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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Statistical mechanics and kinetic theory}}
'''Quantum kinetic theory''' describes the time evolution of many-particle systems when both quantum effects and statistical behavior are important.<ref name="KadanoffBaym">{{cite book
|last1=Kadanoff
|first1=L. P.
|last2=Baym
|first2=G.
|title=Quantum Statistical Mechanics
|publisher=W. A. Benjamin
|year=1962
|isbn=9780805306378
}}</ref><ref name="BonitzQT">{{cite book
|last=Bonitz
|first=M.
|title=Quantum Kinetic Theory
|publisher=Teubner
|year=1998
|isbn=9783519002540
}}</ref> It provides the bridge between [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]], classical kinetic theory, and macroscopic transport theory.<ref name="HaugJauho">{{cite book
|last1=Haug
|first1=H.
|last2=Jauho
|first2=A.-P.
|title=Quantum Kinetics in Transport and Optics of Semiconductors
|publisher=Springer
|year=2008
|isbn=9783540735868
|url=https://link.springer.com/book/10.1007/978-3-540-73564-9
}}</ref>

Instead of tracking the full many-body wavefunction directly, quantum kinetic theory describes systems through reduced distribution functions, density operators, or nonequilibrium Green's functions that evolve in time.<ref name="KadanoffBaym"/><ref name="HaugJauho"/>
[[File:Quantum kinetische theorie visualisatie-1.jpg|thumb|300px|Conceptual illustration of quantum kinetic theory, describing the evolution of distribution functions in phase space with quantum corrections and many-particle interactions]]
==Overview==
Quantum kinetic theory is concerned with nonequilibrium dynamics, relaxation, collisions, transport, and the emergence of macroscopic behavior from microscopic quantum laws.<ref name="BonitzQT"/><ref name="Polkovnikov2011">{{cite journal
|last1=Polkovnikov
|first1=Anatoli
|last2=Sengupta
|first2=Krishnendu
|last3=Silva
|first3=Alessandro
|last4=Vengalattore
|first4=Mukund
|title=Colloquium: Nonequilibrium dynamics of closed interacting quantum systems
|journal=Reviews of Modern Physics
|year=2011
|volume=83
|issue=3
|pages=863–883
|doi=10.1103/RevModPhys.83.863
|url=https://link.aps.org/doi/10.1103/RevModPhys.83.863
}}</ref> It becomes essential when the system is not in equilibrium, when particle statistics matter, or when coherence and interference modify classical transport behavior.<ref name="HaugJauho"/>

Typical applications include semiconductors, plasmas, ultracold gases, quantum optical media, and strongly interacting many-body systems.<ref name="HaugJauho"/><ref name="BonitzQT"/>

==Distribution functions==
In classical kinetic theory, the state of a system is described by a phase-space distribution function

<math>
f(\mathbf{x},\mathbf{p},t),
</math>

which gives the density of particles at position <math>\mathbf{x}</math> with momentum <math>\mathbf{p}</math> at time <math>t</math>.<ref name="Cercignani">{{cite book
|last=Cercignani
|first=C.
|title=The Boltzmann Equation and Its Applications
|publisher=Springer
|year=1988
|isbn=9780387963464
}}</ref>

In quantum theory this concept is generalized through reduced density matrices, Wigner functions, and related quasiprobability distributions.<ref name="BonitzQT"/><ref name="Wigner1932">{{cite journal
|last=Wigner
|first=E.
|title=On the Quantum Correction For Thermodynamic Equilibrium
|journal=Physical Review
|year=1932
|volume=40
|issue=5
|pages=749–759
|doi=10.1103/PhysRev.40.749
|url=https://link.aps.org/doi/10.1103/PhysRev.40.749
}}</ref>

==Wigner function==
A widely used quantum analogue of the classical distribution function is the '''Wigner function''',

<math>
W(x,p)=\frac{1}{2\pi\hbar}\int e^{-ipy/\hbar}\psi^*\left(x+\frac{y}{2}\right)\psi\left(x-\frac{y}{2}\right)\,dy.
</math>

It behaves in many ways like a phase-space distribution, but unlike a classical probability density it can take negative values, reflecting quantum interference and nonclassical correlations.<ref name="Wigner1932"/>

The Wigner formalism is especially useful because it makes the relation between quantum dynamics and the classical phase-space picture transparent.<ref name="BonitzQT"/>

==Quantum kinetic equations==
The evolution of distribution functions is governed by kinetic equations that generalize the classical Boltzmann equation.<ref name="KadanoffBaym"/><ref name="HaugJauho"/>

A generic kinetic equation has the form

<math>
\frac{\partial f}{\partial t}
+
\mathbf{v}\cdot\nabla_x f
+
\mathbf{F}\cdot\nabla_p f
=
\mathcal{Q}[f],
</math>

where <math>\mathbf{F}</math> is an external force and <math>\mathcal{Q}[f]</math> is a collision or interaction term.<ref name="HaugJauho"/>

In quantum systems, the classical structure is modified by:

* Fermi-Dirac or Bose-Einstein statistics
* coherence and phase information
* nonlocality and memory effects
* self-energies and many-body correlations<ref name="KadanoffBaym"/><ref name="HaugJauho"/>

==Collision terms and quantum statistics==
The collision operator determines scattering, relaxation, entropy production, and transport coefficients.<ref name="Cercignani"/><ref name="BonitzQT"/> In quantum systems it must incorporate particle statistics. For fermions, scattering is suppressed by '''Pauli blocking'''; for bosons, it can be enhanced by Bose occupation factors.<ref name="Pathria">{{cite book
|last1=Pathria
|first1=R. K.
|last2=Beale
|first2=Paul D.
|title=Statistical Mechanics
|edition=3
|publisher=Elsevier
|year=2011
|isbn=9780123821881
}}</ref>

These statistical corrections are essential in degenerate electron gases, photon and phonon transport, ultracold atomic systems, and dense plasmas.<ref name="BonitzQT"/><ref name="HaugJauho"/>

==Nonequilibrium Green's functions==
A central formulation of quantum kinetic theory uses nonequilibrium Green's functions (NEGF), developed by Kadanoff, Baym, and Keldysh.<ref name="KadanoffBaym"/><ref name="Keldysh">{{cite journal
|last=Keldysh
|first=L. V.
|title=Diagram technique for nonequilibrium processes
|journal=Soviet Physics JETP
|year=1965
|volume=20
|pages=1018–1026
|url=https://arxiv.org/abs/cond-mat/0506469
}}</ref>

The basic correlation functions,

<math>
G^{<}(x_1,x_2),\qquad G^{>}(x_1,x_2),
</math>

encode occupancies and correlations, while the Kadanoff-Baym equations govern their evolution.<ref name="KadanoffBaym"/><ref name="HaugJauho"/>

This formalism is particularly important for systems with strong interactions, transient dynamics, and memory effects beyond simple Markovian approximations.<ref name="HaugJauho"/>

==Relation to the Boltzmann and Vlasov equations==
Under appropriate approximations, quantum kinetic theory reduces to more familiar kinetic descriptions.<ref name="BonitzQT"/>

In the weak-coupling and semiclassical limit, one obtains the '''quantum Boltzmann equation'''.<ref name="BonitzQT"/> In collisionless mean-field regimes, the collision term may be neglected, leading to the '''Vlasov equation''':

<math>
\frac{\partial f}{\partial t}
+
\mathbf{v}\cdot\nabla_x f
+
\mathbf{F}\cdot\nabla_p f
=
0.
</math>

This equation is widely used in plasma physics and collective many-body dynamics.<ref name="Nicholson">{{cite book
|last=Nicholson
|first=D. R.
|title=Introduction to Plasma Theory
|publisher=John Wiley & Sons
|year=1983
|isbn=9780471090458
}}</ref>

Quantum kinetic theory therefore unifies microscopic quantum dynamics with semiclassical and classical transport models.<ref name="BonitzQT"/><ref name="KadanoffBaym"/>

==Moments and fluid models==
Macroscopic quantities are obtained by taking moments of the distribution function over momentum space.<ref name="Cercignani"/>

The particle density is

<math>
n(\mathbf{x},t)=\int f(\mathbf{x},\mathbf{p},t)\,d^3p,
</math>

the mean velocity is

<math>
\mathbf{u}=\frac{1}{n}\int \mathbf{v}\,f\,d^3p,
</math>

and temperature is related to the kinetic energy of fluctuations about the mean flow.<ref name="Cercignani"/><ref name="Nicholson"/>

These moments lead to hydrodynamic and fluid equations used in plasma theory, semiconductor modeling, and transport theory.<ref name="Nicholson"/><ref name="BonitzQT"/>

==Phonons and quasiparticles==
In condensed-matter applications, quantum kinetic theory is often expressed in terms of quasiparticles such as electrons, holes, excitons, and [[Physics:Phonon|phonons]].<ref name="Mahan">{{cite book
|last=Mahan
|first=G. D.
|title=Many-Particle Physics
|publisher=Springer
|year=1981
|isbn=9780306463389
}}</ref><ref name="GirvinYang">{{cite book
|last1=Girvin
|first1=Steven M.
|last2=Yang
|first2=Kun
|title=Modern Condensed Matter Physics
|publisher=Cambridge University Press
|year=2019
|isbn=9781107137394
}}</ref>

A '''phonon''' is the quantized excitation of a lattice vibration in a crystal or other elastic medium.<ref name="GirvinYang"/><ref name="Kittel">{{cite book
|last=Kittel
|first=Charles
|title=Introduction to Solid State Physics
|edition=8
|publisher=Wiley
|year=2004
|isbn=9780471415268
}}</ref> Phonons play a major role in thermal transport, electrical resistivity, and the relaxation of nonequilibrium carriers in solids.<ref name="AshcroftMermin">{{cite book
|last1=Ashcroft
|first1=Neil W.
|last2=Mermin
|first2=N. David
|title=Solid State Physics
|publisher=Saunders College Publishing
|year=1976
|isbn=9780030839931
}}</ref>

Because phonons are bosonic collective modes, they can be created and annihilated in second-quantized form, and their populations obey Bose-Einstein statistics in equilibrium.<ref name="GirvinYang"/><ref name="Pathria"/> Their dispersion relations determine heat capacity, sound propagation, and lattice-mediated transport processes.<ref name="Kittel"/><ref name="AshcroftMermin"/>

==Acoustic and optical phonons==
Crystals with more than one atom in the primitive cell exhibit both '''acoustic phonons''' and '''optical phonons'''.<ref name="Kittel"/><ref name="AshcroftMermin"/>

Acoustic phonons correspond to collective atomic motion in phase and determine the propagation of sound through solids. Their frequency tends to zero in the long-wavelength limit.<ref name="Kittel"/> Optical phonons correspond to out-of-phase motion of different atoms in the basis and can couple strongly to electromagnetic radiation in ionic crystals.<ref name="GirvinYang"/>

These excitations are important in quantum kinetic descriptions of lattice thermalization, electron-phonon scattering, thermal conductivity, and nonequilibrium solid-state transport.<ref name="Mahan"/><ref name="AshcroftMermin"/>

==Applications to plasma physics==
Quantum kinetic theory is closely related to plasma physics because plasmas consist of particles that are fundamentally quantum yet often described statistically through distribution functions.<ref name="Nicholson"/><ref name="BonitzQT"/>

Key kinetic equations include the Vlasov equation and the Fokker-Planck equation, which describe collective motion, momentum exchange, diffusion, and transport processes.<ref name="Nicholson"/> In fusion research and tokamak modeling, such equations are used to analyze edge transport, drifts, recycling, and asymmetry effects.<ref name="Emdee">{{cite journal
|last1=Emdee
|first1=E. D.
|last2=Stangeby
|first2=P. C.
|last3=Heifetz
|first3=D.
|title=Combined Influence of Rotation and Scrape-Off Layer Drifts on Recycling Asymmetries in Tokamak Plasmas
|journal=Physics of Fluids B
|year=1990
|volume=2
|issue=11
|pages=2680–2687
|doi=10.1063/1.859366
}}</ref>

==Physical interpretation==
'''Quantum kinetic theory explains how'''

* microscopic quantum interactions
* collisions and correlations
* coherence and decoherence
* particle statistics and collective modes

produce macroscopic transport, relaxation, and emergent classical behavior.<ref name="BonitzQT"/><ref name="Polkovnikov2011"/>

It therefore forms one of the main conceptual links between microscopic quantum theory and experimentally observable many-body dynamics.<ref name="KadanoffBaym"/>

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

==References==
{{reflist|3}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]
[[Category:Plasma physics]]
[[Category:Condensed matter physics]]

{{Sourceattribution|Quantum Kinetic theory|1}}

Physics:Quantum Kinetic theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template