Quantum Boltzmann equation

The quantum Boltzmann equation is the quantum mechanical version of the Boltzmann equation, which gives the time evolution of a distribution of free particles out of equilibrium. 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. There is not a single quantum Boltzmann equation; a different equation must be written for each type of particle scattering.

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[1] 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 Poincare recurrence time, which is usually not a severe limitation, because the Poincare 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.[2] 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[3] to approach an equilibrium Maxwell-Boltzmann distribution.

References

  1. ↑ 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.
  2. ↑ 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.
  3. ↑ 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.
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