Kinetic theory of quantum gases: beyond the Boltzmann equation

Alexander E. Meyerovich

Geneviève Tastevin, Pierre-Jean Nacher, Franck Laloë

Xu-Yao Hu, Vladimir Rosenhaus

A bstract We study the kinetic theory of a weakly interacting quantum field. Assuming a state that is close to homogeneous and stationary, we derive a closed kinetic equation for the rate of change of the occupation numbers, perturbatively in the coupling. For a dilute gas, this reproduces the quantum Boltzmann equation, which only accounts for two-to-two scattering processes. Our expression goes beyond this, with terms accounting for multi-particle scattering processes, which are higher order in the density.

A. E. Meyerovich

Quantum coherent phenomena in Boltzmann gases are studied on the basis of the generalized Waldmann-Snider kinetic equation. The equation takes into account the effects of quantum identity of particles. The conditions are formulated for the propagation of different collective modes in gases of particles with arbitrary internal degrees of freedom. Collective coherent effects are macroscopic manifestations of the molecular field for the transverse (off-diagonal) elements of particle distribution in a ``polarized'' system (a system with a nonuniform population of internal energy levels in an unperturbed state). The simplest results are obtained for particles with equidistant internal-energy levels exploiting the analogy with spin-polarized quantum gases. The accurate study of nonlocal interaction terms permits one to compare classical-kinetic and Fermi-liquid approaches to kinetic phenomena in dilute gases, and shows the limitations of both approaches. In the case of collective modes, more accurate schemes of derivation of the kinetic equation must be accompanied by the simultaneous calculations of nonlocal contributions.

Shigeji Fujita

Sašo Grozdanov, Koenraad Schalm, Vincenzo Scopelliti

For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase-space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth is a front propagating into an unstable state in phase space. As in conventional Boltzmann transport, which follows from the dynamics of the net particle number density exchange, the kernel of this kinetic integral equation for chaos is also set by the 2-to-2 scattering rate. This provides a mathematically precise statement of the known fact that in dilute weakly coupled gases, transport and scrambling (or ergodicity) are controlled by the same physics.