Geodesic
Distribution Functions in Quantum Mechanics and Wigner Functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 2, pp. 231-243

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We formulate and solve the problem of finding a distribution function $F(\mathbf r,\mathbf p,t)$ such that calculating statistical averages leads to the same local values of the number of particles, the momentum, and the energy as those in quantum mechanics. The method is based on the quantum mechanical definition of the probability density not limited by the number of particles in the system. The obtained distribution function coincides with the Wigner function only for spatially homogeneous systems. We obtain the chain of Bogoliubov equations, the Liouville equation for quantum distribution functions with an arbitrary number of particles in the system, the quantum kinetic equation with a self-consistent electromagnetic field, and the general expression for the dielectric permittivity tensor of the electron component of the plasma. In addition to the known physical effects that determine the dispersion of longitudinal and transverse waves in plasma, the latter tensor contains a contribution from the exchange Coulomb correlations significant for dense systems. 
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     author = {L. S. Kuz'menkov and S. G. Maksimov},
     title = {Distribution {Functions} in {Quantum} {Mechanics} and {Wigner} {Functions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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L. S. Kuz'menkov; S. G. Maksimov. Distribution Functions in Quantum Mechanics and Wigner Functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 2, pp. 231-243. http://geodesic.mathdoc.fr/item/TMF_2002_131_2_a5/