276:; 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
914:
32:, 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
416:
909:{\displaystyle {\mathcal {Q}}(\mathbf {k} )={\frac {-2}{\hbar (2\pi )^{5}}}\int d\mathbf {q} \int d\mathbf {k_{1}} |{\hat {v}}(\mathbf {q} )|^{2}\delta \left({\frac {\hbar ^{2}}{2m}}(|\mathbf {k-q} |^{2}+|\mathbf {k_{1}+q} |^{2}-\mathbf {k} _{1}^{2}-\mathbf {k} ^{2})\right)\left}
195:
287:
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
50:
411:
267:
243:
382:
360:
331:
219:
47:
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.
245:
is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of
988:
971:
Bao, Weizhu; Markowich, Peter; Pareschi, Lorenzo (2004). "Quantum kinetic theory: Modelling and numerics for Bose-Einstein condensation".
1112:
Filbert, Francis; Hu, Jingwei; Jin, Shi (2012). "A Numerical Scheme for the
Quantum Boltzmann Equation Efficient in the Fluid Regime".
293:
932:
Filbet, Francis; Hu, Jingwei; Jin, Shi (2012). "A Numerical Scheme for the
Quantum Boltzmann Equation Efficient in the Fluid Regime".
1316:
277:
308:
The electron distribution is spatially homogeneous to a reasonable approximation (so all x-dependence may be suppressed)
1151:
Snoke, D.W.; Liu, G.; Girvin, S.M. (2012). "The basis of the Second Law of thermodynamics in quantum field theory".
292:
as a function of time (in picoseconds), measured using a streak camera, has been shown to approach an equilibrium
1030:
Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character
33:
273:
37:
280:, which is usually not a severe limitation, because the Poincaré recurrence time can be many times the
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1026:"On the kinetic method in the new statistics and application in the electron theory of conductivity"
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The gas is sufficiently dilute that three-body interactions between electrons may be ignored.
190:{\displaystyle \leftf(\mathbf {x} ,\mathbf {p} ,t)={\mathcal {Q}}(\mathbf {x} ,\mathbf {p} )}
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The external potential is a function only of position and isotropic in p-space, and so
221:
represents an externally applied potential acting on the gas' p-space distribution and
1310:
1247:
1192:
975:. Modeling and Simulation in Science, Engineering and Technology. pp. 287–320.
1066:
980:
1206:
Snoke, D.W. (2011). "The quantum
Boltzmann equation in semiconductor physics".
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1288:
1098:
272:
The quantum
Boltzmann equation gives irreversible behavior, and therefore an
1239:
1137:
1090:
1050:
1025:
957:
1296:
1261:
Snoke, D. W.; Braun, D.; Cardona, M. (1991). "Carrier thermalization in Cu
304:
A typical model of a semiconductor may be built on the assumptions that:
289:
1222:
1167:
1128:
948:
1067:"Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I"
422:
269:, which depends on the physics of the system to be modeled.
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may be set to zero without losing any further generality
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1024:
Nordhiem, L. W.; Fowler, Ralph Howard (1928-07-02).
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325:
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213:
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1065:Uehling, E. A.; Uhlenbeck, G. E. (1933-04-01).
8:
413:, it is possible to derive the expression
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59:
52:
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566:
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362:between electrons with initial momenta
1006:
996:
340:Considering the exchange of momentum
7:
300:Application to semiconductor physics
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86:
65:
61:
14:
1265:O: Phonon emission by excitons".
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406:{\displaystyle \mathbf {k_{1}} }
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294:Maxwell-Boltzmann distribution
262:{\displaystyle {\mathcal {Q}}}
238:{\displaystyle {\mathcal {Q}}}
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124:
1:
981:10.1007/978-0-8176-8200-2_10
377:{\displaystyle \mathbf {k} }
355:{\displaystyle \mathbf {q} }
326:{\displaystyle \mathbf {F} }
214:{\displaystyle \mathbf {F} }
18:quantum Boltzmann equation,
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22:Uehling-Uhlenbeck equation
1185:10.1016/j.aop.2011.12.016
1289:10.1103/PhysRevB.44.2991
278:Poincaré recurrence time
284:even in small systems.
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1051:10.1098/rspa.1928.0126
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28:modification of the
1281:1991PhRvB..44.2991S
1232:2011AnP...523...87S
1177:2012AnPhy.327.1825S
1083:1933PhRv...43..552U
1042:1928RSPSA.119..689N
673:
282:age of the universe
36:(1928), and by and
1209:Annalen der Physik
906:
657:
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352:
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259:
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30:Boltzmann equation
26:quantum mechanical
20:also known as the
1268:Physical Review B
1154:Annals of Physics
990:978-1-4612-6487-3
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1275:(7): 2991–3000.
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1161:(7): 1825–1851.
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1103:
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1036:(783): 689–698.
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1016:
1007:|journal=
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38:E. A. Uehling
35:
34:L.W. Nordheim
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27:
23:
19:
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1115:Esaim: M2An
935:Esaim: M2An
919:References
1248:119250989
1223:1011.3849
1193:118666925
1168:1112.3009
1129:1009.3352
1099:0031-899X
1009:ignored (
999:cite book
949:1009.3352
877:−
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567:ℏ
556:δ
527:^
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487:∫
471:π
462:ℏ
454:−
108:∇
104:⋅
87:∇
83:⋅
66:∂
62:∂
24:, is the
1311:Category
290:excitons
44:(1933).
1297:9999890
1277:Bibcode
1228:Bibcode
1173:Bibcode
1079:Bibcode
1038:Bibcode
1295:
1246:
1191:
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199:where
1244:S2CID
1218:arXiv
1189:S2CID
1163:arXiv
1124:arXiv
944:arXiv
1293:PMID
1095:ISSN
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985:ISBN
384:and
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