Abstract
This chapter is devoted to the computation of time-dependent observables from the solution of the quantum kinetic equations. Starting from the Markovian Boltzmann-type equation we analyze the conserved quantities–density, momentum and kinetic energy. We demonstrate the irreversibility (increase of entropy , H-theorem ) and obtain the asymptotic solutions in the stationary case which are just the distribution functions of a non-interacting Fermi or Bose gas. This is actually surprising and not satisfactory because we are discussing interacting many-body systems. This puzzle is resolved in the reminder of this chapter where we compute the pair correlation function in equilibrium and continue with the dynamics of macroscopic observables within the non-Markovian quantum kinetic equations that were derived in Chaps. 6 and 7. It is proven that the non-Markovian quantum Landau equation conserves not kinetic energy but total energy (as expected). Furthermore, no H-theorem exists, and the equilibrium distributions differ from a Fermi or Bose distribution.
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Notes
- 1.
We note that Boltzmann did not consider the Landau integral but a similar expression where the pair potential was replaced by a scattering cross section.
- 2.
Naturally, we do not prove this way that this is the only stationary solution.
- 3.
Recall that, in the classical case, \(\mu = k_BT \ln n.\)
- 4.
For a text book discussion, see [267].
- 5.
These results have been obtained in collaboration with D. Semkat.
- 6.
The imaginary part of the correlation function vanishes, because it contains a factor \(x\delta (x)\) which is zero.
- 7.
This is the weak coupling limit of the binary collision (T-matrix) approximation, cf. Chap. 9, where \(g_0^{EQ}\sim e^{-V/k_BT}-1\).
- 8.
We again skip the spin degeneracy factor \(2s+1\).
- 9.
The contributions from the higher energy differences will cancel to a large extent.
- 10.
Cf. the discussion of the Markov limit in Sect. 6.3.
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Bonitz, M. (2016). Properties of the Quantum Kinetic Equation. In: Quantum Kinetic Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-24121-0_8
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DOI: https://doi.org/10.1007/978-3-319-24121-0_8
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