Non-hydrodynamic Modes Research Articles

Gapless nonhydrodynamic modes in relativistic kinetic theory

Gapless nonhydrodynamic modes in relativistic kinetic theory

Dispersion Relations Alone Cannot Guarantee Causality.

We show that linear superpositions of plane waves involving a single-valued, covariantly stable dispersion relation ω(k) always propagate outside the light cone unless ω(k)=a+bk. This implies that there is no notion of causality for individual dispersion relations since no mathematical condition on the function ω(k) (such as the front velocity or the asymptotic group velocity conditions) can serve as a sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emerge from a careful cancellation that occurs when one superimposes all the excitation branches of a physical model. This happens automatically in local theories of matter that are covariantly stable. Hence, we find that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need for antiparticles in relativistic quantum mechanics.

Hydrodynamic and non-hydrodynamic excitations in kinetic theory — a numerical analysis in scalar field theory

Viscous hydrodynamics serves as a successful mesoscopic description of the Quark-Gluon Plasma produced in relativistic heavy-ion collisions. In order to investigate, how such an effective description emerges from the underlying microscopic dynamics we calculate the hydrodynamic and non-hydrodynamic modes of linear response in the sound channel from a first-principle calculation in kinetic theory. We do this with a new approach wherein we discretize the collision kernel to directly calculate eigenvalues and eigenmodes of the evolution operator. This allows us to study the Green’s functions at any point in the complex frequency space. Our study focuses on scalar theory with quartic interaction and we find that the analytic structure of Green’s functions in the complex plane is far more complicated than just poles or cuts which is a first step towards an equivalent study in QCD kinetic theory.

Mesoscopic two-point collective dynamics of glass-forming liquids.

The collective density-density and hydrostatic pressure-pressure correlations of glass-forming liquids are spatiotemporally mapped out using molecular dynamics simulations. It is shown that the sharp rise of structural relaxation time below the Arrhenius temperature coincides with the emergence of slow, nonhydrodynamic collective dynamics on mesoscopic scales. The observed long-range, nonhydrodynamic mode is independent of wave numbers and closely coupled to the local structural dynamics. Below the Arrhenius temperature, it dominates the slow collective dynamics on length scales immediately beyond the first structural peak in contrast to the well-known behavior at high temperatures. These results highlight a key connection between the qualitative change in mesoscopic two-point collective dynamics and the dynamic crossover phenomenon.

Universality of energy-momentum response in kinetic theories

We study the response of the energy-momentum tensor in several kinetic theories, from the simple relaxation time approximation (RTA) to Quantum Chromodynamics (QCD). Irrespective of the differences in microscopic properties, we find a remarkable degree of universality in the response functions from conformal theories. We find that the response to scalar perturbations in kinetic theory can be effectively described by a pair of one hydrodynamic sound mode and one non-hydrodynamic mode. We find that even beyond the range of validity of hydrodynamics, the energy-momentum response in position space can be effectively described by one single mode with non-trivial dispersion relation and residue.

Novel features of attractors and transseries in nonconformal Bjorken flows

In this work we investigate the impact of conformal symmetry breaking on hydrodynamization of a far-from-equilibrium fluid. We find a new kind of transseries solutions for the nonconformal hydrodynamic equations of a longitudinal boost invariant expanding plasma. The new transseries solutions unveil a rich physical structure which arises due to the interplay of different physical scales. In the perfect fluid case the nonconformal speed of sound slows down the cooling of the temperature due to the emergence of logarithmic corrections that depends on the mass of the particle. These terms propagate into the perturbative and nonperturbative sectors of the transseries once viscous corrections are included. The logarithmic mass contributions increase the asymptotic value of the Knudsen number while decreasing the damping rate of the transient nonhydrodynamic modes and thus, yielding to an extremely slow hydrodynamization process where flow lines merge to their forward attractor at extremely late times. The early time free streaming expansion is modified and receives logarithmic mass corrections induced by the shear-bulk couplings. The global flow structure and numerical analyses carried out in our work demonstrate the existence of local attraction at early and late times for the shear viscous tensor and bulk viscous pressure.

Cross effects in spin hydrodynamics: Entropy analysis and statistical operator

We revisit the construction of first-order spin hydrodynamics and find that the constitution relations receive the corrections from the cross effects resulting from spin-orbit coupling. Starting from a routine entropy analysis, we show how to identify cross effects and new cross transport coefficients from the second law of thermodynamics. Interestingly, the conventional transport coefficient heat conductivity $\kappa$ is bounded from below by the product of cross transport coefficients, which means the threshold of heat conduction is changed. With recourse to Zubarev's non-equilibrium statistical operator, we reproduce the construction of first-order spin hydrodynamics and identification of cross effects in a more rigorous way. By seeking the dispersion relations of normal modes, we find that these cross effects suppress the attenuation of sound modes and heat mode appearing in conventional hydrodynamics and also have impacts on the damping of non-hydrodynamic spin modes.

Nonacoustic high-frequency collective excitations in a ZrCuAl metallic glass

We report a combined theoretical and simulation study of collective excitations in the three-component metallic glass ${\mathrm{Zr}}_{46}{\mathrm{Cu}}_{46}{\mathrm{Al}}_{8}$. It is shown in that case how one can combine partial current time correlation functions to represent them via total mass and mass-concentration currents, which for the long-wavelength region separately describe hydrodynamic acoustic and nonhydrodynamic opticlike modes. We present dispersions of the longitudinal and transverse acoustic and optic modes over a wide range of wave numbers. Our theoretical analysis of dispersion and damping of three transverse short-wavelength modes sets the basis for the interpretation of already existing experimental data, where the presence of three modes has been disregarded leading to an overestimation of the alleged transverse mode damping.

Using the nonhydrodynamic mode to study the onset of hydrodynamic behavior in ultraperipheral symmetric nuclear collisions

With the attempts of extending the hydrodynamic framework of heavy-ion collision to proton-proton and other small and low energy systems, we are confronted with the question of how small the system can get and still be safely modelled as a fluid. One of the transport coefficients required in the $2^{nd}$ order relativistic viscous hydrodynamics is the shear relaxation time, inclusion of which solves the causality violation problem in the Navier-Stokes equation. In phenomenological studies this coefficient has been taken as a constant and much attention has gone into finding and fixing the shear viscosity to entropy density ratio, $\eta/s$. This transport coefficient also happens to control the non-hydrodynamic mode of the out-of-equilibrium hydrodynamics theory. It has been predicted that for decreasing system size, observables become sensitive to variation in shear relaxation time as a result of increasing dominance of non-hydrodynamic mode, which could potentially indicate breakdown of hydrodynamics. In this study, we try to test this prediction in the peripheral Pb-Pb collisions at $2.76$ TeV and Au-Au collisions at $200$ GeV, with IPGlasma initial condition and $(2+1)-$Dimensional viscous hydrodynamics. We find that elliptic flow does show adequate sensitivity to variation in relaxation time for decreasing system size. The multiplicity rapidity density limit for applicability of hydrodynamics is found to be around $dN/dy\approx10$, with the possibility of refinement in this value given a way to improve the centrality resolution in experimental data for referencing in peripheral collisions.

Symmetric-hyperbolic quasihydrodynamics

We set up a general framework for systematically building and classifying, in the linear regime, causal and stable dissipative hydrodynamic theories that, alongside with the usual hydrodynamic modes, also allow for an arbitrary number of nonhydrodynamic modes with complex dispersion relation (such theories are often referred to as ``quasihydrodynamic''). To increase the number of nonhydrodynamic modes, one needs to add more effective fields to the model. The system of equations governing this class of quasihydrodynamic theories is symmetric hyperbolic, thermodynamically consistent (i.e., the entropy is a Lyapunov function) and can be derived from an action principle. As a first application of the formalism, we prove that, in the linear regime, the Israel-Stewart theory in the Eckart frame and the Israel-Stewart theory in the Landau frame are exactly the same theory. In addition, with an Onsager-Casimir analysis, we show that in strongly coupled plasmas the nonequilibrium degrees of freedom typically appear in pairs, whose members acquire opposite phase under time reversal. We use this insight to modify Cattaneo's model for diffusion, in a way to make its initial transient consistent with the transient dynamics of holographic plasmas.

Characteristic momentum of Hydro+ and a bound on the enhancement of the speed of sound near the QCD critical point

Near the critical point in the QCD phase diagram, hydrodynamics breaks down at a momentum where the frequency of the fastest hydrodynamic mode becomes comparable with the decay rate of the slowest nonhydrodynamic mode. $\mathrm{Hydro}+$ was developed as a framework which extends the range of validity of hydrodynamics beyond that momentum value. This was achieved through coupling the hydrodynamic modes to the slowest nonhydrodynamic mode. In this work, analyzing the spectrum of linear perturbations in $\mathrm{Hydro}+$, we find that a slow mode falls out of equilibrium if its momentum is greater than a characteristic momentum value. That characteristic momentum turns out to be set by the branch points of the dispersion relations. These branch points occur at the critical momenta of so-called spectral curves and are related to the radius of convergence of the derivative expansion. The existence of such a characteristic momentum scale suggests that a particular class of slow modes has no remarkable effect on the flow of the plasma. Based on these results, we show that there is an enhancement of the speed of sound due to the $\mathrm{Hydro}+$ corrections compared to the known thermodynamic corrections near the critical point. This enhancement can not exceed a temperature-dependent upper bound which we derive near the critical point in the QCD phase diagram.

Upper bound of the charge diffusion constant in holography

We investigate the upper bound of charge diffusion constant in holography. For this purpose, we apply the conjectured upper bound proposal related to the equilibration scales (ωeq, keq) to the Einstein-Maxwell-Axion model. (ωeq, keq) is defined as the collision point between the diffusive hydrodynamic mode and the first non-hydrodynamic mode, giving rise to the upper bound of the diffusion constant D at low temperature T as D={omega}_{mathrm{eq}}/{k}_{mathrm{eq}}^2 . We show that the upper bound proposal also works for the charge diffusion and (ωeq, keq), at low T, is determined by D and the scaling dimension ∆(0) of an infra-red operator as left({omega}_{mathrm{eq}},{k}_{mathrm{eq}}^2right)=left(2pi TDelta (0),{omega}_{mathrm{eq}}/Dright) , as for other diffusion constants. However, for the charge diffusion, we find that the collision occurs at real keq, while it is complex for other diffusions. In order to examine the universality of the conjectured upper bound, we also introduce a higher derivative coupling to the Einstein-Maxwell-Axion model. This coupling is particularly interesting since it leads to the violation of the lower bound of the charge diffusion constant so the correction may also have effects on the upper bound of the charge diffusion. We find that the higher derivative coupling does not affect the upper bound so that the conjectured upper bound would not be easily violated.

The breakdown of magneto-hydrodynamics near AdS2 fixed point and energy diffusion bound

We investigate the breakdown of magneto-hydrodynamics at low temperature (T) with black holes whose extremal geometry is AdS2×R2. The breakdown is identified by the equilibration scales (ωeq, keq) defined as the collision point between the diffusive hydrodynamic mode and the longest-lived non-hydrodynamic mode. We show (ωeq, keq) at low T is determined by the diffusion constant D and the scaling dimension ∆(0) of an infra-red operator: ωeq = 2πT∆(0), {k}_{mathrm{eq}}^2 = ωeq/D, where ∆(0) = 1 in the presence of magnetic fields. For the purpose of comparison, we have analytically shown ∆(0) = 2 for the axion model independent of the translational symmetry breaking pattern (explicit or spontaneous), which is complementary to previous numerical results. Our results support the conjectured universal upper bound of the energy diffusion Dle {omega}_{mathrm{eq}}/{k}_{mathrm{eq}}^2:= {v}_{mathrm{eq}}^2{tau}_{mathrm{eq}} where veq := ωeq/keq and τeq := {omega}_{mathrm{eq}}^{-1} are the velocity and the timescale associated to equilibration, implying that the breakdown of hydrodynamics sets the upper bound of the diffusion constant D at low T.

Breakdown of hydrodynamics from holographic pole collision

We study the breakdown of diffusive hydrodynamics in holographic systems dual to neutral dilatonic black holes with extremal near horizon geometries conformal to AdS2 × R2. We find that at low temperatures by tuning the effective gauge coupling constant in the infra-red, the lowest non-hydrodynamic mode, which collides with the charge diffusive mode and sets the scales at which diffusive hydrodynamics breaks down, could be either an infra-red mode or a slow mode, resulting in different scaling behaviors of the local equilibrium scales. We confirm that the upper bound for the charge diffusion constant is always satisfied using the velocity and timescale of local equilibration from the pole collision. We also examine the breakdown of hydrodynamics at general temperature and find that the convergence radius has nontrivial dependence on temperature, in addition to the effective gauge coupling constant.

Linearized dispersion relations in viscous relativistic hydrodynamics

We compute the dispersion relations for scalar, vector and tensor modes of a viscous relativistic fluid, linearized around an equilibrium solution, for a divergence type theory (which, in the linearized theory, includes Israel-Stewart theory and anisotropic hydrodynamics as particular cases) and contrast them to the corresponding results derived from kinetic theory under the relaxation time approximation, and from causal first order theories. We conclude that all approaches give similar dynamics for the scalar and vector modes, while the particular divergence type theory presented here also contains propagating damped tensor waves, in agreement with kinetic theory. Nonhydrodynamic tensor modes are also a feature of holographic fluids. These results support the application of hydrodynamics in problems involving the interaction between fluids and gravitational waves.

Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation

Using the second law of local thermodynamics and the first-order Palatini formalism, we formulate relativistic spin hydrodynamics for quantum field theories with Dirac fermions, such as QED and QCD, in a torsionful curved background. We work in a regime where spin density, which is assumed to relax much slower than other non-hydrodynamic modes, is treated as an independent degree of freedom in an extended hydrodynamic description. Spin hydrodynamics in our approach contains only three non-hydrodynamic modes corresponding to a spin vector, whose relaxation time is controlled by a new transport coefficient: the rotational viscosity. We study linear response theory and observe an interesting mode mixing phenomenon between the transverse shear and the spin density modes. We propose several field-theoretical ways to compute the spin relaxation time and the rotational viscosity, via the Green-Kubo formula based on retarded correlation functions.

Hydrodynamic attractor in a Hubble expansion

We analytically investigate hydrodynamic attractor solutions in both M\"{u}ller-Israel-Stewart (MIS) and kinetic theories in a viscous fluid system undergoing a Hubble expansion with a fixed expansion rate. We show that the gradient expansion for the MIS theory and the Chapman-Enskog expansion for the Boltzmann equation within the relaxation time approximation are factorially divergent and obtain hydrodynamic attractor solutions by applying the Borel resummation technique to those asymptotic divergent series. In both theories, we find that the hydrodynamic attractor solutions are globally attractive and only the first-order non-hydrodynamic mode exists. We also find that the hydrodynamic attractor solutions in the two theories disagree with each other when gradients become large, and that the speed of the attraction is different. Similarities and differences from hydrodynamic attractors in the Bjorken and Gubser flows are also discussed. Our results push the idea of far-from-equilibrium hydrodynamics in systems undergoing a Hubble expansion.

Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: A direct grid coupling approach

The present study proposes an accurate lattice Boltzmann direct coupling algorithm, well suited for industrial purposes, making it highly valuable for aeroacoustic applications. It is indeed known that the convection of vortical structures across a grid refinement interface, where cell size is abruptly doubled, is likely to generate spurious noise that may corrupt the solution over the whole computational domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. Consequently, any interfering noise that may pollute the acoustic predictions must be reduced.The proposed grid refinement algorithm differs from conventionally used ones, in which an overlapping mesh layer is considered. Instead, it provides a direct connection allowing a tighter link between fine and coarse grids, especially with the use of a coherent equilibrium function shared by both grids. Moreover, the direct coupling makes the algorithm more local and prevents the duplication of points, which might be detrimental for massive parallelization. This work follows our first study (Astoul et al. 2020 [1]) on the deleterious effect of non-hydrodynamic modes crossing mesh transitions, which can be addressed using an appropriate collision model: the hybrid recursive regularization. The grid coupling algorithm is assessed in the context of computational aeroacoustics and compared to a widely-used cell-vertex algorithm. The validation benchmark includes the simulation of (1) an acoustic pulse, (2) a vortex transport by a mean flow, and finally, (3) a turbulent circular cylinder wake flow at high Reynolds number. In the end, the proposed approach is proven to drastically reduced the spurious noise generated at grid interfaces, hence, paving the way for accurate and efficient aeroacoustic simulations based on lattice Boltzmann methods.

Critical behaviour of hydrodynamic series

We investigate the time-dependent perturbations of strongly coupled mathcal{N} = 4 SYM theory at finite temperature and finite chemical potential with a second order phase transition. This theory is modelled by a top-down Einstein-Maxwell-dilaton description which is a consistent truncation of the dimensional reduction of type IIB string theory on AdS5×S5. We focus on spin-1 and spin-2 sectors of perturbations and compute the linearized hydrodynamic transport coefficients up to the third order in gradient expansion. We also determine the radius of convergence of the hydrodynamic mode in spin-1 sector and the lowest non-hydrodynamic modes in spin-2 sector. Analytically, we find that all the hydrodynamic quantities have the same critical exponent near the critical point θ = frac{1}{2} . Moreover, we propose a relation between symmetry enhancement of the underlying theory and vanishing of the only third order hydrodynamic transport coefficient θ1, which appears in the shear dispersion relation of a conformal theory on a flat background.

On the universality of AdS2 diffusion bounds and the breakdown of linearized hydrodynamics

The chase of universal bounds on diffusivities in strongly coupled systems and holographic models has a long track record. The identification of a universal velocity scale, independent of the presence of well-defined quasiparticle excitations, is one of the major challenges of this program. A recent analysis, valid for emergent IR fixed points exhibiting local quantum criticality, and dual to IR AdS2 geometries, suggests to identify such a velocity using the time and length scales at which hydrodynamics breaks down — the equilibration velocity. The latter relates to the radius of convergence of the hydrodynamic expansion and it is extracted from a collision between a hydrodynamic diffusive mode and a non-hydrodynamic mode associated to the IR AdS2 region. In this short note, we confirm this picture for holographic systems displaying the spontaneous breaking of translational invariance. Moreover, we find that, at zero temperature, the lower bound set by quantum chaos and the upper one defined by causality and hydrodynamics exactly coincide, determining uniquely the diffusion constant. Finally, we comment on the meaning and universality of this newly proposed prescription.