Analytic structure of nonhydrodynamic modes in kinetic theory

TL;DR

This work analyzes how nonhydrodynamic modes govern the approach to hydrodynamics in a weakly coupled relativistic kinetic theory with momentum-dependent relaxation time. It develops analytic expressions for retarded correlators, revealing how branch cuts from noncollective excitations interface with hydrodynamic poles and showing that there is no sharp onset of fluid behavior. The study demonstrates dehydrodynamization at late times when nonhydrodynamic modes dominate and shows that gradient expansions are generally asymptotic, yet can be fully recovered via Borel summation from perturbative data. The results illuminate the rich analytic structure of transport in kinetic theory and offer a controlled framework for understanding fluid dynamic emergence and its limitations in real quantum systems.

Abstract

How physical systems approach hydrodynamic behavior is governed by the decay of nonhydrodynamic modes. Here, we start from a relativistic kinetic theory that encodes relaxation mechanisms governed by different timescales thus sharing essential features of generic weakly coupled nonequilib- rium systems. By analytically solving for the retarded correlation functions, we clarify how branch cuts arise generically from noncollective particle excitations, how they interface with poles arising from collective hydrodynamic excitations, and to what extent the appearance of poles remains at best an ambiguous signature for the onset of fluid dynamic behavior. We observe that processes that are slower than the hydrodynamic relaxation timescale can make a system that has already reached fluid dynamic behavior to fall out of hydrodynamics at late times. In addition, the analytical control over this model allows us to explicitly demonstrate how the hydrodynamic gradient expansion of the correlation functions can be Borel resummed such that the full nonperturbative information is recovered using perturbative input only.

Figures (15)

• Figure 1: Analytic structure of the retarded energy momentum correlation function in the shear channel $G^{0x,0x}(\omega, k)$ in the complex frequency plane $\omega$ for the kinetic theory (\ref{['eq1']}). The parts of the cut marked with red crosses correspond to medium constituent particles with lifetimes longer than the hydrodynamical decay time and will eventually dominate the correlation function at late times. The upper complex half plane is analytic by causality whereas for $|{\rm Re}\,\omega | > k$ the correlation function is analytic by locality of the scattering kernel. The nonanalytic features of the function are confined to the grey area.
• Figure 2: Diagram of (\ref{['eq4']}) contributing to a retarded correlation function.
• Figure 3: Left hand side: schematic picture of a perturbation in an equilibrium state that displays sheets of overdensity at wavelength $2\pi/k$. For a massless, free streaming gas at time $t$, the dynamical response at a position $x$ is given by integrating contributions along the circle of radius $c\, t$. Right hand side: Analytic structure of the retarded correlation functions $G_R^{\alpha \beta, \gamma\delta}(\omega,k)$ in the complex frequency plane. The physics of free streaming particles is reflected in a branch cut along the real axis.
• Figure 4: Analytic structure of the retarded shear correlation function $G_R^{0x, 0x}(\omega,k)$ in the complex frequency plane for the kinetic theory with scale-independent relaxation time (\ref{['eq13']}).
• Figure 5: The real (left plot) and imaginary (right plot) part of the shear channel retarded correlation function, $G_R^{0x,0x}(\bar{\omega},\bar{k})$, evaluated for $\bar{k} = 0.4$ and plotted as a function of complex $\bar{\omega}$. The function $G_R^{0x,0x}(\bar{\omega},\bar{k})$ is calculated according to eq. (\ref{['shear']}) with integral moments evaluated according to (\ref{['eq48']}) from the generating function $H_a$ in (\ref{['eq57']}).