Physics:Quantum Boltzmann equation - Revision history

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	~~The~~ '''~~quantum~~ Boltzmann equation,''' also known as the ~~'''~~Uehling–Uhlenbeck equation~~'''~~,<ref>{{cite journal \|last1=Filbet \|first1=Francis \|last2=Hu \|first2=Jingwei \|last3=Jin \|first3=Shi \|title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime \|journal=Esaim: M2An \|year=2012 \|volume=46 \|issue=2 \|pages=443–463\|doi=10.1051/m2an/2011051\|arxiv=1009.3352}}</ref><ref>{{cite book \|last1=Bao \|first1=Weizhu \|last2=Markowich \|first2=Peter \|last3=Pareschi \|first3=Lorenzo\|chapter=Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation \|title=Modeling and Computational Methods for Kinetic Equations \|series=Modeling and Simulation in Science, Engineering and Technology \|year=2004\|pages=287–320\|doi=10.1007/978-0-8176-8200-2_10\|isbn=978-1-4612-6487-3 }}</ref> is the ~~[[Physics:Quantum mechanics\|~~quantum mechanical]] modification of the Boltzmann equation, which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by L.W. Nordheim (1928),<ref>{{Cite journal\|last1=Nordhiem\|first1=L. W.\|last2=Fowler\|first2=Ralph Howard\|date=1928-07-02\|title=On the kinetic method in the new statistics and application in the electron theory of conductivity\|journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character\|volume=119\|issue=783\|pages=689–698\|doi=10.1098/rspa.1928.0126\|bibcode=1928RSPSA.119..689N\|doi-access=free}}</ref> and by and E. A. Uehling and George Uhlenbeck (1933).<ref>{{Cite journal\|last1=Uehling\|first1=E. A.\|last2=Uhlenbeck\|first2=G. E.\|date=1933-04-01\|title=Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I\|url=https://link.aps.org/doi/10.1103/PhysRev.43.552\|journal=Physical Review\|language=en\|volume=43\|issue=7\|pages=552–561\|doi=10.1103/PhysRev.43.552\|bibcode=1933PhRv...43..552U\|issn=0031-899X\|url-access=subscription}}</ref>		'''Boltzmann equation''' the quantum Boltzmann equation, also known as the Uehling–Uhlenbeck equation, is the quantum mechanical modification of the Boltzmann equation, which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation. The quantum Boltzmann equation gives irreversible behavior, and therefore an arrow of time; that is, after a long enough time it gives an equilibrium distribution which no longer changes. Although quantum mechanics is microscopically time-reversible, the quantum Boltzmann equation gives irreversible behavior because phase information is discarded only the average occupation number of the quantum states is kept. The solution of the quantum Boltzmann equation is therefore a good approximation to the exact behavior of the system on time scales short compared to the Poincaré recurrence time, which is usually not a severe limitation, because the Poincaré recurrence time can be many times the age of the universe even in small systems. The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in semiconductor optics. For example, the energy distribution of a gas of excitons as a function of time (in picoseconds), measured using a streak camera, has been shown to approach an equilibrium Maxwell-Boltzmann distribution. A typical model of a semiconductor may be built on the assumptions that:

	In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation.
	~~<math display="block">~~
	~~\left[\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla_x + \mathbf{F} \cdot \nabla_p \right] f(\mathbf{x},\mathbf{p},t) = \mathcal{Q}[f](\mathbf{x},\mathbf{p})~~
	~~</math>~~

	where <math>\mathbf{F}</math> represents an externally applied potential acting on the gas' p-space distribution and <math>\mathcal{Q}</math> is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of <math>\mathcal{Q}</math>, which depends on the physics of the system to be modeled.<ref>{{cite journal \|last1=Filbert \|first1=Francis \|last2=Hu \|first2=Jingwei \|last3=Jin \|first3=Shi \|title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime \|journal=Esaim: M2An \|year=2012 \|volume=46 \|issue=2 \|pages=443–463\|doi=10.1051/m2an/2011051\|arxiv=1009.3352}}</ref>

	The quantum Boltzmann equation gives irreversible behavior, and therefore an arrow of time; that is, after a long enough time it gives an equilibrium distribution which no longer changes. Although quantum mechanics is microscopically time-reversible, the quantum Boltzmann equation gives irreversible behavior because phase information is discarded<ref>{{cite journal \|last1=Snoke \|first1=D.W. \|last2=Liu \|first2=G. \|last3=Girvin \|first3=S.M. \|title=The basis of the Second Law of thermodynamics in quantum field theory \|journal=Annals of Physics \|year=2012 \|volume=327 \|issue=7 \|pages=1825–1851 \|bibcode=2012AnPhy.327.1825S \|doi=10.1016/j.aop.2011.12.016\|arxiv=1112.3009 \|s2cid=118666925 }}</ref> only the average occupation number of the quantum states is kept. The solution of the quantum Boltzmann equation is therefore a good approximation to the exact behavior of the system on time scales short compared to the Poincaré recurrence time, which is usually not a severe limitation, because the Poincaré recurrence time can be many times the age of the universe even in small systems.

	The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in semiconductor optics.<ref>{{cite journal \|last1=Snoke \|first1=D.W. \|title=The quantum Boltzmann equation in semiconductor physics \|journal=Annalen der Physik \|year=2011 \|volume=523 \|issue=1–2 \|pages=87–100 \|bibcode=2011AnP...523...87S \|doi=10.1002/andp.201000102\|arxiv=1011.3849 \|s2cid=119250989 }}</ref> For example, the energy distribution of a gas of excitons as a function of time (in picoseconds), measured using a streak camera, has been shown<ref>{{cite journal \|last1=Snoke \|first1=D. W. \|last2=Braun \|first2=D. \|last3=Cardona \|first3=M. \|title=Carrier thermalization in Cu<sub>2</sub>O: Phonon emission by excitons \|journal=Physical Review B \|year=1991 \|volume=44 \|issue=7 \|pages=2991–3000 \|bibcode=1991PhRvB..44.2991S \|doi=10.1103/PhysRevB.44.2991\|pmid=9999890 }}</ref> to approach an equilibrium Maxwell-Boltzmann distribution.
	</div>		</div>

Maintenance script: Clean Book I red links, intro, and image slots

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	{{Short description\|Quantum Collection topic on Quantum Boltzmann equation}}		{{Short description\|Quantum Collection topic on Quantum Boltzmann equation}}

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}
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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	The '''quantum Boltzmann equation,''' also known as the '''Uehling–Uhlenbeck equation''',<ref>{{cite journal \|last1=Filbet \|first1=Francis \|last2=Hu \|first2=Jingwei \|last3=Jin \|first3=Shi \|title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime \|journal=Esaim: M2An \|year=2012 \|volume=46 \|issue=2 \|pages=443–463\|doi=10.1051/m2an/2011051\|arxiv=1009.3352}}</ref><ref>{{cite book \|last1=Bao \|first1=Weizhu \|last2=Markowich \|first2=Peter \|last3=Pareschi \|first3=Lorenzo\|chapter=Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation \|title=Modeling and Computational Methods for Kinetic Equations \|series=Modeling and Simulation in Science, Engineering and Technology \|year=2004\|pages=287–320\|doi=10.1007/978-0-8176-8200-2_10\|isbn=978-1-4612-6487-3 }}</ref> is the [[Physics:Quantum mechanics\|quantum mechanical]] modification of the [[Boltzmann equation]], which gives the nonequilibrium ~~[[Time evolution\|~~time evolution]] of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by ~~[[Biography:Lothar Wolfgang Nordheim\|~~L.W. Nordheim]] (1928),<ref>{{Cite journal\|last1=Nordhiem\|first1=L. W.\|last2=Fowler\|first2=Ralph Howard\|date=1928-07-02\|title=On the kinetic method in the new statistics and application in the electron theory of conductivity\|journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character\|volume=119\|issue=783\|pages=689–698\|doi=10.1098/rspa.1928.0126\|bibcode=1928RSPSA.119..689N\|doi-access=free}}</ref> and by and E. A. Uehling and ~~[[Biography:George Uhlenbeck\|~~George Uhlenbeck]] (1933).<ref>{{Cite journal\|last1=Uehling\|first1=E. A.\|last2=Uhlenbeck\|first2=G. E.\|date=1933-04-01\|title=Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I\|url=https://link.aps.org/doi/10.1103/PhysRev.43.552\|journal=Physical Review\|language=en\|volume=43\|issue=7\|pages=552–561\|doi=10.1103/PhysRev.43.552\|bibcode=1933PhRv...43..552U\|issn=0031-899X\|url-access=subscription}}</ref>		The '''quantum Boltzmann equation,''' also known as the '''Uehling–Uhlenbeck equation''',<ref>{{cite journal \|last1=Filbet \|first1=Francis \|last2=Hu \|first2=Jingwei \|last3=Jin \|first3=Shi \|title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime \|journal=Esaim: M2An \|year=2012 \|volume=46 \|issue=2 \|pages=443–463\|doi=10.1051/m2an/2011051\|arxiv=1009.3352}}</ref><ref>{{cite book \|last1=Bao \|first1=Weizhu \|last2=Markowich \|first2=Peter \|last3=Pareschi \|first3=Lorenzo\|chapter=Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation \|title=Modeling and Computational Methods for Kinetic Equations \|series=Modeling and Simulation in Science, Engineering and Technology \|year=2004\|pages=287–320\|doi=10.1007/978-0-8176-8200-2_10\|isbn=978-1-4612-6487-3 }}</ref> is the [[Physics:Quantum mechanics\|quantum mechanical]] modification of the Boltzmann equation, which gives the nonequilibrium time evolution of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by L.W. Nordheim (1928),<ref>{{Cite journal\|last1=Nordhiem\|first1=L. W.\|last2=Fowler\|first2=Ralph Howard\|date=1928-07-02\|title=On the kinetic method in the new statistics and application in the electron theory of conductivity\|journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character\|volume=119\|issue=783\|pages=689–698\|doi=10.1098/rspa.1928.0126\|bibcode=1928RSPSA.119..689N\|doi-access=free}}</ref> and by and E. A. Uehling and George Uhlenbeck (1933).<ref>{{Cite journal\|last1=Uehling\|first1=E. A.\|last2=Uhlenbeck\|first2=G. E.\|date=1933-04-01\|title=Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I\|url=https://link.aps.org/doi/10.1103/PhysRev.43.552\|journal=Physical Review\|language=en\|volume=43\|issue=7\|pages=552–561\|doi=10.1103/PhysRev.43.552\|bibcode=1933PhRv...43..552U\|issn=0031-899X\|url-access=subscription}}</ref>

	In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation.		In full generality (including the p-space and x-space drift terms, which are often neglected) the equation is represented analogously to the Boltzmann equation.
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	where <math>\mathbf{F}</math> represents an externally applied potential acting on the gas' p-space distribution and <math>\mathcal{Q}</math> is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of <math>\mathcal{Q}</math>, which depends on the physics of the system to be modeled.<ref>{{cite journal \|last1=Filbert \|first1=Francis \|last2=Hu \|first2=Jingwei \|last3=Jin \|first3=Shi \|title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime \|journal=Esaim: M2An \|year=2012 \|volume=46 \|issue=2 \|pages=443–463\|doi=10.1051/m2an/2011051\|arxiv=1009.3352}}</ref>		where <math>\mathbf{F}</math> represents an externally applied potential acting on the gas' p-space distribution and <math>\mathcal{Q}</math> is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of <math>\mathcal{Q}</math>, which depends on the physics of the system to be modeled.<ref>{{cite journal \|last1=Filbert \|first1=Francis \|last2=Hu \|first2=Jingwei \|last3=Jin \|first3=Shi \|title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime \|journal=Esaim: M2An \|year=2012 \|volume=46 \|issue=2 \|pages=443–463\|doi=10.1051/m2an/2011051\|arxiv=1009.3352}}</ref>

	The quantum Boltzmann equation gives irreversible behavior, and therefore an ~~[[Physics:Entropy (~~arrow of time~~)\|arrow of time]]~~; that is, after a long enough time it gives an equilibrium distribution which no longer changes. Although quantum mechanics is microscopically time-reversible, the quantum Boltzmann equation gives irreversible behavior because phase information is discarded<ref>{{cite journal \|last1=Snoke \|first1=D.W. \|last2=Liu \|first2=G. \|last3=Girvin \|first3=S.M. \|title=The basis of the Second Law of thermodynamics in quantum field theory \|journal=~~[[Physics:~~Annals of Physics~~\|Annals of Physics]]~~ \|year=2012 \|volume=327 \|issue=7 \|pages=1825–1851 \|bibcode=2012AnPhy.327.1825S \|doi=10.1016/j.aop.2011.12.016\|arxiv=1112.3009 \|s2cid=118666925 }}</ref> only the average occupation number of the quantum states is kept. The solution of the quantum Boltzmann equation is therefore a good approximation to the exact behavior of the system on time scales short compared to the ~~[[Poincaré recurrence theorem\|~~Poincaré recurrence time]], which is usually not a severe limitation, because the Poincaré recurrence time can be many times the ~~[[Astronomy:Age of the universe\|~~age of the universe]] even in small systems.		The quantum Boltzmann equation gives irreversible behavior, and therefore an arrow of time; that is, after a long enough time it gives an equilibrium distribution which no longer changes. Although quantum mechanics is microscopically time-reversible, the quantum Boltzmann equation gives irreversible behavior because phase information is discarded<ref>{{cite journal \|last1=Snoke \|first1=D.W. \|last2=Liu \|first2=G. \|last3=Girvin \|first3=S.M. \|title=The basis of the Second Law of thermodynamics in quantum field theory \|journal=Annals of Physics \|year=2012 \|volume=327 \|issue=7 \|pages=1825–1851 \|bibcode=2012AnPhy.327.1825S \|doi=10.1016/j.aop.2011.12.016\|arxiv=1112.3009 \|s2cid=118666925 }}</ref> only the average occupation number of the quantum states is kept. The solution of the quantum Boltzmann equation is therefore a good approximation to the exact behavior of the system on time scales short compared to the Poincaré recurrence time, which is usually not a severe limitation, because the Poincaré recurrence time can be many times the age of the universe even in small systems.

	The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in ~~[[Physics:Semiconductor\|~~semiconductor]] optics.<ref>{{cite journal \|last1=Snoke \|first1=D.W. \|title=The quantum Boltzmann equation in semiconductor physics \|journal=~~[[Physics:Annalen der Physik\|~~Annalen der Physik]] \|year=2011 \|volume=523 \|issue=1–2 \|pages=87–100 \|bibcode=2011AnP...523...87S \|doi=10.1002/andp.201000102\|arxiv=1011.3849 \|s2cid=119250989 }}</ref> For example, the energy distribution of a gas of ~~[[Physics:Exciton\|exciton]]s~~ as a function of time (in picoseconds), measured using a ~~[[Engineering:Streak camera\|~~streak camera]], has been shown<ref>{{cite journal \|last1=Snoke \|first1=D. W. \|last2=Braun \|first2=D. \|last3=Cardona \|first3=M. \|title=Carrier thermalization in Cu<sub>2</sub>O: Phonon emission by excitons \|journal=~~[[Physics:Physical Review B\|~~Physical Review B]] \|year=1991 \|volume=44 \|issue=7 \|pages=2991–3000 \|bibcode=1991PhRvB..44.2991S \|doi=10.1103/PhysRevB.44.2991\|pmid=9999890 }}</ref> to approach an equilibrium ~~[[Maxwell–Boltzmann distribution\|~~Maxwell-Boltzmann distribution]].		The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in semiconductor optics.<ref>{{cite journal \|last1=Snoke \|first1=D.W. \|title=The quantum Boltzmann equation in semiconductor physics \|journal=Annalen der Physik \|year=2011 \|volume=523 \|issue=1–2 \|pages=87–100 \|bibcode=2011AnP...523...87S \|doi=10.1002/andp.201000102\|arxiv=1011.3849 \|s2cid=119250989 }}</ref> For example, the energy distribution of a gas of excitons as a function of time (in picoseconds), measured using a streak camera, has been shown<ref>{{cite journal \|last1=Snoke \|first1=D. W. \|last2=Braun \|first2=D. \|last3=Cardona \|first3=M. \|title=Carrier thermalization in Cu<sub>2</sub>O: Phonon emission by excitons \|journal=Physical Review B \|year=1991 \|volume=44 \|issue=7 \|pages=2991–3000 \|bibcode=1991PhRvB..44.2991S \|doi=10.1103/PhysRevB.44.2991\|pmid=9999890 }}</ref> to approach an equilibrium Maxwell-Boltzmann distribution.
	</div>		</div>

Maintenance script: Normalize Book I Quantum page structure

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	\end{aligned}		\end{aligned}
	</math>		</math>

			== See also ==
			{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	== References ==		== References ==
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	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

	{{Sourceattribution\|Quantum Boltzmann equation}}		{{Author\|Harold Foppele}}

			{{Sourceattribution\|Physics:Quantum Boltzmann equation\|1}}

Harold at 23:13, 18 May 2026

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	<div style="width:300px;">		<div style="width:300px;">
	[[File:~~not found~~]]		[[File:Quantum_Boltzmann_equation_yellow.png\|thumb\|240px]]
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@@ Line 1: / Line 1: @@
 The '''quantum Boltzmann equation,''' also known as the '''Uehling–Uhlenbeck equation''',<ref>{{cite journal |last1=Filbet |first1=Francis |last2=Hu |first2=Jingwei |last3=Jin |first3=Shi |title=A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime |journal=Esaim: M2An |year=2012 |volume=46 |issue=2 |pages=443–463|doi=10.1051/m2an/2011051|arxiv=1009.3352}}</ref><ref>{{cite book |last1=Bao |first1=Weizhu |last2=Markowich |first2=Peter |last3=Pareschi |first3=Lorenzo|chapter=Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation |title=Modeling and Computational Methods for Kinetic Equations |series=Modeling and Simulation in Science, Engineering and Technology |year=2004|pages=287–320|doi=10.1007/978-0-8176-8200-2_10|isbn=978-1-4612-6487-3 }}</ref> is the [[Physics:Quantum mechanics|quantum mechanical]] modification of the [[Boltzmann equation]], which gives the nonequilibrium [[Time evolution|time evolution]] of a gas of quantum-mechanically interacting particles. Typically, the quantum Boltzmann equation is given as only the “collision term” of the full Boltzmann equation, giving the change of the momentum distribution of a locally homogeneous gas, but not the drift and diffusion in space. It was originally formulated by [[Biography:Lothar Wolfgang Nordheim|L.W. Nordheim]] (1928),<ref>{{Cite journal|last1=Nordhiem|first1=L. W.|last2=Fowler|first2=Ralph Howard|date=1928-07-02|title=On the kinetic method in the new statistics and application in the electron theory of conductivity|journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character|volume=119|issue=783|pages=689–698|doi=10.1098/rspa.1928.0126|bibcode=1928RSPSA.119..689N|doi-access=free}}</ref> and by and E. A. Uehling and [[Biography:George Uhlenbeck|George Uhlenbeck]] (1933).<ref>{{Cite journal|last1=Uehling|first1=E. A.|last2=Uhlenbeck|first2=G. E.|date=1933-04-01|title=Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I|url=https://link.aps.org/doi/10.1103/PhysRev.43.552|journal=Physical Review|language=en|volume=43|issue=7|pages=552–561|doi=10.1103/PhysRev.43.552|bibcode=1933PhRv...43..552U|issn=0031-899X|url-access=subscription}}</ref>
@@ Line 11: / Line 22: @@
 The quantum Boltzmann equation has been verified by direct comparison to time-resolved experimental measurements, and in general has found much use in [[Physics:Semiconductor|semiconductor]] optics.<ref>{{cite journal |last1=Snoke |first1=D.W. |title=The quantum Boltzmann equation in semiconductor physics |journal=[[Physics:Annalen der Physik|Annalen der Physik]] |year=2011 |volume=523 |issue=1–2 |pages=87–100 |bibcode=2011AnP...523...87S |doi=10.1002/andp.201000102|arxiv=1011.3849 |s2cid=119250989 }}</ref> For example, the energy distribution of a gas of [[Physics:Exciton|exciton]]s as a function of time (in picoseconds), measured using a [[Engineering:Streak camera|streak camera]], has been shown<ref>{{cite journal |last1=Snoke |first1=D. W. |last2=Braun |first2=D. |last3=Cardona |first3=M. |title=Carrier thermalization in Cu<sub>2</sub>O: Phonon emission by excitons |journal=[[Physics:Physical Review B|Physical Review B]] |year=1991 |volume=44 |issue=7 |pages=2991–3000 |bibcode=1991PhRvB..44.2991S |doi=10.1103/PhysRevB.44.2991|pmid=9999890 }}</ref> to approach an equilibrium [[Maxwell–Boltzmann distribution|Maxwell-Boltzmann distribution]].
 == Application to semiconductor physics ==

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	{{Short description\|Quantum Collection topic on Quantum Boltzmann equation}}		{{Short description\|Quantum Collection topic on Quantum Boltzmann equation}}

	{{Quantum book backlink\|Statistical mechanics and kinetic theory}}		{{Quantum book backlink\|Statistical mechanics and kinetic theory}}
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