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| {{for|other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}}
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| {{redirect|BTE}}
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| In [[physics]], specifically [[non-equilibrium statistical mechanics]], the '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in [[thermodynamic equilibrium]]. It was devised by [[Ludwig Boltzmann]] in 1872.<ref name="Encyclopaediaof">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>
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| The classic example is a [[fluid]] with [[temperature gradient]]s in space causing heat to flow from hotter regions to colder ones, by the random (and biased) transport of [[particle]]s. In the modern literature the term Boltzmann equation is often used in a more general sense and refers to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. | |
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| The equation arises not by [[statistical analysis]] of all the individual [[position vector|position]]s and [[momenta]] of each particle in the fluid; rather by considering the [[probability]] that a number of particles all occupy a [[infinitesimal|very small]] region of space (mathematically written ''d''<sup>3</sup>'''r''', where ''d'' means "[[differential of a function|differential]]", a very small change) centered at the tip of the position vector '''r''', and have very nearly equal small changes in momenta from a momentum vector '''p''', at an instant of time.
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| The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport, and other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]] also [[electrical conductivity]] (by treating the charge carriers in a material as a gas) can be derived.<ref name="Encyclopaediaof" /> See also [[convection-diffusion equation]].
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| The equation is a [[linear differential equation|linear]] [[stochastic partial differential equation]], since the function to solve the equation for is a [[continuous random variable]]. In fact - the problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.<ref>{{cite journal | last1=DiPerna | first1= R. J. |last2 = Lions | first2 = P.-L. | title= On the Cauchy problem for Boltzmann equations: global existence and weak stability | journal= Ann. Of Math. (2) | volume=130 | pages= 321–366 | year=1989 | doi=10.2307/1971423 | issue=2}}
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| </ref><ref>{{cite journal |author= Philip T. Gressman and Robert M. Strain |year=2010 |title= Global classical solutions of the Boltzmann equation with long-range interactions |journal= Proceedings of the National Academy of Sciences |volume=107 |pages= 5744–5749 | doi = 10.1073/pnas.1001185107 |bibcode = 2010PNAS..107.5744G |arxiv = 1002.3639 |issue= 13 }}</ref>
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| ==Overview==
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| ===The phase space and density function===
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| The set of all possible positions '''r''' and momenta '''p''' is called the [[phase space]] of the system; in other words a set of three [[coordinates]] for each position coordinate ''x, y, z'', and three more for each momentum component ''p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written ''d''<sup>3</sup>'''r'''''d''<sup>3</sup>'''p''' = ''dxdydzdp<sub>x</sub>dp<sub>y</sub>dp<sub>z</sub>''.
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| Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within ''d''<sup>3</sup>'''r'''''d''<sup>3</sup>'''p''' is in question, at the heart of the equation is a quantity ''f'' which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time ''t''. This is a [[probability density function]]: ''f''('''r''', '''p''', ''t''), defined so that,
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| :<math>dN = f (\mathbf{r},\mathbf{p},t)\,d^3\mathbf{r}\,d^3\mathbf{p}</math>
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| is the number of molecules which ''all'' have positions lying within a volume element ''d''<sup>3</sup>'''r''' about '''r''' and momenta lying within a [[momentum space]] element ''d''<sup>3</sup>'''p''' about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |location=New York |publisher=Wiley |isbn=0-471-81518-7 |page=53 |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:
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| :<math>N = \int\limits_\mathrm{positions} d^3\mathbf{r} \int\limits_\mathrm{momenta} d^3\mathbf{p} f (\mathbf{r},\mathbf{p},t) = \iiint\limits_\mathrm{positions} \quad \iiint\limits_\mathrm{momenta} f (x,y,z,p_x,p_y,p_z,t) dxdydz dp_xdp_ydp_z
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| </math>
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| which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
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| It is assumed the particles in the system are identical (so each has an identical [[mass]] ''m''). For a mixture of more than one [[chemical species]], one distribution is needed for each, see below.
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| ===Principal statement===
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| The general equation can then be written:<ref name="McGrawHill">McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
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| :<math>\frac{\partial f}{\partial t} = \left(\frac{\partial f}{\partial t}\right)_\mathrm{force} + \left(\frac{\partial f}{\partial t}\right)_\mathrm{diff}+ \left(\frac{\partial f}{\partial t}\right)_\mathrm{coll}</math>
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| where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the [[diffusion]] of particles, and "coll" is the [[collision]] term - accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.<ref name="McGrawHill" />
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| Note that some authors use the particle velocity '''v''' instead of momentum '''p'''; they are related in the definition of momentum by '''p''' = ''m'''''v'''.
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| ==The force and diffusion terms==
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| Consider particles described by ''f'', each experiencing an ''external'' force '''F''' not due to other particles (see the collision term for the latter treatment).
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| Suppose at time ''t'' some number of particles all have position '''r''' within element ''d''<sup>3</sup>'''r''' and momentum '''p''' within ''d''<sup>3</sup>'''p'''. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy
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| :<math>
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| f \left (\mathbf{r}+\frac{\mathbf{p}}{m} \Delta t,\mathbf{p}+\mathbf{F}\Delta t,t+\Delta t \right )\,d^3\mathbf{r}\,d^3\mathbf{p} =
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| f(\mathbf{r},\mathbf{p},t)\,d^3\mathbf{r}\,d^3\mathbf{p}
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| </math>
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| Note that we have used the fact that the phase space volume element ''d''<sup>3</sup>'''r'''''d''<sup>3</sup>'''p''' is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume ''d''<sup>3</sup>'''r'''''d''<sup>3</sup>'''p''' changes, so
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| {{NumBlk|:|
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| <math>\begin{align}
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| dN_\mathrm{coll} & = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}\Delta td^3\mathbf{r} d^3\mathbf{p} \\
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| & = f \left (\mathbf{r}+\frac{\mathbf{p}}{m}\Delta t,\mathbf{p} + \mathbf{F}\Delta t,t+\Delta t \right)d^3\mathbf{r}d^3\mathbf{p}
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| - f(\mathbf{r},\mathbf{p},t)d^3\mathbf{r}d^3\mathbf{p} \\
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| & = \Delta f d^3\mathbf{r}d^3\mathbf{p}
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| \end{align}</math>
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| |{{EquationRef|1}}}}
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| where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by ''d''<sup>3</sup>'''r'''''d''<sup>3</sup>'''p'''Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have
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| {{NumBlk|:|
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| <math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
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| |{{EquationRef|2}}}}
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| The total [[differential of a function|differential]] of ''f'' is:
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| {{NumBlk|:|
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| <math>\begin{align}
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| d f & = \frac{\partial f}{\partial t}dt
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| +\left(\frac{\partial f}{\partial x}dx
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| +\frac{\partial f}{\partial y}dy
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| +\frac{\partial f}{\partial z}dz
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| \right)
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| +\left(\frac{\partial f}{\partial p_x}dp_x
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| +\frac{\partial f}{\partial p_y}dp_y
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| +\frac{\partial f}{\partial p_z}dp_z
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| \right)\\
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| & = \frac{\partial f}{\partial t}dt +\nabla f \cdot d\mathbf{r} + \frac{\partial f}{\partial \mathbf{p}}\cdot d\mathbf{p} \\
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| & = \frac{\partial f}{\partial t}dt +\nabla f \cdot \frac{\mathbf{p}dt}{m} + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F}dt
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| \end{align}</math>
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| |{{EquationRef|3}}}}
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| where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],
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| :<math>
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| \frac{\partial f}{\partial \mathbf{p}} = \mathbf{\hat{e}}_x\frac{\partial f}{\partial p_x} + \mathbf{\hat{e}}_y\frac{\partial f}{\partial p_y}+\mathbf{\hat{e}}_z\frac{\partial f}{\partial p_z}= \nabla_\mathbf{p}f
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| </math>
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| is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|cartesian]] [[unit vector]]s.
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| ===Final statement===
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| Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:
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| :<math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F}\cdot\frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
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| In this context, '''F'''('''r''', ''t'') is the [[Force field (chemistry)|force field]] acting on the particles in the fluid, and ''m'' is the [[mass]] of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation is often mistakenly called the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (the Liouville Equation is a [[Many-body theory|many-particle]] equation).
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| This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved for unless the collision term in ''f'' is known. This term cannot be found as easily or generally as the others - it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the [[Maxwell-Boltzmann distribution|Maxwell-Boltzmann]], [[Fermi-Dirac distribution|Fermi-Dirac]] or [[Bose-Einstein distribution|Bose-Einstein]] distributions.
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| ==The collision term (Stosszahlansatz) and molecular chaos==
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| A key insight applied by [[Ludwig Boltzmann|Boltzmann]] was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "Stosszahlansatz", and is also known as the "[[molecular chaos]] assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:<ref name="Encyclopaediaof" />
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| :<math>
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| \left(\frac{\partial f}{\partial t} \right)_{\mathrm{coll}} = \iint gI(g, \Omega)[f(\mathbf{p'}_A,t) f(\mathbf{p'}_B,t) - f(\mathbf{p}_A,t) f(\mathbf{p}_B,t)] \,d\Omega\,d^3\mathbf{p}_A.
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| </math>
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| where '''p'''<sub>''A''</sub> and '''p'''<sub>''B''</sub> are the momenta of any two particles (labeled as ''A'' and ''B'' for convenience) before a collision, '''p′'''<sub>''A''</sub> and '''p′'''<sub>''B''</sub> are the momenta after the collision,
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| :<math>g = |\mathbf{p}_B - \mathbf{p}_A| = |\mathbf{p'}_B - \mathbf{p'}_A|</math>
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| is the magnitude of the relative momenta (see [[relative velocity]] for more on this concept), and ''I''(''g'', Ω) is the [[differential cross section]] of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the [[solid angle]] ''d''Ω, due to the collision.
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| ==General equation (for a mixture)==
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| For a mixture of chemical species labelled by indices ''i'' = 1,2,3...,''n'' the equation for species ''i'' is:<ref name="Encyclopaediaof" />
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| :<math>\frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i}\cdot\nabla f_i + \mathbf{F}\cdot\frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\mathrm{coll}</math>
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| where ''f<sub>i</sub>'' = ''f<sub>i</sub>''('''r''', '''p'''<sub>''i''</sub>, ''t''), and the collision term is
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| :<math>
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| \left(\frac{\partial f_i}{\partial t} \right)_{\mathrm{coll}} = \sum_{j=1}^n \iint g_{ij} I_{ij}(g_{ij}, \Omega)[f'_i f'_j - f_if_j] \,d\Omega\,d^3\mathbf{p'}.
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| </math>
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| where ''f′'' = ''f′''('''p′'''<sub>''i''</sub>, ''t''), the magnitude of the relative momenta is
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| :<math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|</math>
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| and ''I<sub>ij</sub>'' is the differential cross-section as before, between particles ''i'' and ''j''. The integration is over the momentum components in the integrand (which are labelled ''i'' and ''j''). The sum of integrals describes the entry and exit of particles of species ''i'' in or out of the phase space element.
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| ==Applications and extensions==
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| ===Conservation equations===
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| The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum and energy<ref name="dG1984">{{cite book |last1=de Groot |first1=S.R.|last2=Mazur|first2=P.|title=Non-Equilibrium Thermodynamics |url=http://www.amazon.com/Non-Equilibrium-Thermodynamics-Dover-Books-Physics/dp/0486647412 |accessdate=2013-01-31 |year=1984 |publisher=Dover Publications Inc. |location=New York |isbn=0-486-64741-2}}</ref>{{rp|p 163}}. For a fluid consisting of only one kind of particle, the number density ''n'' is given by:
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| :<math>n=\int f\,d^3p</math>
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| The average value of any function ''A'' is:
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| :<math>\langle A \rangle=\frac{1}{n}\int A f\,d^3p</math>
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| Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus <math>\mathbf{x}\rightarrow x_i</math> and <math>\mathbf{p}\rightarrow p_i = m w_i</math> where <math>w_i</math> is the particle velocity vector. Define <math>g(p_i)</math> as some function of momentum <math>p_i</math> only, which is conserved in a collision. Assume also that the force <math>F_i</math> is a function of position only, and that ''f'' is zero for <math>p_i\rightarrow\pm \infty</math>. Multiplying the Boltzmann equation by ''g'' and integrating over momentum yields four terms which, using integration by parts, can be expressed as:
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| :<math>\int g \frac{\partial f}{\partial t}\,d^3p=\frac{\partial }{\partial t} (n\langle g \rangle)</math>
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| :<math>\int \frac{p_j g}{m}\frac{\partial f}{\partial x_j}\,d^3p=\frac{1}{m}\frac{\partial}{\partial x_j}(n\langle g p_j \rangle)</math>
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| :<math>\int g F_j \frac{\partial f}{\partial p_j}\,d^3p=-nF_j\left\langle \frac{\partial g}{\partial p_j}\right\rangle</math>
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| :<math>\int g \left(\frac{\partial f}{\partial t}\right)_{coll}\,d^3p=0</math>
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| where the last term is zero since ''g'' is conserved in a collision. Letting <math>g=m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation<ref name="dG1984" />{{rp|pp 12,168}}:
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| :<math>\frac{\partial}{\partial t}\rho
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| + \frac{\partial}{\partial x_j}(\rho V_j)
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| =0</math>
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| where <math>\rho=mn</math> is the mass density and <math>V_i=\langle w_i\rangle</math> is the average fluid velocity.
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| Letting <math>g=m w_i</math>, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation<ref name="dG1984" />{{rp|pp 15,169}}:
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| :<math>\frac{\partial}{\partial t}(\rho V_i)
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| + \frac{\partial}{\partial x_j}(\rho V_i V_j+P_{ij})
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| - nF_i=0</math>
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| where <math>P_{ij}=\rho\langle (w_i-V_i) (w_j-V_j) \rangle</math> is the pressure tensor. (The [[viscous stress tensor]] plus the hydrostatic [[pressure]].)
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| Letting <math>g=\tfrac{1}{2}m w_i w_i</math>, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation<ref name="dG1984" />{{rp|pp 19,169}}:
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| :<math>\frac{\partial}{\partial t}(u+\tfrac{1}{2}\rho V_i V_i)
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| + \frac{\partial}{\partial x_j}(uV_j+\tfrac{1}{2}\rho V_i V_i V_j + J_{qj}+P_{ij}V_i)-nF_iV_i
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| =0</math>
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| where <math>u=\tfrac{1}{2}\rho\langle (w_i-V_i) (w_i-V_i) \rangle</math> is the kinetic thermal energy density and <math>J_{qi}=\tfrac{1}{2}\rho\langle (w_i-V_i)(w_k-V_k)(w_k-V_k)\rangle</math> is the heat flux vector.
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| ===Hamiltonian mechanics===
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| In [[Hamiltonian mechanics]], the Boltzmann equation is often written more generally as
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| :<math>\hat{\mathbf{L}}[f]=\mathbf{C}[f], \, </math>
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| where '''L''' is the [[Liouville operator]] describing the evolution of a phase space volume and '''C''' is the collision operator. The non-relativistic form of '''L''' is
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| :<math>\hat{\mathbf{L}}_\mathrm{NR} = \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,.</math>
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| ===Quantum theory and violation of particle number ===
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| It is possible to write down [[theory of relativity|relativistic]] Boltzmann equations for [[theory of relativity|relativistic]] [[quantum systems]] in which the number of particles is not conserved in collisions. This has several applications in [[physical cosmology]],<ref name=KolbTurner>{{cite book|last=Edward Kolb and Michael Turner|title=The Early Universe|year=1990|publisher=Westview Press|isbn=9780201626742}}</ref> including the formation of the light elements in [[big bang nucleosynthesis]], the production of [[dark matter]] and [[baryogenesis]]. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density ''f''. However, for a wide class of applications a well-defined generalization of ''f'' exists which is the solution of an effective Boltzmann equation that can be derived from first principles of [[quantum field theory]].<ref name=BEfromQFT>{{cite journal|last=M. Drewes, C. Weniger, S. Mendizabal|journal=Phys. Lett. B|date=8 January 2013|volume=718|issue=3|pages=1119–1124|doi=10.1016/j.physletb.2012.11.046|arxiv = 1202.1301 |bibcode = 2013PhLB..718.1119D }}</ref>
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| ===General relativity and astronomy===
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| The Boltzmann equation is also often used in dynamics, especially galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by ''f''; in galaxies, physical collisions between the stars are very rare, and the effect of ''gravitational collisions'' can be neglected for times far longer than the [[age of the universe]].
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| The generalization to [[general relativity]] is
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| :<math>\hat{\mathbf{L}}_\mathrm{GR}=p^\alpha\frac{\partial}{\partial x^\alpha}-\Gamma^\alpha{}_{\beta\gamma}p^\beta p^\gamma\frac{\partial}{\partial p^\alpha},</math>
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| where Γ<sup>α</sup><sub>βγ</sub> is the [[Christoffel symbol]] of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (''x<sup>i</sup>, p<sub>i</sub>'') phase space as opposed to fully contravariant (''x<sup>i</sup>, p<sup>i</sup>'') phase space.<ref>{{cite journal
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| | last = Debbasch
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| | first = Fabrice
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| | coauthors = Willem van Leeuwen
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| | title = General relativistic Boltzmann equation I: Covariant treatment
| |
| | journal = Physica A
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| | volume = 388
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| | issue = 7
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| | pages = 1079–1104
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| | year = 2009|bibcode = 2009PhyA..388.1079D |doi = 10.1016/j.physa.2008.12.023 }}</ref><ref>{{cite journal
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| | last = Debbasch | |
| | first = Fabrice
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| | coauthors = Willem van Leeuwen
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| | title = General relativistic Boltzmann equation II: Manifestly covariant treatment
| |
| | journal = Physica A
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| | volume = 388
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| | issue = 9
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| | pages = 1818–34
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| | year = 2009|bibcode = 2009PhyA..388.1818D |doi = 10.1016/j.physa.2009.01.009 }}</ref>
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| In [[physical cosmology]], the study of processes in the [[early universe]] often requires to take into account the effects of [[quantum mechanics]] and [[general relativity]].<ref name=KolbTurner /> In the very dense medium formed by the primordial plasma after the [[big bang]] particles are continuously created and annihilated. In such an environment [[quantum coherence]] and the spatial extension of the [[wavefunction]] can affect the dynamics, making it questionable whether the classical phase space distribution ''f'' that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of [[quantum field theory]].<ref name=BEfromQFT /> This includes the formation of the light elements in [[big bang nucleosynthesis]], the production of [[dark matter]] and [[baryogenesis]].
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| ==Extension of the relativistic differential equations of classical fluid dynamics to shorter distances==
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| {{Empty section|date=June 2013}}
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| ==See also==
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| *[[H-theorem]]
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| *[[Fokker–Planck equation]]
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| *[[Navier–Stokes equations]]
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| *[[Vlasov equation]]
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| *[[Double layer (plasma)#The Vlasov–Poisson equation|Vlasov–Poisson equation]]
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| *[[Lattice Boltzmann methods]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite journal
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| | last1= Arkeryd
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| | first1= Leif
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| |author1-link= Leif Arkeryd
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| | title= On the Boltzmann equation part II: The full initial value problem | journal= Arch. Rational Mech. Anal. | volume= 45 | pages= 17–34 | year= 1972 | doi = 10.1007/BF00253393 | bibcode = 1972ArRMA..45...17A }}
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| *{{cite journal | last1= Arkeryd | first1= Leif | title= On the Boltzmann equation part I: Existence | journal= Arch. Rational Mech. Anal. | volume= 45 | pages= 1–16 | year= 1972 | doi = 10.1007/BF00253392 | bibcode = 1972ArRMA..45....1A }}
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| *{{cite journal | last1=DiPerna | first1= R. J. |last2 = Lions | first2 = P.-L. | title= On the Cauchy problem for Boltzmann equations: global existence and weak stability | journal= Ann. Of Math. (2) | volume=130 | pages= 321–366 | year=1989 | doi=10.2307/1971423}}
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| ==External links==
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| * [http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html The Boltzmann Transport Equation by Franz Vesely]
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| * [http://www.upenn.edu/pennnews/news/university-pennsylvania-mathematicians-solve-140-year-old-boltzmann-equation-gaseous-behaviors Boltzmann gaseous behaviors solved]
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| [[Category:Partial differential equations]]
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| [[Category:Statistical mechanics]]
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| [[Category:Transport phenomena]]
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| [[Category:Equations of physics]]
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