Kinetic theory for classical and quantum many-body chaos

TL;DR

The paper establishes a precise link between many-body quantum chaos and Boltzmann-type kinetics in dilute, weakly coupled theories by showing that the late-time growth of the out-of-time-ordered correlator can be captured by a gross energy-exchange kinetic equation with an energy-weighting function that becomes $\mathcal{E}(E)\to 1/E$ at high temperature. The authors derive this kinetic equation from both a Boltzmann framework and the OTOC via Bethe-Salpeter equations, revealing that the chaos kernel shares the same underlying $2\to2$ scattering physics as transport, but with an instability that yields Lyapunov exponents $\lambda_L=-\xi_m$. They demonstrate a near-one-to-one mapping between the OTOC dynamics and the gross-exchange kinetic description, showing that the exponential growth spectrum of scrambling matches the chaos-encoded part of the kinetic equation up to a similarity transformation. The work also discusses analytic chaos signatures, the role of the Ehrenfest time, and how the Lyapunov exponents can be computed from microscopic scattering data, providing a practical route to quantify chaos in perturbative quantum field theories.

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.

Figures (1)

• Figure 1: The spectra of the kernel $\mathcal{L}(\mathbf{p},\mathbf{l})$ for the linearized Boltzmann equation (and also of $\left\langle T^{xy}(k_z), T^{xy}(-k_z) \right\rangle_R$, cf. Eq. \ref{['Txyxy']}) (top left) and of the kernel $\mathcal{L}_{EX}(\mathbf{p},\mathbf{l})$ for the kinetic equation for the OTOC (top right) are plotted over the complex $\omega$ plane and in the limit of $\beta m \to 0$. In the lower half of the complex $\omega$ plane, there is a dense sequence of numerically obtained poles. In both spectra, these poles are believed to be the signature of a branch cut. See Moore:2018mma and also Romatschke:2015gicGrozdanov:2016vggKurkela:2017xisSolana:2018pbk. In the upper half of the complex $\omega$ plane, only the kernel $\mathcal{L}_{ EX}(\mathbf{p},\mathbf{l})$ has distinct poles which are identified with the Lyapunov exponents, as explained below equation \ref{['eq:label']}. The dependence of these two Lyapunov exponents and the branch cuts on $\beta m$ is depicted in the inlay (bottom). For large values of $\beta m$, the Lyapunov exponents decay exponentially. The plots are obtained by diagonalizing the kernels of the integral equations \ref{['BSE-Boltzmann1']} and \ref{['eq:kineticOTOC']} after a discretization with $N=1000$ grid points on the domain $p\in[m/N, N\times m]$. The discretization is not uniform. This is done in order for the diagonalization to appropriately account for the contributions of both the soft momenta and collinear momenta $\mathbf{p}\approx \mathbf{l}$, which are not negligible even when both $\mathbf{p}$ and $\mathbf{l}$ are large Jeon:1994ifExtended. The finite size of the branch cuts, i.e. its end point for large Im($\omega$), is related to finite domain of the discretization procedure.