Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation

Consistent formulations of relativistic viscous hydrodynamics involve short lived modes, leading to asymptotic rather than convergent gradient expansions. In this Letter we consider the Mueller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identify hydrodynamics as a universal attractor without invoking the gradient expansion. We give strong evidence for the existence of this attractor and then show that it can be recovered from the divergent gradient expansion by Borel summation. This requires careful accounting for the short-lived modes which leads to an intricate mathematical structure known from the theory of resurgence.Comment: Presentation improved, typos fixed; roughly matches the published versio

How does relativistic kinetic theory remember about initial conditions?

Understanding hydrodynamization in microscopic models of heavy-ion collisions has been an important topic in current research. Many lessons obtained within the strongly-coupled (holographic) models originate from the properties of transient excitations of equilibrium encapsulated by short-lived quasinormal modes of black holes. This paper aims to develop similar intuition for expanding plasma systems described by a simple model from the weakly-coupled domain, the Boltzmann equation in the relaxation time approximation. We show that in this kinetic theory setup there are infinitely many transient modes carrying information about the initial distribution function. They all have the same exponential damping set by the relaxation time but are distinguished by different power-law suppressions and different frequencies of oscillations, logarithmic in proper time. We also analyze the resurgent interplay between the hydrodynamics and transients in this setup.Comment: 11 pages, 4 figures; Published in Physical Review

Hydrodynamization in kinetic theory: Transient modes and the gradient expansion

We explore the transition to hydrodynamics in a weakly-coupled model of quark-gluon plasma given by kinetic theory in the relaxation time approximation with conformal symmetry. We demonstrate that the gradient expansion in this model has a vanishing radius of convergence due to the presence of a transient (nonhydrodynamic) mode, in a way similar to results obtained earlier in strongly-coupled gauge theories. This suggests that the mechanism by which hydrodynamic behaviour emerges is the same, which we further corroborate by a novel comparison between solutions of different weakly and strongly coupled models. However, in contrast with other known cases, we find that not all the singularities of the analytic continuation of the Borel transform of the gradient expansion correspond to transient excitations of the microscopic system: some of them reflect analytic properties of the kinetic equation when the proper time is continued to complex values.Comment: 6 pages, 2 figures, v2: author added, major rewrite, mysterious off real axis singularities in the Borel plane explained (!), see also arXiv:1802.08225 [nucl-th] by Heller and Svensson; v3: references added, minor improvements in the text, first 426 terms from Eq. (8) included in the submission; v4: title changed, matches published versio

Coupling hydrodynamics to nonequilibrium degrees of freedom in strongly interacting quark-gluon plasma

Relativistic hydrodynamics simulations of quark-gluon plasma play a pivotal role in our understanding of heavy ion collisions at RHIC and LHC. They are based on a phenomenological description due to Mueller, Israel, Stewart (MIS) and others, which incorporates viscous effects and ensures a well-posed initial value problem. Focusing on the case of conformal plasma we propose a generalization which includes, in addition, the dynamics of the least damped far-from-equilibrium degree of freedom found in strongly coupled plasmas through the AdS/CFT correspondence. We formulate new evolution equations for general flows and then test them in the case of N=4 super Yang-Mills plasma by comparing their solutions alongside solutions of MIS theory with numerical computations of isotropization and boost-invariant flow based on holography. In these tests the new equations reproduce the results of MIS theory when initialized close to the hydrodynamic stage of evolution, but give a more accurate description of the dynamics when initial conditions are set in the pre-equilibrium regime.Comment: Minor improvements; references adde

Entropy Production, Hydrodynamics, and Resurgence in the Primordial Quark-Gluon Plasma from Holography

Microseconds after the Big Bang quarks and gluons formed a strongly-coupled non-conformal liquid driven out-of-equilibrium by the expansion of the Universe. We use holography to determine the non-equilibrium behavior of this liquid in a Friedmann-Lemaitre-Robertson-Walker Universe and develop an expansion for the corresponding entropy production in terms of the derivatives of the cosmological scale factor. We show that the resulting series has zero radius of convergence and we discuss its resurgent properties. Finally, we compute the resummed entropy production rate in de Sitter Universe at late times and show that the leading order approximation given by bulk viscosity effects can strongly overestimate/underestimate the rate depending on the microscopic parameters.Comment: 7 pages, 1 figure; v2: various improvements in presentation, title changed by journal, matches the published versio

Hydrodynamic gradient expansion in gauge theory plasmas

We utilize the fluid-gravity duality to investigate the large order behavior of hydrodynamic gradient expansion of the dynamics of a gauge theory plasma system. This corresponds to the inclusion of dissipative terms and transport coefficients of very high order. Using the dual gravity description, we calculate numerically the form of the stress tensor for a boost-invariant flow in a hydrodynamic expansion up to terms with 240 derivatives. We observe a factorial growth of gradient contributions at large orders, which indicates a zero radius of convergence of the hydrodynamic series. Furthermore, we identify the leading singularity in the Borel transform of the hydrodynamic energy density with the lowest nonhydrodynamic excitation corresponding to a `nonhydrodynamic' quasinormal mode on the gravity side.Comment: v2: 4+2 pages, 2 figures, title changed by journal, supplemental material incorporated into the preprint, energy density coefficients up to 240th order included in the submission (change in normalization with respect to v1), matches published versio

sQGP as hCFT

We examine the proposal to make quantitative comparisons between the strongly coupled quark-gluon plasma and holographic descriptions of conformal field theory. In this note, we calculate corrections to certain transport coefficients appearing in second-order hydrodynamics from higher curvature terms to the dual gravity theory. We also clarify how these results might be consistently applied in comparisons with the sQGP.Comment: 13 page

Equilibration rates in a strongly coupled nonconformal quark-gluon plasma

We initiate the study of equilibration rates of strongly coupled quark-gluon plasmas in the absence of conformal symmetry. We primarily consider a supersymmetric mass deformation within ${\cal N}=2^{*}$ gauge theory and use holography to compute quasinormal modes of a variety of scalar operators, as well as the energy-momentum tensor. In each case, the lowest quasinormal frequency, which provides an approximate upper bound on the thermalization time, is proportional to temperature, up to a pre-factor with only a mild temperature dependence. We find similar behaviour in other holographic plasmas, where the model contains an additional scale beyond the temperature. Hence, our study suggests that the thermalization time is generically set by the temperature, irrespective of any other scales, in strongly coupled gauge theories.Comment: 6 pages, 7 figure

Towards Complexity for Quantum Field Theory States

We investigate notions of complexity of states in continuous quantum-many body systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of Multiscale Entanglement Renormalization Ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum preserving quadratic generators which form $\mathfrak{su}(1,1)$ algebras. On the manifold of Gaussian states generated by these operations the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and holographic complexity proposals.Comment: 6+7 pages, 6 appendices, 2 figures; v2: references added; acknowledgments expanded; appendix F added, reviewing similarities and differences with hep-th/1707.08570; v3: version published in PR