Kinetic theory for classical and quantum many-body chaos

TL;DR

This paper derives a kinetic equation describing quantum chaos in many-body systems, linking exponential growth in particle exchange to the Lyapunov exponent and unifying transport and scrambling physics.

Contribution

It provides a concrete derivation of a Boltzmann-type kinetic equation for quantum chaos, connecting out-of-time-ordered correlators to particle exchange dynamics.

Findings

Exponential growth in gross particle exchange determines the Lyapunov exponent.

The kinetic equation kernel is set by the 2-to-2 scattering rate.

Transport and scrambling are governed by the same scattering physics.

Abstract

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 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.