Integration of the quantum kinetic equation

M. Bönitz

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.

A. N. Temchin

A. Ziya Akcasu

Kinetic equations for the phase---space-time correlation functions contain memory functions that involve projection operators. It is shown that these memory functions can be represented by integral equations involving only real-time correlation functions, thereby eliminating the projection operators completely in the kinetic description of correlation functions. The weak-coupling and density expansions of the memory functions have been obtained through these integral equations.

Pierre-Jean Nacher, Geneviève Tastevin, Franck Laloë

Abstract We apply the formalism of the free Wigner transform to the derivation of a kinetic equation valid for dilute quantum gases; this formalism is compatible with a precise treatment of two‐body correlations and allows the inclusion of second virial corrections. We generalize the results obtained in previous articles where the calculation of the collision integral was limited to zero (“local”) and first order gradient terms; here we use a more general (“non‐local”) form for the interaction term which is an integral of higher dimension (18). The conservation laws are studied in this formalism, which implies non‐linear expressions for the local hydrodynamic quantities in terms of the function appearing in the kinetic equation. The pressure at thermal equilibrium is then evaluated and found to be equal to the value predicted by the Beth‐Uhlenbeck formula, without adjusting any free parameter of the theory.

William C. Schieve, Lawrence Paul Horwitz