我们研究了一维硬杆气体中准粒子的随机动力学，重点分析其均值、方差和自相关函数在两种初始态下的演化：(i) 具有长程相关（LR）的初始态和 (ii) 无长程相关的初始态。我们推导了前者相空间密度相关性的解析结果，以补充后者已知的结果。这些结果使我们能够获得准粒子的均值、方差和自相关函数的表达式，适用于两种初始态。长程相关引入了均值欧拉广义水动力学（GHD）方程的扩散尺度修正，修改了标准的局部平衡形式，我们的发现揭示了修正项的形式取决于初始态中存在的长程相关。
英文摘要：
We investigate the stochastic dynamics of quasiparticles within a gas of hard rods in one dimension for two choices of initial states: one with long-range correlations and the other without it. We derive analytical results for the phase space density correlations in the former case to complement the known results for the latter case. These results enable us to obtain explicit expressions for the mean, variance, autocorrelation and cross-correlation of individual quasiparticles, extending previous results to inhomogeneous cases. We also propose two methods for simulating an ensemble of trajectories of a quasiparticle with fixed initial position and velocity, and verify our analytical results on correlations. The long-range correlations introduce a diffusive-scale correction to the mean Euler generalized hydrodynamic equations, modifying the standard local equilibrium form, and our findings reveal that the form of the correction term depends on the long-range correlations present in the system.
We extend beyond the Euler scales the hydrodynamic theory for quantum and classical integrable models developed in recent years, accounting for diffusive dynamics and local entropy production. We review how the diffusive scale can be reached via a gradient expansion of the expectation values of the conserved fields and how the coefficients of the expansion can be computed via integrated steady-state two-point correlation functions, emphasising that PT-symmetry can fully fix the inherent ambiguity in the definition of conserved fields at the diffusive scale. We develop a form factor expansion t…
We address the hydrodynamics of operator spreading in interacting integrable lattice models. In these models, operators spread through the ballistic propagation of quasiparticles, with an operator front whose velocity is locally set by the fastest quasiparticle velocity. In interacting integrable systems, this velocity depends on the density of the other quasiparticles, so equilibrium density fluctuations cause the front to follow a biased random walk, and therefore to broaden diffusively. Ballistic front propagation and diffusive front broadening are also generically present in non-integrable…
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…
We use a semiclassical approach to study out of equilibrium dynamics and transport in quantum systems with massive quasiparticle excitations having internal quantum numbers. In the universal limit of low energy quasiparticles, the system is described in terms of a classical gas of colored hard-core particles. Starting from an inhomogeneous initial state, in this limit we give analytic expressions for the space and time dependent spin density and spin current profiles. Depending on the initial state, the spin transport is found to be ballistic or diffusive. In the ballistic case we identify a `…
We derive a large-scale hydrodynamic equation, including diffusive and dissipative effects, for systems with generic static position-dependent driving forces coupling to local conserved quantities. We show that this equation predicts entropy increase and thermal states as the only stationary states. The equation applies to any hydrodynamic system with any number of local, PT-symmetric conserved quantities, in arbitrary dimension. It is fully expressed in terms of elements of an extended Onsager matrix. In integrable systems, this matrix admits an expansion in the density of excitations. We eva…
The emergence of superdiffusive spin dynamics in integrable classical and quantum magnets is well established by now, but there is no generally valid theoretical explanation for this phenomenon. A fundamental difficulty is that the hydrodynamic fluctuations of conserved quasiparticle modes are purely diffusive. We argue that in isotropic integrable magnets, a complete hydrodynamic description must include soft "gauge" degrees of freedom, that arise from spontaneous breaking of the Bethe pseudovacuum symmetry. We show that the coarse-grained time evolution of these modes lies in the Kardar-Pari…
We show that hydrodynamic diffusion is generically present in many-body interacting integrable models. We extend the recently developed generalised hydrodynamic (GHD) to include terms of Navier-Stokes type which lead to positive entropy production and diffusive relaxation mechanisms. These terms provide the subleading diffusive corrections to Euler-scale GHD for the large-scale non-equilibrium dynamics of integrable systems, and arise due to two-body scatterings among quasiparticles. We give exact expressions for the diffusion coefficients. Our results apply to a large class of integrable mode…
The sine-Gordon model appears as the low-energy effective field theory of various one-dimensional gapped quantum systems. Here we investigate the dynamics of generic, non-integrable systems belonging to the sine-Gordon family at finite temperature within the semiclassical approach. Focusing on time scales where the effect of nontrivial quasiparticle scatterings becomes relevant, we obtain universal results for the long-time behavior of dynamical correlation functions. We find that correlation functions of vertex operators behave neither ballistically nor diffusively but follow a stretched expo…
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…
For a decade the fate of a one-dimensional gas of interacting bosons in an external trapping potential remained mysterious. We here show that whenever the underlying integrability of the gas is broken by the presence of the external potential, the inevitable diffusive rearrangements between the quasiparticles, quantified by the diffusion constants of the gas, eventually lead the system to thermalise at late times. We show that the full thermalising dynamics can be described by the generalised hydrodynamics with diffusion and force terms, and we compare these predictions with numerical simulati…
