更改

Consider particles described by <math> f</math> , each experiencing an ''external'' force '''<math> F</math>''' not due to other particles (see the collision term for the latter treatment).

Consider particles described by <math> f</math> , each experiencing an ''external'' force '''<math> F</math>''' not due to other particles (see the collision term for the latter treatment).

考虑一群以<nowiki><math>f</math></nowiki>分布的粒子。每个粒子均受到外力<nowiki><math>\mathbf{F}</math></nowiki>的作用（不包括粒子间作用力。粒子间的作用见后面对“coll”项的处理)。

考虑一群以<math>f</math>分布的粒子。每个粒子均受到外力<math>\mathbf{F}</math>的作用（不包括粒子间作用力。粒子间的作用见后面对“coll”项的处理)。

Suppose at time <math> t</math> some number of particles all have position '''<math> r</math>''' within element <math> d^3\bf{r}</math> and momentum '''<math> p</math>''' within <math> d^3\bf{p}</math>. If a force '''<math> F</math>''' instantly acts on each particle, then at time <math> t+\Delta t</math> their position will be <math> \mathbf{r}+\Delta \mathbf{r}= \textbf{r}+\frac{\textbf{p}}{m}\Delta t</math> and momentum <math> \mathbf{p}+\Delta \mathbf{p}= \mathbf{p}+\mathbf{F}\Delta t</math>. Then, in the absence of collisions, <math> f</math> must satisfy

Suppose at time <math>t</math> some number of particles all have position '''<math>r</math>''' within element <math> d^3\bf{r}</math> and momentum '''<math> p</math>''' within <math> d^3\bf{p}</math>. If a force '''<math> F</math>''' instantly acts on each particle, then at time <math> t+\Delta t</math> their position will be <math> \mathbf{r}+\Delta \mathbf{r}= \textbf{r}+\frac{\textbf{p}}{m}\Delta t</math> and momentum <math> \mathbf{p}+\Delta \mathbf{p}= \mathbf{p}+\mathbf{F}\Delta t</math>. Then, in the absence of collisions, <math> f</math> must satisfy

假设在时间 <nowiki><math>t</math></nowiki>，一定数量的粒子都有位置 <nowiki><math>\mathbf{r}</math></nowiki>（于微元 <nowiki><math> d^3\mathbf{r}</math></nowiki> 内），和动量 <nowiki><math>\mathbf{p}</math></nowiki>（于微元 <nowiki><math> d^3\mathbf{p}</math></nowiki> 内）。如果此时有一个力<nowiki><math>\mathbf{F}</math></nowiki>在这一瞬作用在每个颗粒上，那么在时间 <nowiki><math>t + \Delta\,t</math></nowiki>，它们的位置将会是<nowiki><math>\mathbf{r} + \Delta\,\mathbf{r} = \mathbf{r} + \mathbf{p} \Delta\,t/m</math></nowiki>，动量将变成 <nowiki><math>\mathbf{p} + \Delta\,\mathbf{p} = \mathbf{p} + \mathbf{F}\Delta\,t</math></nowiki>。在没有碰撞的情况下，<nowiki><math>f</math></nowiki>必须满足

假设在时间 <nowiki><math>t</math></nowiki>，一定数量的粒子都有位置 <nowiki><math>\mathbf{r}</math></nowiki>（于微元 <nowiki><math> d^3\mathbf{r}</math></nowiki> 内），和动量 <nowiki><math>\mathbf{p}</math></nowiki>（于微元 <nowiki><math> d^3\mathbf{p}</math></nowiki> 内）。如果此时有一个力<nowiki><math>\mathbf{F}</math></nowiki>在这一瞬作用在每个颗粒上，那么在时间 <nowiki><math>t + \Delta\,t</math></nowiki>，它们的位置将会是<nowiki><math>\mathbf{r} + \Delta\,\mathbf{r} = \mathbf{r} + \mathbf{p} \Delta\,t/m</math></nowiki>，动量将变成 <nowiki><math>\mathbf{p} + \Delta\,\mathbf{p} = \mathbf{p} + \mathbf{F}\Delta\,t</math></nowiki>。在没有碰撞的情况下，<nowiki><math>f</math></nowiki>必须满足

is a shorthand for the momentum analogue of ∇, and '''ê'''_''x'', '''ê'''_''y'', '''ê'''_''z'' are [[cartesian coordinates|Cartesian]] [[unit vector]]s.

is a shorthand for the momentum analogue of ∇, and '''ê'''_''x'', '''ê'''_''y'', '''ê'''_''z'' are [[cartesian coordinates|Cartesian]] [[unit vector]]s.

=== Final statement===

===Final statement===

Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:

Dividing ({{EquationNote|3}}) by ''dt'' and substituting into ({{EquationNote|2}}) gives:

This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved 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.

This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved 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.

==The collision term (Stosszahlansatz) and molecular chaos==

==The collision term (Stosszahlansatz) and molecular chaos==

===Two-body collision term ===

===Two-body collision term===

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" />

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" />

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:

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>

:<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.

For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />  

For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />  

: <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>

:<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_i'' = ''f_i''('''r''', '''p'''_''i'', ''t''), and the collision term is

where ''f_i'' = ''f_i''('''r''', '''p'''_''i'', ''t''), and the collision term is

: <math>

:<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'},

  \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>

</math>

where ''f&#x2032;'' = ''f&#x2032;''('''p&#x2032;'''_''i'', ''t''), the magnitude of the relative momenta is

where ''f&#x2032;'' = ''f&#x2032;''('''p&#x2032;'''_''i'', ''t''), the magnitude of the relative momenta is

: <math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|,</math>

:<math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|,</math>

and ''I_ij'' 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.  

and ''I_ij'' 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==

==应用与推广==

=== Conservation equations ===

===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

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

: <math>n = \int f \,d^3p.</math>

:<math>n = \int f \,d^3p.</math>

The average value of any function ''A'' is

The average value of any function ''A'' is

: <math>\langle A \rangle = \frac 1 n \int A f \,d^3p.</math>

:<math>\langle A \rangle = \frac 1 n \int A f \,d^3p.</math>

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

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>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</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 \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 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>\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 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:<ref name="dG1984" />{{rp|pp 12,168}}

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>

:<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.

where <math>\rho = mn</math> is the mass density, and <math>V_i = \langle w_i\rangle</math> is the average fluid velocity.

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}}

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>\frac{\partial}{\partial t}(\rho V_i) + \frac{\partial}{\partial x_j}(\rho V_i V_j+P_{ij}) - nF_i = 0,</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  <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]]).

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}}

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>

:<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.

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 ===

===Hamiltonian mechanics===

In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as

In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as

: <math>\hat{\mathbf{L}}[f]=\mathbf{C}[f], \, </math>

:<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

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>

:<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 ===

===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>

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 ===

===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]].

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]].

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

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>

:<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^α_βγ 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 (''xⁱ, p_i'') phase space as opposed to fully contravariant (''xⁱ, pⁱ'') 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>

where Γ^α_βγ 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 (''xⁱ, p_i'') phase space as opposed to fully contravariant (''xⁱ, pⁱ'') 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>

相反，数值方法(包括有限元)通常用于寻找各种形式的玻尔兹曼方程的近似解。示例应用范围从稀薄气流中的高超音速空气动力学到等离子流。

相反，数值方法(包括有限元)通常用于寻找各种形式的玻尔兹曼方程的近似解。示例应用范围从稀薄气流中的高超音速空气动力学到等离子流。

Close to [[wikipedia:Non-equilibrium_thermodynamics#Local_thermodynamic_equilibrium|local equilibrium]], solution of the Boltzmann equation can be represented by an [[wikipedia:Asymptotic_expansion|asymptotic expansion]] in powers of [[wikipedia:Knudsen_number|Knudsen number]] (the [[wikipedia:Chapman–Enskog_theory|Chapman-Enskog]] expansion<ref>Sydney Chapman; Thomas George Cowling The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, 1970. ISBN 0-521-40844-X </ref>). The first two terms of this expansion give the [[wikipedia:Euler_equations_(fluid_dynamics)|Euler equations]] and the [[wikipedia:Navier-Stokes_equations|Navier-Stokes equations]]. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of [[wikipedia:Hilbert's_sixth_problem|Hilbert's sixth problem]].<ref>{{cite journal|doi=10.1098/rsta/376/2118|volume=376|year=2018|journal=Philosophical Transactions of the Royal Society A|title=Theme issue 'Hilbert's sixth problem'|issue=2118|doi-access=free}}</ref>

Close to [[wikipedia:Non-equilibrium_thermodynamics#Local_thermodynamic_equilibrium|local equilibrium]], solution of the Boltzmann equation can be represented by an [[wikipedia:Asymptotic_expansion|asymptotic expansion]] in powers of [[wikipedia:Knudsen_number|Knudsen number]] (the [[wikipedia:Chapman–Enskog_theory|Chapman-Enskog]] expansion<ref>Sydney Chapman; Thomas George Cowling The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, 1970. [[index.php?title=Special:BookSources/052140844X|ISBN 0-521-40844-X]] </ref>). The first two terms of this expansion give the [[wikipedia:Euler_equations_(fluid_dynamics)|Euler equations]] and the [[wikipedia:Navier-Stokes_equations|Navier-Stokes equations]]. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of [[wikipedia:Hilbert's_sixth_problem|Hilbert's sixth problem]].<ref>{{cite journal|doi=10.1098/rsta/376/2118|volume=376|year=2018|journal=Philosophical Transactions of the Royal Society A|title=Theme issue 'Hilbert's sixth problem'|issue=2118|doi-access=free}}</ref>

在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。

在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。

<references />{{reflist|40em}}

<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 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|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|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=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}}

*{{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 ==

== 外部链接==

* [http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html The Boltzmann Transport Equation by Franz Vesely]

*[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]

*[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]]