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We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${mathbb Z}^n$, where the jump rates are asymmetric and long-range of order $|x|^{-(n+alpha)}$ for a particle displacement of order $|x|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<alpha<1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $alphageq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.
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We study the hydrodynamic limits of the simple exclusion processes and the zero range processes on crystal lattices. For a periodic realization of crystal lattice, we derive the hydrodynamic limit for the exclusion processes and the zero range processes, which depends on both the structure of crystal lattice and the periodic realization. Even through the crystal lattices have inhomogeneous local structure, for all periodic realizations, we apply the entropy method to derive the hydrodynamic limits. Also, we discuss how the limit equation depends on the choices of the realizations.
In this paper we consider three classes of interacting particle systems on $mathbb Z$: independent random walks, the exclusion process, and the inclusion process. We allow particles to switch their jump rate (the rate identifies the type of particle) between $1$ (fast particles) and $epsilonin[0,1]$ (slow particles). The switch between the two jump rates happens at rate $gammain(0,infty)$. In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by $N^{-1}$, time by $N^2$, the switching rate by $N^{-2}$, and letting $Ntoinfty$. The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Ficks law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on $[N]={1,ldots,N}$, adding boundary reservoirs at sites $1$ and $N$ of fast and slow particles, respectively. Inside $[N]$ particles move as before, but now particles are injected and absorbed at sites $1$ and $N$ with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Ficks law made possible by the switching between types.
The presence of non-local and long-range interactions in quantum systems induces several peculiar features in their equilibrium and out-of-equilibrium behavior. In current experimental platforms control parameters such as interaction range, temperature, density and dimension can be changed. The existence of universal scaling regimes, where diverse physical systems and observables display quantitative agreement, generates a common framework, where the efforts of different research communities can be -- in some cases rigorously -- connected. Still, the application of this general framework to particular experimental realisations requires the identification of the regimes where the universality phenomenon is expected to appear. In the present review we summarise the recent investigations of many-body quantum systems with long-range interactions, which are currently realised in Rydberg atom arrays, dipolar systems, trapped ion setups and cold atoms in cavity experiments. Our main aim is to present and identify the common and (mostly) universal features induced by long-range interactions in the behaviour of quantum many-body systems. We will discuss both the case of very strong non-local couplings, i.e. the non-additive regime, and the one in which energy is extensive, but nevertheless low-energy, long wavelength properties are altered with respect to the short-range limit. Cases of competition with other local effects in the above mentioned setups are also reviewed.
We prove almost sure Euler hydrodynamics for a large class of attractive particle systems on $Z$ starting from an arbitrary initial profile. We generalize earlier works by Seppalainen (1999) and Andjel et al. (2004). Our constructive approach requires new ideas since the subadditive ergodic theorem (central to previous works) is no longer effective in our setting.
Harnessing power-law interactions ($1/r^alpha$) in a large variety of physical systems are increasing. We study the dynamics of chiral spin chains as a possible multi-directional quantum channel. This arises from the nonlinear character of the dispersion with complex quantum interference effects. Using complementary numerically and analytical techniques, we engineer models to transfer quantum states. We illustrate our approach using the long-range XXZ model modulated by Dzyaloshinskii-Moriya (DM) interaction. With exploring non-equilibrium dynamics after a local quantum quench, we identify at fully nonlocal regime (which breaks generalized Lieb-Robinson bounds ) the interplay of interaction range $alpha$ and Dzyaloshinskii-Moriya coupling gives rise to spatially asymmetric spin excitations transport. This could be interesting for quantum information protocols to transfer quantum states and maybe testable with current trapped-ion experiments. We further explore the growth of block entanglement entropy in these systems and the order of magnitude reduction distinguished. A possible effective interaction induces by DM coupling and integrability breaking in these systems is discussed.
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