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→Simplifications to the collision term
玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)<ref name=":2" />|链接=Special:FilePath/StairsOfReduction.svg]]
玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)<ref name=":2" />|链接=Special:FilePath/StairsOfReduction.svg]]
The '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in a state of [[Thermodynamic equilibrium|equilibrium]], 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> The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.
The '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in a state of [[Thermodynamic equilibrium|equilibrium]], 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> The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.
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.
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.
===Simplifications to the collision term===
=== Simplifications to the collision term ===
Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook.<ref>{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref> The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form:
: <math>\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f),</math>
where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.
where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.
==通用方程（对于混合物）==
For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof2">Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R. G. Lerner]], G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref>
: <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)_\text{coll},</math>
where ''f<sub>i</sub>'' = ''f<sub>i</sub>''('''r''', '''p'''<sub>''i''</sub>, ''t''), and the collision term is
: <math>
\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'},
</math>
where ''f′'' = ''f′''('''p′'''<sub>''i''</sub>, ''t''), the magnitude of the relative momenta is
: <math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|,</math>
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.
==应用与推广==
=== Conservation equations ===
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|year=1984|publisher=Dover Publications Inc.|location=New York|isbn=978-0-486-64741-8}}</ref>{{rp|p 163}} For a fluid consisting of only one kind of particle, the number density ''n'' is given by
<noinclude>
: <math>n = \int f \,d^3p.</math>
The average value of any function ''A'' is
==通用方程（对于混合物）==
: <math>\langle A \rangle = \frac 1 n \int A f \,d^3p.</math>
==应用与推广==
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} \mapsto x_i</math> and <math>\mathbf{p} \mapsto p_i = m w_i</math>, where <math>w_i</math> is the particle velocity vector. Define <math>A(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 \to \pm\infty</math>. Multiplying the Boltzmann equation by ''A'' and integrating over momentum yields four terms, which, using integration by parts, can be expressed as
: <math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math>
<math>n = \int f \,d^3p.</math>
: <math>\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),</math>
: <math>\int A F_j \frac{\partial f}{\partial p_j} \,d^3p = -nF_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle,</math>
: <math>\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 0,</math>
<math>\langle A \rangle = \frac 1 n \int A f \,d^3p.</math>
where the last term is zero, since ''A'' is conserved in a collision. Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:<ref name="dG1984" />{{rp|pp 12,168}}
: <math>\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x_j}(\rho V_j) = 0,</math>
where <math>\rho = mn</math> is the mass density, and <math>V_i = \langle w_i\rangle</math> is the average fluid velocity.
<math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math>
Letting <math>A = p_i</math>, the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:<ref name="dG1984" />{{rp|pp 15,169}}
<math>\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),</math>
: <math>\frac{\partial}{\partial t}(\rho V_i) + \frac{\partial}{\partial x_j}(\rho V_i V_j+P_{ij}) - nF_i = 0,</math>
where the last term is zero, since A is conserved in a collision. Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:
where <math>P_{ij} = \rho \langle (w_i-V_i) (w_j-V_j) \rangle</math> is the pressure tensor (the [[wikipedia:Viscous_stress_tensor|viscous stress tensor]] plus the hydrostatic [[wikipedia:Pressure|pressure]]).
Letting <math>A =\frac{p_i p_i}{2m}</math>, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:<ref name="dG1984" />{{rp|pp 19,169}}
: <math>\frac{\partial}{\partial t}(u + \tfrac{1}{2}\rho V_i V_i) + \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 = 0,</math>
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.
=== Hamiltonian mechanics ===
In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as
: <math>\hat{\mathbf{L}}[f]=\mathbf{C}[f], \, </math>
where '''L''' is the [[wikipedia:Liouville_operator|Liouville operator]] (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and '''C''' is the collision operator. The non-relativistic form of '''L''' is
: <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>
=== Quantum theory and violation of particle number conservation ===
It is possible to write down relativistic [[wikipedia:Quantum_Boltzmann_equation|quantum Boltzmann equations]] for [[wikipedia:Quantum_field_theory|relativistic]] quantum systems in which the number of particles is not conserved in collisions. This has several applications in [[wikipedia:Physical_cosmology|physical cosmology]],<ref name="KolbTurner">{{cite book|author1=Edward Kolb|author2=Michael Turner|name-list-style=amp|title=The Early Universe|year=1990|publisher=Westview Press|isbn=9780201626742}}</ref> including the formation of the light elements in [[wikipedia:Big_Bang_nucleosynthesis|Big Bang nucleosynthesis]], the production of [[wikipedia:Dark_matter|dark matter]] and [[wikipedia:Baryogenesis|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 [[wikipedia:Quantum_field_theory|quantum field theory]].<ref name="BEfromQFT">{{cite journal|author1=M. Drewes|author2=C. Weniger|author3=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|title=The Boltzmann equation from quantum field theory|s2cid=119253828}}</ref>
=== General relativity and astronomy ===
The Boltzmann equation is of use in 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 [[wikipedia:Age_of_the_universe|age of the universe]].
<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>
Its generalization in [[wikipedia:General_relativity|general relativity]].<ref>Ehlers J (1971) General Relativity and Cosmology (Varenna), R K Sachs (Academic Press NY);Thorne K S (1980) Rev. Mod. Phys., 52, 299; Ellis G F R, Treciokas R, Matravers D R, (1983) Ann. Phys., 150, 487}</ref> is
: <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>
where Γ<sup>α</sup><sub>βγ</sub> is the [[wikipedia:Christoffel_symbol|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|last=Debbasch|first=Fabrice|author2=Willem van Leeuwen|title=General relativistic Boltzmann equation I: Covariant treatment|journal=Physica A|volume=388|issue=7|pages=1079–1104|year=2009|bibcode=2009PhyA..388.1079D|doi=10.1016/j.physa.2008.12.023}}</ref><ref>{{cite journal|last=Debbasch|first=Fabrice|author2=Willem van Leeuwen|title=General relativistic Boltzmann equation II: Manifestly covariant treatment|journal=Physica A|volume=388|issue=9|pages=1818–34|year=2009|bibcode=2009PhyA..388.1818D|doi=10.1016/j.physa.2009.01.009}}</ref>
In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity.
In [[wikipedia:Physical_cosmology|physical cosmology]] the fully covariant approach has been used to study the cosmic microwave background radiation.<ref>Maartens R, Gebbie T, Ellis GFR (1999). "Cosmic microwave background anisotropies: Nonlinear dynamics". Phys. Rev. D. 59 (8): 083506</ref> More generically the study of processes in the [[wikipedia:Early_universe|early universe]] often attempt to take into account the effects of [[wikipedia:Quantum_mechanics|quantum mechanics]] and [[wikipedia:General_relativity|general relativity]].<ref name="KolbTurner" /> In the very dense medium formed by the primordial plasma after the [[wikipedia:Big_Bang|Big Bang]], particles are continuously created and annihilated. In such an environment [[wikipedia:Quantum_coherence|quantum coherence]] and the spatial extension of the [[wikipedia:Wavefunction|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 [[wikipedia:Quantum_field_theory|quantum field theory]].<ref name="BEfromQFT" /> This includes the formation of the light elements in [[wikipedia:Big_Bang_nucleosynthesis|Big Bang nucleosynthesis]], the production of [[wikipedia:Dark_matter|dark matter]] and [[wikipedia:Baryogenesis|baryogenesis]].
==方程求解==
==方程求解==
this analytical approach provides insight, but is not generally usable in practical problems.
Exact solutions to the Boltzmann equations have been proven to exist in some cases;<ref>{{cite journal|doi=10.1090/S0894-0347-2011-00697-8|arxiv=1011.5441|title=Global Classical Solutions of the Boltzmann Equation without Angular Cut-off|authors=Philip T. Gressman, Robert M. Strain|journal=Journal of the American Mathematical Society|volume=24|issue=3|year=2011|page=771|s2cid=115167686}}</ref> this analytical approach provides insight, but is not generally usable in practical problems.
这种分析方法提供了洞察力，但在实际问题中通常不能使用。
这种分析方法提供了洞察力，但在实际问题中通常不能使用。
==另见==
==另见==
{{Div col|colwidth=20em}}
* [[Vlasov equation]]
<!--*[[BGK equation]]
*[[linear Boltzmann equation]]
*[[neutron transport equation]]
*[[photon transport equation]]-->
* [[H-theorem]]
* [[Fokker–Planck equation]]
* [[Williams spray equation|Williams-Boltzmann equation]]
* [[Navier–Stokes equations]]
* [[Vlasov equation#The Vlasov–Poisson equation|Vlasov–Poisson equation]]
* [[Lattice Boltzmann methods]]
{{Div col end}}
==注释==
==注释==
<references />
<references />{{reflist|40em}}
==参考文献==
==参考文献==
* {{cite book|last1=Harris|first1=Stewart|title=An introduction to the theory of the Boltzmann equation|publisher=Dover Books|pages=221|year=1971|isbn=978-0-486-43831-3|url=https://books.google.com/books?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like [[wikipedia:Fokker–Planck_equation|Fokker–Planck]] or [[wikipedia:Landau_equation|Landau equations]].
* {{cite journal|last1=Arkeryd|first1=Leif|title=On the Boltzmann equation part I: Existence|journal=Arch. Rational Mech. Anal.|volume=45|issue=1|pages=1–16|year=1972|doi=10.1007/BF00253392|bibcode=1972ArRMA..45....1A|s2cid=117877311}}
* {{cite journal|last1=Arkeryd|first1=Leif|author1-link=Leif Arkeryd|title=On the Boltzmann equation part II: The full initial value problem|journal=Arch. Rational Mech. Anal.|volume=45|issue=1|pages=17–34|year=1972|doi=10.1007/BF00253393|bibcode=1972ArRMA..45...17A|s2cid=119481100}}
* {{cite journal|last1=Arkeryd|first1=Leif|title=On the Boltzmann equation part I: Existence|journal=Arch. Rational Mech. Anal.|volume=45|issue=1|pages=1–16|year=1972|doi=10.1007/BF00253392|bibcode=1972ArRMA..45....1A|s2cid=117877311}}
* {{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.|series=2|volume=130|issue=2|pages=321–366|year=1989|doi=10.2307/1971423|jstor=1971423}}
== 外部链接 ==
* [http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html The Boltzmann Transport Equation by Franz Vesely]
* [https://web.archive.org/web/20151123214334/http://www.upenn.edu/pennnews/news/university-pennsylvania-mathematicians-solve-140-year-old-boltzmann-equation-gaseous-behaviors Boltzmann gaseous behaviors solved]
[[分类:偏微分方程]]
[[分类:偏微分方程]]
[[分类:统计力学]]
[[分类:统计力学]]