Anomalous diffusion and singular transport from hydrodynamic recoupling
Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎高等师范学院及巴黎文理研究大学）；Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气与计算机工程系）；Department of Theoretical Physics, University of Geneva（日内瓦大学理论物理系）；Geneva Quantum Center, University of Geneva（日内瓦大学日内瓦量子中心）
In charge neutral fluids, such as the Dirac fluid in graphene at the Dirac point, charge transport remains diffusive despite the presence of ballistically propagating sound waves: sound waves ``hydrodynamically decouple'' from the slower charge fluctuations. For quasi-one-dimensional charge neutral fluids, we show that this convective charge diffusion is not smoothly connected to the normal diffusion that arises when momentum conservation is broken by noise (or static impurities). Instead, the charge diffusion constant is a discontinuous function of noise, which (in the weak-noise limit) depends only on the ratio of momentum and energy relaxation rates. In the special limit of momentum-conserving noise (e.g., spatially uniform fluctuations of the Hamiltonian), the diffusion constant diverges in the presence of noise. We describe the resulting superdiffusion in terms of coupled Burgers equations. We present a general mechanism---hydrodynamic recoupling---by which weak noise can induce singular changes in transport coefficients. Our results highlight the limits of zero-noise extrapolation for predicting dynamical quantities like diffusion constants.
Anomalous hydrodynamic fluctuations in the quantum XXZ spin chain
King’s College London（伦敦国王学院）；University of Oxford（牛津大学）；New York University（纽约大学）；CNRS, Université Paris-Dauphine, Université PSL（法国国家科学研究中心，巴黎第九大学，巴黎文理研究大学）；University of Ljubljana（卢布尔雅那大学）
A recently developed non-linear fluctuating hydrodynamics theory has been quite successful in describing various features of anomalous energy transport. However the diffusion and the noise terms present in this theory are not derived from microscopic descriptions but rather added phenomenologically. We here derive these hydrodynamic equations with explicit calculation of the diffusion and noise terms in a one-dimensional model. We show that in this model the energy current scales anomalously with system size $L$ as $\sim L^{-2/3}$ in the leading order with a diffusive correction of order $\sim…
Anomalous Diffusion in Driven Electrolytes due to Hydrodynamic Fluctuations
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）
The problem of anomalous scaling in magnetohydrodynamics turbulence is considered within the framework of the kinematic approximation, in the presence of a large-scale background magnetic field. Field theoretic renormalization group methods are applied to the Kazantsev-Kraichnan model of a passive vector advected by the Gaussian velocity field with zero mean and correlation function $\propto δ(t-t')/k^{d+ε}$. Inertial-range anomalous scaling for the tensor pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain "dangerous" c…
We study the nonequilibrium dynamics of random spin chains that remain integrable (i.e., solvable via Bethe ansatz): because of correlations in the disorder, these systems escape localization and feature ballistically spreading quasiparticles. We derive a generalized hydrodynamic theory for dynamics in such random integrable systems, including diffusive corrections due to disorder, and use it to study non-equilibrium energy and spin transport. We show that diffusive corrections to the ballistic propagation of quasiparticles can arise even in noninteracting settings, in sharp contrast with clea…
Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, high-temperature spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticl…
Atypical eigenstates in the form of quantum scars and fragmentation of Hilbert space due to conservation laws provide obstructions to thermalization in the absence of disorder. In certain models with dipole and $U(1)$ conservation, the fragmentation results in subdiffusive transport. In this paper we study the interplay between scarring and weak fragmentation giving rise to anomalous hydrodynamics in a class of one-dimensional spin-$1$ frustration-free projector Hamiltonians, known as deformed Motzkin chain. The ground states and low-lying excitations of these chains exhibit large entanglement…
Anomalous diffusion, manifest as a nonlinear temporal evolution of the position mean square displacement, and/or non-Gaussian features of the position statistics, is prevalent in biological transport processes. Likewise, collective behavior is often observed to emerge spontaneously from the mutual interactions between constituent motile units in biological systems. Examples where these phenomena can be observed simultaneously have been identified in recent experiments on bird flocks, fish schools and bacterial swarms. These results pose an intriguing question, which cannot be resolved by exist…
A classical wave-particle entity in the form of a millimetric walking droplet can emerge on the free surface of a vertically vibrating liquid bath. Such wave-particle entities have been shown to exhibit hydrodynamic analogs of quantum systems. Using an idealized theoretical model of this wave-particle entity in a tilted potential, we explore its transport behavior. The integro-differential equation of motion governing the dynamics of the wave-particle entity transforms to a Lorenz-like system of ordinary differential equations (ODEs) that drives the particle's velocity. Several anomalous trans…
Collective motion is often modeled within the framework of active fluids, where the constituent active particles, when interactions with other particles are switched off, perform normal diffusion at long times. However, in biology, single-particle superdiffusion and fat-tailed displacement statistics are also widespread. The collective properties of interacting systems exhibiting such anomalous diffusive dynamics, which we call active Lévy matter, cannot be captured by current active fluid theories. Here, we formulate a hydrodynamic theory of active Lévy matter by coarse-graining a microscopic…
We introduce a class of one dimensional deterministic models of energy-volume conserving interfaces. Numerical simulations show that these dynamics are genuinely super-diffusive. We then modify the dynamics by adding a conservative stochastic noise so that it becomes ergodic. System of conservation laws are derived as hydrodynamic limits of the modified dynamics. Numerical evidence shows these models are still super-diffusive. This is proven rigorously for harmonic potentials.
This review summarizes recent advances in our understanding of anomalous transport in spin chains, viewed through the lens of integrability. Numerical advances, based on tensor-network methods, have shown that transport in many canonical integrable spin chains -- most famously the Heisenberg model -- is anomalous. Concurrently, the framework of generalized hydrodynamics has been extended to explain some of the mechanisms underlying anomalous transport. We present what is currently understood about these mechanisms, and discuss how they resemble (and differ from) the mechanisms for anomalous tr…
We present a thermo-kinetic description of anomalous diffusion of single-particles and clusters in a viscoelastic medium in terms of a non-Markovian diffusion equation involving memory functions. The scaling behaviour of these functions is analyzed by considering hydrodynamics and cluster-size space random walk arguments. We explain experimental results on diffusion of Brownian particles in the cytoskeleton, in cluster-cluster aggregation and in a suspension of micelles.
Hydrodynamic tails in chaotic spin chains with quantum group symmetry
University of Chicago（芝加哥大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Ningbo University（宁波大学）；Università di Genova and INFN, Sezione di Genova（热那亚大学及意大利国家核物理研究所热那亚分部）
The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimension. Here we study one of the simplest classical interacting lattice models, the nearest-neighbour Heisenberg chain, with temperature as tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin super-diffusion or an extremely slow crossover to diffusion. This difference also governs the equilibration after a quenc…
Linear fluctuating hydrodynamics is a useful and versatile tool for describing fluids, as well as other systems with conserved fields, on a mesoscopic scale. In one spatial dimension, however, transport is anomalous, which requires to develop a nonlinear extension of fluctuating hydrodynamics. The relevant nonlinearity turns out to be the quadratic part of the Euler currents when expanding relative to a uniform background. We outline the theory and compare with recent molecular dynamics simulations.
In this paper, we discuss relativistic hydrodynamics for a massless Dirac fermion in $(2+1)$ dimensions, which has the parity anomaly -- a global 't Hooft anomaly between $\mathrm{U}(1)$ and parity symmetries. We investigate how hydrodynamics implements the party anomaly, particularly focusing on the transport phenomena at the boundary. Based on the parity anomaly matching and the second law of local thermodynamics, we find $\mathrm{U}(1)$ and entropy currents localized at the boundary as well as the bulk anomalous current with vanishing divergence. These edge currents are similar to the $(1+1…
We present a new approach to determine the small-scale statistical behavior of hydrodynamic turbulence by means of lattice simulations. Using the functional integral representation of the random-force-driven Burgers equation we show that high-order moments of velocity differences satisfy anomalous scaling. The general applicability of Monte Carlo methods provides the opportunity to study also other systems of interest within this framework.
We compute the spin structure factor of XXZ spin chains in the Heisenberg and gapped (Ising) regimes in the high-temperature limit for nonzero magnetization, within the framework of generalized hydrodynamics including diffusive corrections. The structure factor shows a hierarchy of timescales in the gapped phase, owing to $s$-spin magnon bound states (`strings') of various sizes. Although short strings move ballistically, long strings move primarily diffusively as a result of their collisions with short strings. The interplay between these effects gives rise to anomalous power-law decay of the…
We show that the simplest universality classes of fracton hydrodynamics in more than one spatial dimension, including isotropic theories of charge and dipole conservation, can exhibit hidden "quasiconservation laws", in which certain higher multipole moments can only decay due to dangerously irrelevant corrections to hydrodynamics. We present two simple examples of this phenomenon. Firstly, an isotropic dipole-conserving fluid in the infinite plane conserves an infinite number of "harmonic multipole charges" within linear response; we calculate the decay or growth of these charges due to dange…
A planar colloidal monolayer exhibits anomalous collective diffusion due to the hydrodynamic interactions. We investigate how this behavior is affected by the curvature of the monolayer when it resides on the interface of a spherical droplet. It is found that the characteristic times of the dynamics still exhibit the same anomalous scaling as in the planar case. The spatial distribution, however, shows a difference due to the relevance of the radius of the droplet. Since for the droplet this is both a global magnitude, i.e., pertaining the spatial extent of the spherical surface, and a local o…
Hydrodynamic fluctuations of stochastic charged cellular automata
All Souls College, Oxford（牛津大学万灵学院）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·派尔斯理论物理中心）；Department of Physics, New York University（纽约大学物理系）
Bacterial swarms display intriguing dynamical states like active turbulence. Using a hydrodynamic model we now show that such dense active suspensions manifest super-diffusion, via Lévy walks, which masquerades as a crossover from ballistic to diffusive scaling in measurements of mean-squared-displacements, and is tied to the emergence of hitherto undetected oscillatory streaks in the flow. Thus, while laying the theoretical framework of an emergent advantageous strategy in the collective behaviour of microorganisms, our study underlines the essential differences between active and inertial tu…
Giant number fluctuations (GNF) are an anomaly universally observed in active fluids with polar or nematic order. In this paper, we show that GNF arise in the fluid phase of active Brownian particles (ABP), where the polar order is absent. GNF in ABP extends over a large but finite length which characterizes the growing velocity correlations. To suppress unwanted phase separation and allow ones to explore the disordered fluid phase at large activities, we impart the inertia, or the mass, to the ABP. A linearized hydrodynamic theory captures our findings, but only qualitatively. We find numeric…
After reviewing the main features of anomalous energy transport in 1D systems, we report simulations performed with chains of noisy anharmonic oscillators. The stochastic terms are added in such a way to conserve total energy and momentum, thus keeping the basic hydrodynamic features of these models. The addition of this "conservative noise" allows to obtain a more efficient estimate of the power-law divergence of heat conductivity kappa(L) ~ L^alpha in the limit of small noise and large system size L. By comparing the numerical results with rigorous predictions obtained for the harmonic chain…
We discuss anomalous relaxation processes in Davydov one-dimensional chain molecule that consists of an exciton and an acoustic phonon field as a thermal reservoir in the chain. We derive a kinetic equation for the exciton using the complex spectral representation of the Liouville-von Neumann operator. Due to the one-dimensionality, the momentum space separates into infinite sets of disjoint irreducible subspaces dynamically independent of one another. Hence, momentum relaxation occurs only within each subspace toward the Maxwell distribution. We obtain a hydrodynamic mode with transport coeff…
We study the non-equilibrium steady-states of a one-dimensional ($1D1V$) fluid in a finite space region of length $L$. Particles interact among themselves by multi-particle collisions and are in contact with two thermal-wall heat reservoirs, located at the boundaries of the region. After an initial ballistic regime, we find a crossover from a normal (kinetic) transport regime to an anomalous (hydrodynamic) one, above a characteristic size $L_*$. We argue that $L_*$ is proportional to the cube of the collision time among particles. Motivated by the physics of emissive divertors in fusion plasma…
