For a real massless scalar field in general relativity with a negative cosmological constant, we uncover a large class of spherically symmetric initial conditions that are close to anti-de Sitter space (AdS) but whose numerical evolution does not result in black hole formation. According to the AdS/conformal field theory (CFT) dictionary, these bulk solutions are dual to states of a strongly interacting boundary CFT that fail to thermalize at late times. Furthermore, as these states are not stationary, they define dynamical CFT configurations that do not equilibrate. We develop a two-time-scale perturbative formalism that captures both direct and inverse cascades of energy and agrees with our fully nonlinear evolutions in the appropriate regime. We also show that this formalism admits a large class of quasiperiodic solutions. Finally, we demonstrate a striking parallel between the dynamics of AdS and the classic Fermi-Pasta-Ulam-Tsingou problem.

In a recent Comment [1], Bizoń and Rostworowski present a few criticisms of our recent Letter [2]. In particular, they present three arguments: (1) that their own evolutions of the two-mode initial data we had studied collapse to a black hole around a time t ≈ 1080 whereas we found no collapse for times up to roughly t = 1500, (2) that our two-timescale framework (TTF) "cannot be even used to infer stability," and (3) that our "claims about stability islands" may not be correct.We have studied evolutions of this initial data with resolutions higher than we had originally. As displayed in Fig. 1, we do find that higher resolutions display higher concentrations of energy, but we nevertheless have still not observed collapse to a black hole. We note that our code demonstrates convergence to a unique solution even at the late times in question (i.e., around t ≈ 1080). We also emphasize that we have demonstrated that the solution to which our numerical results converge is in fact a solution of the scalar anti-de Sitter (AdS) system we seek to solve. This latter property (consistency) is demonstrated with tests that the constraint residuals and mass loss converge to zero (see our Supplementary Material). And so, while it is possible that the continuum solution does in fact collapse around t ≈ 1080 (and indeed, the work of [3] does see collapse in a code for which both convergence and consistency are verified), we cannot confirm this (in time for the publication of [1]) and hence their claim (1).However even if collapse does take place at a time t ≈ 1080, the main claims of our paper still stand. We could not know if collapse occurred after the time which we ran our code, and so collapse for some time soon after t = 1500 was always a possibility. And so if instead collapse occurs at t ≈ 1080, there is no change to our claims. Within that time one still finds both direct and inverse cascades which is now confirmed by the Comment.Another important point is that, were we to decrease the initial amplitude , we would certainly observe any possible collapse pushed to times later than t ≈ 1080.Regarding their claims (2) and (3), there seems to be some misunderstanding of what we tried to communicate in our Letter. Determining whether some scalar perturbation of AdS with arbitrarily small amplitude collapses to a black hole is not possible with numerical evolutions which require some finite , finite resolution, and finite evolution time t.More particularly regarding their claim (2), we agree that the TTF cannot be used to infer stability beyond times ∝ −2 . Indeed, we stressed that we have carried our TTF analysis to O( 3 ) so that one can only trust predictions within times scaling as −2 . However, for shorter times, we disagree with the claims of the Comment that our truncation to j max = 47 "does not suffice to capture the dynamics of the turbulent cascade" for 2-mode initial data. As is clear from Fig. 4 of [2], the vast majority of the energy remains in the lowest modes of the system during evolution, and the dynamics of ...

Flow-induced anisotropic thermal conduction in a polymer liquid is studied using force Rayleigh scattering. Time-dependent measurements of the complete thermal diffusivity tensor, which includes one off-diagonal and three diagonal components, are reported on an entangled polymer melt subjected to a uniform shear deformation. These data, in conjunction with mechanical measurements of the stress, provide the first direct evidence that the thermal conductivity tensor and the stress tensor are linearly related in a deformed polymer liquid.

Flow-induced anisotropic thermal conduction in a polyisobutylene melt subjected to shear deformations is studied experimentally. Time-dependent measurements of the full (four components) thermal diffusivity tensor following step shear strain flow are presented. These data were obtained with a novel experimental setup based on the optical technique of forced Rayleigh scattering. Birefringence and stress measurements are made for the same flow, and the well-known stress-thermal rule is found to be satisfied. Thermal diffusivity data and stress data are used to test directly the stress-thermal rule, which is also satisfied for the two cases considered. Consideration of stress-thermal coefficients from the present and previous studies gives preliminary evidence that flow-induced anisotropic thermal conduction is a universal phenomenon for flexible polymers.

Almost no experimental data exist to test theories for the nonisothermal flow of complex fluids. To provide quantitative tests for newly proposed theories, we have developed a holographic grating technique to study energy transport in an amorphous polymer melt subject to flow. Polyisobutylene with weight-averaged molecular mass of 85 kDa is sheared at a rate of 10 s ؊1 , and all nonzero components of the thermal conductivity tensor are measured as a function of time, after cessation. Our results are consistent with proposed generalizations to the energy balance for microstructural fluids, including a generalized Fourier's law for anisotropic media. The data are also consistent with a proposed stress-thermal rule for amorphous polymer melts. Confirmation of the universality of these results would allow numerical modelers to make quantitative predictions for the nonisothermal flow of polymer melts.N early all biological and advanced synthetic materials can exhibit molecular order or structure spontaneously, or be induced by flow. For example, rod-like molecules can form liquid crystals or membranes at equilibrium, and surfactants can self-assemble into worm-like shapes (1). These materials with microstructure exhibit nonequilibrium behavior much more complex than what is observed for simple, low-molecular-weight liquids. Upon flow, they may orient, crystallize, and show stresses many orders of magnitude larger than water. What is more, these large stresses relax back to equilibrium on time scales from seconds to minutes when the flow is stopped (2).The governing dynamics of simple (low-molecular-weight) f luids are well understood. Their derivation begins with straightforward balance equations for mass, momentum, and energy (and possibly angular momentum conservation) applied to a continuum (3). Mass conservation leads to an evolution equation for density , the continuity equation. The second and third balances, however, are not closed; we have too many unknowns and not enough equations. For closure we need additional equations called constitutive relations. For the momentum balance, we need an equation to relate stress to velocity in the f luid, and, in the case of the energy balance, we require two additional equations: a relationship between energy and measurable quantities, usually temperature T and velocity v; and a relationship between heat f lux and temperature. For simple f luids, these constitutive relations are well known. For the stress tensor we use the Newtonian constitutive equation, which states that the stress is linearly related to the instantaneous velocity gradient ٌv, through the viscosity. The energy density is a sum of internal energy density u, and kinetic energy density 1͞2 v 2 . The third relation is Fourier's law for the heat f lux q, which is linearly related to the temperature gradient by a scalar thermal conductivity.Using these three relations, we then have a closed set of evolution equations for the density , the velocity field v, and the temperature field T. If material parameters, ...