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In our contribution we study stochastic models in one space dimension with two conservation laws. One model is the coupled continuum stochastic Burgers equation, for which each current is a sum of quadratic non-linearities, linear diffusion, and spacetime white noise. The second model is a two-lane stochastic lattice gas. As distinct from previous studies, the two conserved densities are tuned such that the flux Jacobian, a $2 \times 2$ matrix, has coinciding eigenvalues. In the steady state, investigated are spacetime correlations of the conserved fields and the time-integrated currents at the origin. For a particular choice of couplings the dynamical exponent 3/2 is confirmed. Furthermore, at these couplings, continuum stochastic Burgers equation and lattice gas are demonstrated to be in the same universality class., Comment: 34 pages, 9 figures

Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behaviour. Although such behaviour has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and non-integrable) when coupled to two reservoirs at two different temperatures/magnetization. In both the integrable case and broken-integrability (but spin-symmetry preserving), we report anomalous scaling of spin current with system size ($\mathbb{J}^s \propto L^{-\mu}$) with an exponent, $\mu \approx 2/3$, falling under the KPZ universality class. On the other hand, it is noteworthy that energy current remains ballistic ($\mathbb{J}^e \propto L^{-\eta}$ with $\eta \approx 0$) in the purely integrable case and there is departure from ballistic behaviour ($\eta > 0$) when integrability is broken regardless of spin-symmetry. Under nonequilibrium conditions, we have thoroughly investigated spatial profiles of local magnetization and energy. We find interesting nonlinear spatial profiles which are hallmarks of anomalous transport. We also unravel subtle striking differences between the equilibrium and nonequilibrium steady state through the lens of spatial spin-spin correlations., Comment: 13 pages, 10 figures (including supplementary material)

The out-of-time-ordered correlator (OTOC) has emerged as an interesting object in both classical and quantum systems for probing the spatial spread and temporal growth of initially local perturbations in spatially extended chaotic systems. Here, we study the (classical) OTOC and its ``light-cone'' in the nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical simulations. We also show that the linearized KS equation exhibits a qualitatively similar OTOC and light-cone, which can be understood via a saddle-point analysis of the linearly unstable modes. Given the deep connection between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ which is stochastic) equations, we also explore the OTOC in the KPZ equation. While our numerical results in the KS case are expected to hold in the continuum limit, for the KPZ case it is valid in a discretized version of the KPZ equation. More broadly, our work unravels the intrinsic interplay between noise/instability, nonlinearity and dissipation in partial differential equations (deterministic or stochastic) through the lens of OTOC., Comment: 8 pages (including supplemental material), 5 figures, updated

Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g. generalized hydrodynamics) have highlighted the role of integrability as well as spin-symmetry in KPZ behaviour. However understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have deep analogy with the quantum spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behaviour prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin-symmetry plays a central role even in the non-perturbative regime. On the other hand, in the non-perturbative regime, we find that energy correlations exhibit clear diffusive behaviour. We also study the classical analog of out-of-time-ordered correlator (OTOC) and Lyapunov exponents. We find significant presence of chaos for the integrability-broken cases even though KPZ behaviour remains robust. The robustness of KPZ behaviour is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms., Comment: 10 pages, 9 figures (including supplementary material)

In this article, we investigate a multispecies generalization of the single-species asymmetric simple exclusion process defined on an open one-dimensional lattice. We devise an exact projection scheme to find the phase diagram in terms of densities and currents of all species. In most of the phases, one or more species are absent in the system due to dynamical expulsion. We observe shocks as well in some regions of the phase diagram. We explain the density profiles using a generalized shock structure that is substantiated by numerical simulations., Comment: 16 pages, 2 figures

Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) \textit{stochastic} partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state (NESS) of the one-dimensional Kuramoto-Sivashinsky (KS) \textit{deterministic} PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile $h(x,t)$ for different initial conditions. We establish, therefore, that the statistical properties of the 1D KS PDE in this state are in the 1D KPZ universality class.

We study a multispecies generalization of a left-permeable asymmetric exclusion process (LPASEP) in one dimension with open boundaries. We determine all phases in the phase diagram using an exact projection to the LPASEP solved by us in a previous work. In most phases, we observe the phenomenon of dynamical expulsion of one or more species. We explain the density profiles in each phase using interacting shocks. This explanation is corroborated by simulations., Comment: 19 pages, 11 figures, minor improvements, references added

We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the `slower' species but the right boundary is not. We find a matrix product solution for the stationary state, and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero., Comment: 22 pages, 6 figures, introduction improved, typos fixed, final version

The asymmetric exclusion process is an idealised stochastic model of transport, whose exact solution has given important insight into a general theory of nonequilibrium statistical physics. In this work, we consider a totally asymmetric exclusion process with multiple species of particles on a one-dimensional lattice in contact with reservoirs. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: when the density of a species becomes zero throughout the system, and dynamical localisation when the density of a species is nonzero only within an interval far from the boundaries. We give a complete explanation of the macroscopic features of the phase diagram using what we call nested fat shocks., Comment: Scientific Reports document style, 9 pages, 5 figures. Stylistic changes made, typos corrected, supplementary material added in the ancillary directory, final version