Quantum Boltzmann equation

The quantum Boltzmann equation, also known as the Uehling-Uhlenbeck equation, ^{[1] [2]} 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. It was originally formulated by L.W. Nordheim (1928), ^[3] and by and E. A. Uehling and George Uhlenbeck (1933). ^[4]

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.

$${\displaystyle \left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla _{x}+\mathbf {F} \cdot \nabla _{p}\right]f(\mathbf {x} ,\mathbf {p} ,t)={\mathcal {Q}}[f](\mathbf {x} ,\mathbf {p} )}$$

where ${\displaystyle \mathbf {F} }$ represents an externally applied potential acting on the gas' p-space distribution and ${\displaystyle {\mathcal {Q}}}$ is the collision operator, accounting for the interactions between the gas particles. The quantum mechanics must be represented in the exact form of ${\displaystyle {\mathcal {Q}}}$, which depends on the physics of the system to be modeled. ^[5]

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 ^[6] 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. ^[7] 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 ^[8] to approach an equilibrium Maxwell-Boltzmann distribution.

Application to semiconductor physics

A typical model of a semiconductor may be built on the assumptions that:

1. The electron distribution is spatially homogeneous to a reasonable approximation (so all x-dependence may be suppressed)
2. The external potential is a function only of position and isotropic in p-space, and so ${\displaystyle \mathbf {F} }$ may be set to zero without losing any further generality
3. The gas is sufficiently dilute that three-body interactions between electrons may be ignored.

Considering the exchange of momentum ${\displaystyle \mathbf {q} }$ between electrons with initial momenta ${\displaystyle \mathbf {k} }$ and ${\displaystyle \mathbf {k_{1}} }$, it is possible to derive the expression

$${\displaystyle {\mathcal {Q}}[f](\mathbf {k} )={\frac {-2}{\hbar (2\pi )^{5}}}\int d\mathbf {q} \int d\mathbf {k_{1}} |{\hat {v}}(\mathbf {q} )|^{2}\delta \left({\frac {\hbar ^{2}}{2m}}(|\mathbf {k-q} |^{2}+|\mathbf {k_{1}+q} |^{2}-\mathbf {k} _{1}^{2}-\mathbf {k} ^{2})\right)\left[f_{\mathbf {k} }f_{\mathbf {k_{1}} }(1-f_{\mathbf {k-q} })(1-f_{\mathbf {k_{1}+q} })-f_{\mathbf {k-q} }f_{\mathbf {k_{1}+q} }(1-f_{\mathbf {k} })(1-f_{\mathbf {k_{1}} })\right]}$$

References

1. Filbet, Francis; Hu, Jingwei; Jin, Shi (2012). "A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime". Esaim: M2An . 46 (2): 443–463. arXiv: 1009.3352 . doi:10.1051/m2an/2011051.
2. Bao, Weizhu; Markowich, Peter; Pareschi, Lorenzo (2004). "Quantum kinetic theory: modelling and numerics for Bose-Einstein condensation". Modeling and Computational Methods for Kinetic Equations, Series: Modeling and Simulation in Science, Engineering and Technology . Modeling and Simulation in Science, Engineering and Technology: 287–320. doi:10.1007/978-0-8176-8200-2_10. ISBN   978-1-4612-6487-3.
3. Nordhiem, L. W.; Fowler, Ralph Howard (1928-07-02). "On the kinetic method in the new statistics and application in the electron theory of conductivity". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 119 (783): 689–698. Bibcode:1928RSPSA.119..689N. doi: 10.1098/rspa.1928.0126 .
4. Uehling, E. A.; Uhlenbeck, G. E. (1933-04-01). "Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I". Physical Review. 43 (7): 552–561. Bibcode:1933PhRv...43..552U. doi:10.1103/PhysRev.43.552. ISSN   0031-899X.
5. Filbert, Francis; Hu, Jingwei; Jin, Shi (2012). "A Numerical Scheme for the Quantum Boltzmann Equation Efficient in the Fluid Regime". Esaim: M2An . 46 (2): 443–463. arXiv: 1009.3352 . doi:10.1051/m2an/2011051.
6. Snoke, D.W.; Liu, G.; Girvin, S.M. (2012). "The basis of the Second Law of thermodynamics in quantum field theory". Annals of Physics . 327 (7): 1825–1851. arXiv: 1112.3009 . Bibcode:2012AnPhy.327.1825S. doi:10.1016/j.aop.2011.12.016. S2CID   118666925.
7. Snoke, D.W. (2011). "The quantum Boltzmann equation in semiconductor physics". Annalen der Physik . 523 (1–2): 87–100. arXiv: 1011.3849 . Bibcode:2011AnP...523...87S. doi:10.1002/andp.201000102. S2CID   119250989.
8. Snoke, D. W.; Braun, D.; Cardona, M. (1991). "Carrier thermalization in Cu₂O: Phonon emission by excitons". Physical Review B . 44 (7): 2991–3000. Bibcode:1991PhRvB..44.2991S. doi:10.1103/PhysRevB.44.2991. PMID   9999890.