Classical and quantum kinetic equations with exact conservation laws

D. Kremp, Th. Bornath, M. Bönitz +1 more

Martin L R Fürst, Markus Kotulla, Christian B. Mendl +1 more

We derive and analyze an effective quantum Boltzmann equation in the kinetic regime for the interactions of four distinguishable types of fermionic spin- particles, starting from a general quantum field Hamiltonian. Each particle type is described by a time-dependent, 2 × 2 spin-density ('Wigner') matrix. We show that density and energy conservation laws as well as the H-theorem hold, and enumerate additional conservation laws depending on the interaction. The conserved quantities characterize the thermal (Fermi–Dirac) equilibrium state. We illustrate the approach to equilibrium by numerical simulations in the isotropic three-dimensional setting.

H. W. Wyld, Burton D. Fried

Vir B. Bulchandani

Abstract Recently, a hydrodynamic description of local equilibrium dynamics in quantum integrable systems was discovered. In the diffusionless limit, this is equivalent to a certain ‘Bethe–Boltzmann’ kinetic equation, which has the form of an integro-differential conservation law in ( 1 + 1 ) D. The purpose of the present work is to investigate the sense in which the Bethe–Boltzmann equation defines an ‘integrable kinetic equation’. To this end, we study a class of N dimensional systems of evolution equations that arise naturally as finite-dimensional approximations to the Bethe–Boltzmann equation. We obtain non-local Poisson brackets and Hamiltonian densities for these equations and derive an infinite family of first integrals, parameterized by N functional degrees of freedom. We find that the conserved charges arising from quantum integrability map to Casimir invariants of the hydrodynamic bracket and their group velocities map to Hamiltonian flows. Some results from the finite-dimensional setting extend to the underlying integro-differential equation, providing evidence for its integrability in the hydrodynamic sense.

Eugene Paul Wigner

The relation between the conservation laws and the symmetry properties of space and time is examined. If the simplest possible equation of motion takes the place of Newton’s equation, the usual conservation laws are not obtained. In the quantum physics, on the contrary, there is a superbundance of conservation laws and their physical significance is not obvious.