>a>0\,</math>|
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| support =<math>x\in [0;\infty)</math>|
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| pdf =<math>\sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3}</math>|
| |
| cdf =<math>\textrm{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \frac{x e^{-x^2/(2a^2)}}{a} </math> where erf is the [[Error function]] |
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| mean =<math>\mu=2a \sqrt{\frac{2}{\pi}}</math>|
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| median =|
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| mode =<math>\sqrt{2} a</math>|
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| variance =<math>\sigma^2=\frac{a^2(3 \pi - 8)}{\pi}</math>|
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| skewness =<math>\gamma_1=\frac{2 \sqrt{2} (16 -5 \pi)}{(3 \pi - 8)^{3/2}}</math>|
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| kurtosis =<math>\gamma_2=4\frac{(-96+40\pi-3\pi^2)}{(3 \pi - 8)^2}</math>|
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| entropy =<math>\frac{1}{2}-\gamma-\ln(a\sqrt{2\pi})</math>|
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| mgf =|
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| char =|
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| }}
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| }}
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| {{distinguish|Maxwell–Boltzmann statistics}}
| | それはすぐに、赤色の「色」のタッチは静か分より弱多く、急速な拡大の最後の激怒の目を表面化、ナマの赤」の色が [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ] '他の二つである、再びローリング雷雲ですヤンかなり「色」。<br><br>色」！ヤンの三種類に達するの「 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 ソーラー] '3'色 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 時計] 'それは！'<br><br>が完全に見えた三万円、「カラー」、少し静かな広場を浮上し、突然沸騰し、再び、多くの人が感動を流し直面し、時間のこの短い期間に一日は、彼らがある3 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 時計 価格] 3「カラー」Danlei、この壮大な他のを見て、本当にまれな世紀と見なされる！<br><br>このその後、Danleiの3ヤン「色」の存在を見て、高齢者のムーの骨が3つだけ「カラー」Danleiであれば、高いプラットフォームは、いくつかのDanta長老の心は、非常に緊張したアップとなっている回償還が奇数シャオヤンと清華は、競争する強さを持って存在する |
| | | 相关的主题文章： |
| In [[physics]], particularly [[statistical mechanics]], the '''Maxwell–Boltzmann distribution''' or '''Maxwell speed distribution''' describes particle speeds in idealized [[gas]]es where the particles move freely inside a stationary container without interacting with one another, except for very brief [[collision]]s in which they exchange energy and momentum with each other or with their thermal environment. Particle in this context refers to either gaseous [[atoms]] or [[molecules]], and the system of particles is assumed to have reached [[thermodynamic equilibrium]].<ref name="StatisticalPhysics">''Statistical Physics'' (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, ISBN 9780471915331</ref>
| | <ul> |
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| The distribution is a [[probability distribution]] for the ''speed'' of a particle within the gas - the [[Magnitude (mathematics)|magnitude]] of its [[velocity]]. This probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The distribution depends on the [[temperature]] of the system and the mass of the particle.<ref>University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, ISBN (10-) 0-321-50130-6, ISBN (13-) 978-0-321-50130-1</ref>
| | <li>[http://www.xinruihuayuan.com/home.php?mod=space&uid=91961 http://www.xinruihuayuan.com/home.php?mod=space&uid=91961]</li> |
| | | |
| The Maxwell–Boltzmann distribution applies to the classical [[ideal gas]], which is an idealization of real gases. In real gases, there are various effects (e.g., [[van der Waals interaction]]s, [[special relativity|relativistic]] speed limits, and [[quantum mechanics|quantum]] [[exchange interaction]]s) that make their speed distribution sometimes very different from the Maxwell–Boltzmann form. That said, [[rarefied]] gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. Thus, it forms the basis of the [[kinetic theory of gases]], which provides a simplified explanation of many fundamental gaseous properties, including [[pressure]] and [[diffusion]].<ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>
| | <li>[http://www.xiaopaomuli.com/plus/feedback.php?aid=3 http://www.xiaopaomuli.com/plus/feedback.php?aid=3]</li> |
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| The distribution is named after [[James Clerk Maxwell]] and [[Ludwig Boltzmann]]. While the distribution was first derived by Maxwell in 1860 on basic grounds,<ref>Maxwell, J.C. (1860) Illustrations of the dynamical theory of gases. ''Philosophical Magazine'' 19, 19-32 and ''Philosophical Magazine'' 20, 21-37.</ref> Boltzmann later carried out significant investigations into the physical origins of this distribution.
| | <li>[http://jirostyle.com/bbs/index.cgi http://jirostyle.com/bbs/index.cgi]</li> |
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| == Distribution function ==
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| [[Image:MaxwellBoltzmann-en.svg|right|thumb|360px|The speed probability density functions of the speeds of a few [[noble gas]]es at a temperature of 298.15 K (25 °C). The ''y''-axis is in s/m so that the area under any section of the curve (which represents the probability of the speed being in that range) is dimensionless.]]
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| The Maxwell–Boltzmann distribution is the function
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| :<math> f(v) = \sqrt{\left(\frac{m}{2 \pi kT}\right)^3}\, 4\pi v^2 \exp \left(- \frac{mv^2}{2kT}\right), </math>
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| where <math>m</math> is the particle mass and <math>kT</math> is the product of [[Boltzmann's constant]] and [[thermodynamic temperature]].
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| This [[probability density function]] gives the probability, per unit speed, of finding the particle with a speed near <math>v</math>. This equation is simply the Maxwell distribution (given in the infobox) with distribution parameter <math>a=\sqrt{kT/m}</math>. In probability theory the Maxwell–Boltzmann distribution is a [[chi distribution]] with three degrees of freedom and [[scale parameter]] <math>a=\sqrt{kT/m}</math>.
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| ==Typical speeds==
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| The mean speed, most probable speed (mode), and root-mean-square can be obtained from properties of the Maxwell distribution.
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| <ul>
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| <li>
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| The most probable speed, ''v''<sub>''p''</sub>, is the speed most likely to be possessed by any molecule (of the same mass ''m'') in the system and corresponds to the maximum value or [[mode (statistics)|mode]] of ''f''(''v''). To find it, we calculate ''df''/''dv'', set it to zero and solve for ''v'':
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| :<math>\frac{df(v)}{dv} = 0</math>
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| which yields:
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| :<math>v_p = \sqrt { \frac{2kT}{m} } = \sqrt { \frac{2RT}{M} }</math>
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| where ''R'' is the [[gas constant]] and ''M'' = [[Avogadro constant|''N<sub>A</sub>'']] ''m'' is the [[molar mass]] of the substance.
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| For diatomic nitrogen (N<sub>2</sub>, the primary component of [[air]]) at [[room temperature]] (300 [[degrees Kelvin|K]]), this gives <math>v_p = 422 </math>m/s
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| </li>
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| <li>The mean speed is the mathematical average of the speed distribution
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| :<math> \langle v \rangle = \int_0^{\infty} v \, f(v) \, dv= \sqrt { \frac{8kT}{\pi m}}= \sqrt { \frac{8RT}{\pi M}} = \frac{2}{\sqrt{\pi}} v_p </math>
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| </li>
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| <li> | |
| The [[root mean square speed]] is the square root of the average squared speed:
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| :<math> \sqrt{\langle v^2 \rangle} = \left(\int_0^{\infty} v^2 \, f(v) \, dv \right)^{1/2}= \sqrt { \frac{3kT}{m}}= \sqrt { \frac{3RT}{M} } = \sqrt{ \frac{3}{2} } v_p </math> | |
| </li> | |
| </ul>
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| The typical speeds are related as follows:
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| :<math> 0.886 \langle v \rangle = v_p < \langle v \rangle < \sqrt{\langle v^2 \rangle} = 1.085 \langle v \rangle. </math>
| | </ul> |
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| ==Derivation and related distributions==
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| The original derivation by [[James Clerk Maxwell|Maxwell]] assumed all three directions would behave in the same fashion, but a later derivation by [[Boltzmann]] dropped this assumption using [[kinetic theory]]. The Maxwell–Boltzmann distribution (for energies) can now most readily be derived from the [[Boltzmann distribution]] for energies (see also the [[Maxwell–Boltzmann statistics]] of [[statistical mechanics]]):<ref name="StatisticalPhysics" /><ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
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| {{NumBlk|:|<math>
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| \frac{N_i}{N} = \frac{g_i \exp\left(-E_i/kT \right) } { \sum_{j}^{} g_j \,{\exp\left(-E_j/kT\right)} }</math>
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| |{{EquationRef|1}}}}
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| where:
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| * ''i'' is the [[Microstate (statistical mechanics)|microstate]] (indicating one configuration particle [[quantum state]]s - see [[Partition function (statistical mechanics)|partition function]]).
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| * ''E''<sub>''i''</sub> is the energy level of microstate ''i''.
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| * ''T'' is the equilibrium temperature of the system.
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| * ''g<sub>i</sub>'' is the degeneracy factor, or number of [[Degenerate energy level|degenerate]] microstates which have the same energy level
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| * ''k'' is the [[Boltzmann constant]].
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| * ''N''<sub>''i''</sub> is the number of molecules at equilibrium temperature ''T'', in a state ''i'' which has energy ''E''<sub>''i''</sub> and degeneracy ''g<sub>i</sub>''.
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| * ''N'' is the total number of molecules in the system.
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| Note that sometimes the above equation is written without the degeneracy factor ''g''<sub>''i''</sub>. In this case the index ''i'' will specify an individual state, rather than a set of ''g''<sub>''i''</sub> states having the same energy ''E''<sub>''i''</sub>. Because velocity and speed are related to energy, Equation ({{EquationNote|1}}) can be used to derive relationships between temperature and the speeds of molecules in a gas. The denominator in this equation is known as the canonical [[partition function (statistical mechanics)|partition function]].
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| ===Distribution for the momentum vector===
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| The following is a derivation wildly different from the derivation described by [[James Clerk Maxwell]] and later described with fewer assumptions by [[Ludwig Boltzmann]]. Instead it is close to Boltzmann's later approach of 1877.
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| For the case of an "ideal gas" consisting of non-interacting atoms in the ground state, all energy is in the form of kinetic energy, and ''g''<sub>i</sub> is constant for all ''i''. The relationship between [[Kinetic energy#Kinetic energy of rigid bodies|kinetic energy and momentum]] for massive particles is
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| {{NumBlk|:|<math>E=\frac{p^2}{2m}</math>|{{EquationRef|2}}}}
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| where ''p''<sup>2</sup> is the square of the momentum vector
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| '''p''' = [''p''<sub>''x''</sub>, ''p''<sub>''y''</sub>, ''p''<sub>''z''</sub>]. We may therefore rewrite Equation ({{EquationNote|1}}) as:
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| {{NumBlk|:|<math>
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| \frac{N_i}{N} =
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| \frac{1}{Z}
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| \exp \left[
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| -\frac{p_{i, x}^2 + p_{i, y}^2 + p_{i, z}^2}{2mkT}
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| \right]</math>
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| |{{EquationRef|3}}}}
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| where ''Z'' is the [[partition function (statistical mechanics)|partition function]], corresponding to the denominator in Equation ({{EquationNote|1}}). Here ''m'' is the molecular mass of the gas, ''T'' is the thermodynamic temperature and ''k'' is the [[Boltzmann constant]]. This distribution of ''N''<sub>'''i'''</sub>/''N'' is [[Proportionality (mathematics)|proportional]] to the [[probability density function]] ''f''<sub>'''p'''</sub> for finding a molecule with these values of momentum components, so:
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| {{NumBlk|:|<math>
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| f_\mathbf{p} (p_x, p_y, p_z) =
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| \frac{c}{Z}
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| \exp \left[
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| -\frac{p_x^2 + p_y^2 + p_z^2}{2mkT}
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| \right]</math>|{{EquationRef|4}}}}
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| The [[normalizing constant]] ''c'', can be determined by recognizing that the probability of a molecule having ''some'' momentum must be 1. Therefore the integral of equation ({{EquationNote|4}}) over all ''p''<sub>''x''</sub>, ''p''<sub>''y''</sub>, and ''p''<sub>''z''</sub> must be 1.
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| It can be shown that:
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| {{NumBlk|:|<math>
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| c = \frac{Z}{(2 \pi mkT)^{3/2}}</math>|{{EquationRef|5}}}}
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| Substituting Equation ({{EquationNote|5}}) into Equation ({{EquationNote|4}}) gives:
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| {{NumBlk|:|<math>
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| f_\mathbf{p} (p_x, p_y, p_z) =
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| \left( \frac{1}{2 \pi mkT} \right)^{3/2}
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| \exp \left[
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| -\frac{p_x^2 + p_y^2 + p_z^2}{2mkT}
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| \right]</math>|{{EquationRef|6}}}}
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| The distribution is seen to be the product of three independent [[normal distribution|normally distributed]] variables <math>p_x</math>, <math>p_y</math>, and <math>p_z</math>, with variance <math>mkT</math>. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with <math>a=\sqrt{mkT}</math>.
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| The Maxwell–Boltzmann distribution for the momentum (or equally for the velocities) can be obtained more fundamentally using the [[H-theorem]] at equilibrium within the [[kinetic theory]] framework.
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| ===Distribution for the energy===
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| Using ''p''² = 2''mE'', and the distribution function for the magnitude of the momentum (see [[#Distribution for the speed|below]]), we get the energy distribution:
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| {{NumBlk|:|<math>
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| f_E\,dE=f_p\left(\frac{dp}{dE}\right)\,dE =2\sqrt{\frac{E}{\pi}} \left(\frac{1}{kT} \right)^{3/2}\exp\left[\frac{-E}{kT}\right]\,dE.
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| </math>|{{EquationRef|7}}}}
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| Since the energy is proportional to the sum of the squares of the three normally distributed momentum components, this distribution is a [[gamma distribution]]; in particular, it is a [[chi-squared distribution]] with three degrees of freedom.
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| By the [[equipartition theorem]], this energy is evenly distributed among all three degrees of freedom, so that the energy per degree of freedom is distributed as a chi-squared distribution with one degree of freedom:<ref>{{cite book
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| |title=Statistical thermodynamics: fundamentals and applications
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| |first1=Normand M.
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| |last1=Laurendeau
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| |publisher=Cambridge University Press
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| |year=2005
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| |isbn=0-521-84635-8
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| |page=434
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| |url=http://books.google.com/books?id=QF6iMewh4KMC}}, [http://books.google.com/books?id=QF6iMewh4KMC&pg=PA434 Appendix N, page 434]
| |
| </ref>
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| :<math>
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| f_\epsilon(\epsilon)\,d\epsilon= \sqrt{\frac{1}{\epsilon \pi kT}}~\exp\left[\frac{-\epsilon}{kT}\right]\,d\epsilon
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| </math>
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| where <math>\epsilon</math> is the energy per degree of freedom. At equilibrium, this distribution will hold true for any number of degrees of freedom. For example, if the particles are rigid mass dipoles, they will have three translational degrees of freedom and two additional rotational degrees of freedom. The energy in each degree of freedom will be described according to the above chi-squared distribution with one degree of freedom, and the total energy will be distributed according to a chi-squared distribution with five degrees of freedom. This has implications in the theory of the [[specific heat]] of a gas.
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| The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a type of [[gas in a box|quantum gas]].
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| ===Distribution for the velocity vector===
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| Recognizing that the velocity probability density ''f''<sub>'''v'''</sub> is proportional to the momentum probability density function by
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| :<math>
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| f_\mathbf{v} d^3v = f_\mathbf{p} \left(\frac{dp}{dv}\right)^3 d^3v
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| </math>
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| and using '''p''' = m'''v''' we get
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| :<math>
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| f_\mathbf{v} (v_x, v_y, v_z) =
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| \left(\frac{m}{2 \pi kT} \right)^{3/2}
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| \exp \left[-
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| \frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT}
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| \right],
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| </math>
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| which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element [''dv''<sub>''x''</sub>, ''dv''<sub>''y''</sub>, ''dv''<sub>''z''</sub>] about velocity '''v''' = [''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, ''v''<sub>''z''</sub>] is
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| :<math>
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| f_\mathbf{v} \left(v_x, v_y, v_z\right)\, dv_x\, dv_y\, dv_z.
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| </math>
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| Like the momentum, this distribution is seen to be the product of three independent [[normal distribution|normally distributed]] variables <math>v_x</math>, <math>v_y</math>, and <math>v_z</math>, but with variance <math>\frac{kT}{m}</math>. It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity
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| [''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, ''v''<sub>''z''</sub>] is the product of the distributions for each of the three directions:
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| :<math>
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| f_v \left(v_x, v_y, v_z\right) = f_v (v_x)f_v (v_y)f_v (v_z)
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| </math>
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| where the distribution for a single direction is
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| :<math>
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| f_v (v_i) =
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| \sqrt{\frac{m}{2 \pi kT}}
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| \exp \left[
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| \frac{-mv_i^2}{2kT}
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| \right].
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| </math>
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| Each component of the velocity vector has a [[normal distribution]] with mean <math>\mu_{v_x} = \mu_{v_y} = \mu_{v_z} = 0</math> and standard deviation <math>\sigma_{v_x} = \sigma_{v_y} = \sigma_{v_z} = \sqrt{\frac{kT}{m}}</math>, so the vector has a 3-dimensional normal distribution, also called a "multinormal" distribution, with mean <math> \mu_{\mathbf{v}} = {\mathbf{0}} </math> and standard deviation <math>\sigma_{\mathbf{v}} = \sqrt{\frac{3kT}{m}}</math>.
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| The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is
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| :<math>v = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>
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| and the increment of volume is
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| :<math> dv_x\, dv_y\, dv_z = v^2 \sin \theta\, dv\, d\theta\, d\phi </math>
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| where <math>\phi</math> and <math>\theta</math> are the "course" (azimuth of the velocity vector) and "path angle" (elevation angle of the velocity vector). Integration of the normal probability density function of the velocity, above, over the course (from 0 to <math>2\pi</math>) and path angle (from 0 to <math>\pi</math>), with substitution of the speed for the sum of the squares of the vector components, yields the speed distribution.
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| ==See also==
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| * [[Maxwell–Boltzmann statistics]]
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| * [[Maxwell-Jüttner distribution]]
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| * [[Boltzmann distribution]]
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| * [[Boltzmann factor]]
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| * [[Rayleigh distribution]]
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| * [[Ideal gas law]]
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| * [[James Clerk Maxwell]]
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| * [[Boltzmann|Ludwig Eduard Boltzmann]]
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| * [[Kinetic theory]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7
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| * Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009, ISBN (13-) 978-1-4200-7368-3
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| * Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
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| * Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974, ISBN 0-201-05229-6
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| * Ward, CA & Fang, G 1999, 'Expression for predicting liquid evaporation flux: Statistical rate theory approach', Physical Review E, vol. 59, no. 1, pp. 429-40.
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| * Rahimi, P & Ward, CA 2005, 'Kinetics of Evaporation: Statistical Rate Theory Approach', Int. J. of Thermodynamics, vol. 8, no. 9, pp. 1-14.
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| ==External links==
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| * [http://demonstrations.wolfram.com/TheMaxwellSpeedDistribution/ "The Maxwell Speed Distribution"] from The Wolfram Demonstrations Project at [[Mathworld]]
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{DEFAULTSORT:Maxwell-Boltzmann Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Gases]]
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| [[Category:James Clerk Maxwell]]
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| [[Category:Normal distribution]]
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| [[Category:Particle distributions]]
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