We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.
We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable syste…
These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.
We obtain the exact generalised hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorised scattering of Toda particles, we obtain the generalised free energy and exact average currents, and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta, and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By expl…
We consider the out-of-equilibrium dynamics of an interacting integrable system in the presence of an external dephasing noise. In the limit of large spatial correlation of the noise, we develop an exact description of the dynamics of the system based on a hydrodynamic formulation. This results in an additional term to the standard generalized hydrodynamics theory describing diffusive dynamics in the momentum space of the quasiparticles of the system, with a time- and momentum-dependent diffusion constant. Our analytical predictions are then benchmarked in the classical limit by comparison wit…
The large-scale properties of homogeneous states after quantum quenches in integrable systems have been successfully described by a semiclassical picture of moving quasiparticles. Here we consider the generalisation for the entanglement evolution after an inhomogeneous quench in noninteracting systems in the framework of generalised hydrodynamics. We focus on the protocol where two semi-infinite halves are initially prepared in different states and then joined together, showing that a proper generalisation of the quasiparticle picture leads to exact quantitative predictions. If the system is i…
We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally non-local and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the…
A generalised phase-space kinetic Boltzmann equation for highly non-equilibrium charged particle transport via localised and delocalised states is used to develop continuity, momentum and energy balance equations, accounting explicitly for scattering, trapping/detrapping and recombination loss processes. Analytic expressions detail the effect of these microscopic processes on the mobility and diffusivity. Generalised Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field-dependence of the mobility. Interesting phen…
The generalised hydrodynamic theory of an electron gas, which does not rely on an assumption of a local equilibrium, is derived as the long-wave limit of a kinetic equation. Apart from the common hydrodynamics variables the theory includes the tensor fields of the higher moments of the distribution function. In contrast to the Bloch hydrodynamics, the theory leads to the correct plasmon dispersion and in the low frequency limit recovers the Navies-Stocks hydrodynamics. The linear approximation to the generalised hydrodynamics is closely related to the theory of highly viscous fluids.
We investigate the non-equilibrium dynamics of the symmetry-resolved Rényi entropies in a one-dimensional gas of non-interacting spinless fermions by means of quantum generalised hydrodynamics, which recently allowed to obtain very accurate results for the total entanglement in inhomogeneous quench settings. Although our discussion is valid for any quench setting accessible with quantum generalised hydrodynamics, we focus on the case of a quantum gas initially prepared in a bipartite fashion and subsequently let evolve unitarily with a hopping Hamiltonian. For this system, we characterise the …
We present a general phase-space kinetic model for charged particle transport through combined localised and delocalised states, capable of describing scattering collisions, trapping, detrapping and losses. The model is described by a generalised Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. The velocity of the centre of mass (CM) and the diffusivity about it are determined analytically, together with the flux transport coefficients. Transient negative values of the free particle CM transport coefficients can be observed due to the trapping to, and det…
Unstable particles rarely feature in conjunction with integrability in 1+1D quantum field theory. However, the family of homogenous sine-Gordon models provides a rare example where both stable and unstable bound states are present in the spectrum whilst the scattering matrix is diagonal and solves the usual bootstrap equations. In the standard scattering picture, unstable particles result from complex poles of the $S$-matrix located in the unphysical sheet of rapidity space. Since they are not part of the asymptotic spectrum, their presence is only felt through the effect they have on physical…
The generalised Boltzmann equation which treats the combined localised and delocalised nature of transport present in certain materials is extended to accommodate time-dependent fields. In particular, AC fields are shown to be a means to probe the trapping and detrapping rates of materials under certain conditions. Conditions leading to dispersive transport are considered, and the signature of fractional/anomalous diffusion under AC electric fields is presented.
We use a Generalised Langevin Equation (GLE) scheme to study the thermal transport of low dimensional systems. In this approach, the central classical region is connected to two realistic thermal baths kept at two different temperatures [H. Ness et al., Phys. Rev. B {\bf 93}, 174303 (2016)]. We consider model Al systems, i.e. one-dimensional atomic chains connected to three-dimensional baths. The thermal transport properties are studied as a function of the chain length $N$ and the temperature difference $ΔT$ between the baths. We calculate the transport properties both in the linear response …
Quantum integrable systems, such as the interacting Bose gas in one dimension and the XXZ quantum spin chain, have an extensive number of local conserved quantities that endow them with exotic thermalization and transport properties. We discuss recently introduced hydrodynamic approaches for such integrable systems from the viewpoint of kinetic theory and extend the previous works by proposing a numerical scheme to solve the hydrodynamic equations for finite times and arbitrary locally equilibrated initial conditions. We then discuss how such methods can be applied to describe non-equilibrium …
It is shown that strong fluctuations preclude a hydrodynamic description of transport phenomena in helimagnets, such as MnSi, at T>0. This breakdown of hydrodynamics is analogous to the one in chiral liquid crystals. Mode-mode coupling effects lead to infinite renormalizations of various transport coefficients, and the actual macroscopic description is nonlocal. At T=0 these effects are weakened due to the fluctuation-dissipation theorem, and the renormalizations remain finite. Observable consequences of these results, as manifested in the neutron scattering cross-section, are discussed
The last few years have seen an explosion of interest in hydrodynamic effects in interacting electron systems in ultra-pure materials. In this paper we briefly review the recent advances, both theoretical and experimental, in the hydrodynamic approach to electronic transport in graphene, focusing on viscous phenomena, Coulomb drag, non-local transport measurements, and possibilities for observing nonlinear effects.
We propose a general formalism, within large deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the "extended fluctuation relations" of Bernard and Doyon follow from …
We study hydrodynamic phonon heat transport in two-dimensional (2D) materials. Starting from the Peierls-Boltzmann equation within the Callaway model, we derive a 2D Guyer-Krumhansl-like equation describing non-local hydrodynamic phonon transport, taking into account the quadratic dispersion of flexural phonons. In additional to Poiseuille flow, second sound propagation, the equation predicts heat current vortices and negative nonlocal thermal conductance in 2D materials, common in classical fluid but scarcely considered in phonon transport. Our results also illustrate the universal transport …
Directed transport of self-propelled particles is numerically investigated in a three-dimensional asymmetric potential. Beside the steric repulsive forces, hydrodynamic interactions between particles have been taken into account in an approximate way. From numerical simulations, we find that hydrodynamic interactions can strongly affect the rectified transport of self-propelled particles. Hydrodynamic interactions enhance the performance of the rectified transport when particles can easily pass across the barrier of the potential, and reduce the rectified transport when particles are mainly tr…
The equations of hydrodynamics including mass, linear momentum, angular momentum, and energy are derived by coarse-graining the microscopic equations of motion for systems consisting of rotary dumbbells driven by internal torques.
We demonstrate that "moderate" departures from the no-slip hydrodynamic boundary condition (hydrodynamic slip lengths in the nanometer range) can result in a very large enhancement - up to two orders of magnitude- of most interfacially driven transport phenomena. We study analytically and numerically the case of neutral solute diffusio-osmosis in a slab geometry to account for non-trivial couplings between interfacial structure and hydrodynamic slip. Possible outcomes are fast transport of particles in externally applied or self-generated gradient, and flow enhancement in nano- or micro-fluidi…
We present a new framework for computing low frequency transport properties of strongly correlated, ergodic systems. Our main assumption is that, when a thermalizing diffusive system is driven at frequency $ω$, domains of size $ξ\sim\sqrt{D/ω}$ can be considered as internally thermal, but weakly coupled with each other. We calculate the transport coefficients to lowest order in the coupling, assuming incoherent transport between such domains. Our framework naturally captures the sub-leading non analytic corrections to the transport coefficients, known as hydrodynamic long time tails. In additi…
We introduce an extension of the generalised $T\bar{T}$-deformation described by Smirnov-Zamolodchikov, to include the complete set of extensive charges. We show that this gives deformations of S-matrices beyond CDD factors, generating arbitrary functional dependence on momenta. We further derive from basic principles of statistical mechanics the flow equations for the free energy and all free energy fluxes. From this follows, without invoking the microscopic Bethe ansatz or other methods from integrability, that the thermodynamics of the deformed models are described by the integral equations…
The Green-Kubo-Nakano formula should be modified in relativistic hydrodynamics because of the problem of acausality and the breaking of sum rules. In this work, we propose a formula to calculate the transport coefficients of causal hydrodynamics based on the projection operator method. As concrete examples, we derive the expressions for the diffusion coefficient, the shear viscosity coefficient, and corresponding relaxation times.
We study the out-of-equilibrium dynamics of the quantum cellular automaton Rule 54 using a time-channel approach. We exhibit a family of (non-equilibrium) product states for which we are able to describe exactly the full relaxation dynamics. We use this to prove that finite subsystems relax to a one-parameter family of Gibbs states. We also consider inhomogeneous quenches. Specifically, we show that when the two halves of the system are prepared in two different solvable states, finite subsystems at finite distance from the centre eventually relax to the non-equilibrium steady state (NESS) pre…
We obtain hydrodynamic descriptions of a broad class of conserved-mass transport processes on a ring. These processes are governed by chipping, diffusion and coalescence of masses, where microscopic probability weights in their nonequilibrium steady states, having nontrivial correlations, are not known. In these processes, we analytically calculate two transport coefficients, the bulk-diffusion coefficient and the conductivity. We, remarkably, find that the two transport coefficients obey an equilibriumlike Einstein relation, although the microscopic dynamics does not satisfy detailed balance …
We find the hydrodynamic equations of a system of particles constrained to be in the lowest Landau level. We interpret the hydrodynamic theory as a Hamiltonian system with the Poisson brackets between the hydrodynamic variables determined from the noncommutativity of space. We argue that the most general hydrodynamic theory can be obtained from this Hamiltonian system by allowing the Righi-Leduc coefficient to be an arbitrary function of thermodynamic variables. We compute the Righi-Leduc coefficients at high temperatures and show that it satisfies the requirements of particle-hole symmetry, w…
We investigate the effects of stochastic interactions on hydrodynamic correlation functions using the Schwinger-Keldysh effective field theory. We identify new "stochastic transport coefficients" that are invisible in the classical constitutive relations, but nonetheless affect the late-time behaviour of hydrodynamic correlation functions through loop corrections. These results indicate that classical transport coefficients do not provide a universal characterisation of long-distance, late-time correlations even within the framework of fluctuating hydrodynamics.
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…
The hydrodynamics of granular gases of viscoelastic particles, whose collision is described by an impact-velocity dependent coefficient of restitution, is developed using a modified Chapman-Enskog approach. We derive the hydrodynamic equations and the according transport coefficients with the assumption that the shape of the velocity distribution function follows adiabatically the decaying temperature. We show numerically that this approximation is justified up to intermediate dissipation. The transport coefficients and the coefficient of cooling are expressed in terms of the elastic and dissi…
Thermal transport in classical fluids is analyzed in terms of a Higher-Order Generalized Hydrodynamics (or Mesoscopic Hydro-Thermodynamics), that is, depending on the evolution of the energy density and its fluxes of all orders. It is derived in terms of a Kinetic Theory based on the Non-Equilibrium Statistical Ensemble Formalism. The general system of coupled evolution equations is derived. Maxwell times - which are of large relevance to determine the character of the motion - are derived. They also have a quite important role for the choice of the contraction of description (limitation in th…
Using the method of nonequilibrium statistical operator by Zubarev, an approach is proposed for the description of kinetics which takes into account the nonlinear hydrodynamic fluctuations for a quantum Bose system. Non-equilibrium statistical operator is presented which consistently describes both the kinetic and nonlinear hydrodynamic processes. Both a kinetic equation for the nonequilibrium one-particle distribution function and a generalized Fokker-Planck equation for nonequilibrium distribution function of hydrodynamic variables (densities of momentum, energy and particle number) are obta…
Looking at bare transport coefficients in fluctuating hydrodynamics
Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）；Faculty of Engineering, Gifu University（岐阜大学工学部）；Department of Physics, Kyoto University（京都大学物理学系）
A new formula to calculate the transport coefficients of the causal dissipative hydrodynamics is derived by using the projection operator method (Mori-Zwanzig formalism) in [T. Koide, Phys. Rev. E75, 060103(R) (2007)]. This is an extension of the Green-Kubo-Nakano (GKN) formula to the case of non-Newtonian fluids, which is the essential factor to preserve the relativistic causality in relativistic dissipative hydrodynamics. This formula is the generalization of the GKN formula in the sense that it can reproduce the GKN formula in a certain limit. In this work, we extend the previous work so as…
Using the analytical Fick-Jacobs approximation formalism and extensive Brownian dynamics simulations we study particle transport through two-dimensional periodic channels with triangularly shaped walls. Directed motion is caused by the interplay of constant bias acting along the channel axis and a pressure-driven flow. In particular, we analyze the particle mobility and the effective diffusion coefficient. The mechanisms of entropic rectification is revealed in channels with a broken spatial reflection symmetry in presence of hydrodynamically enforced entropic trapping. Due to the combined act…
The generalized transport equations for a consistent description of kinetic and hydrodynamic processes in dense gases and liquids are considered. The inner structure of the generalized transport kernels for these equations is established. It is shown how in this approach to obtain the transport equation of molecular hydrodynamics. For the model potential of interaction presented as a sum of the hard spheres potential and certain long-range potential a spectrum of collective modes in the system is investigated.
In the present work we compute the enhancement in the long time transport coefficients due to correlated motion of fluid particles at high density. The fully wave vecor dependent extended mode coupling model is studied with the inclusion of an additional slow variable of the defect density for the amorphous system. We use the extremely slow relaxation of the density correlation function observed in the light scattering experiments on colloids to estimate the input parameters for the model The ratio of long time to short time diffusion coefficient is studied around the the peak of the structure…
Near equilibrium, Green-Kubo relations provide microscopic expressions for macroscopic transport coefficients in terms of equilibrium correlation functions. At their core, they are based on the intimate relationship between response and fluctuations as embodied by the equilibrium fluctuation-dissipation theorem, a connection generically broken far from equilibrium. In this work, we identify a class of perturbations whose response around far-from-equilibrium steady states is linked to steady-state correlation functions via an equilibrium-like fluctuation-response equality. We then utilize this …
We investigate the hydrodynamic stability and transport of magnetic microswimmers in an external field using a kinetic theory framework. Combining linear stability analysis and nonlinear 3D continuum simulations, we show that for sufficiently large activity and magnetic field strengths, a homogeneous polar steady state is unstable for both puller and pusher swimmers. This instability is caused by the amplification of anisotropic hydrodynamic interactions due to the external alignment and leads to a partial depolarization and a reduction of the average transport speed of the swimmers in the fie…
In this paper, we present a Hamiltonian and thermodynamic theory of heat transport on various levels of description. Transport of heat is formulated within kinetic theory of polarized phonons, kinetic theory of unpolarized phonons, hydrodynamics of polarized phonons, and hydrodynamics of unpolarized phonons. These various levels of description are linked by Poisson reductions, where no linearizations are made. Consequently, we obtain a new phonon hydrodynamics that contains convective terms dependent on vorticity of the heat flux, which are missing in the standard theories of phonon hydrodynam…
