We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of $\mathbb{R}^d$ with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman--Kac equation underlying the evolution of generating functions. Our results generalize recent works on density-type observables, and are illustrated for an $N$-particle single-file diffusion on a ring, which can be mapped to a reflected $N$-dimensional diffusion.
We investigate one of the simplest multi-species generalizations of the one dimensional exclusion process with reflective boundaries. The Markov matrix governing the dynamics of the system splits into blocks (sectors) specified by the number of particles of each kind. We find matrices connecting the blocks in a matrix product form. The procedure (generalized matrix ansatz) to verify that a matrix intertwines blocks of the Markov matrix was introduced in the periodic boundary condition, which starts with a local relation [Arita et al, J. Phys. A 44, 335004 (2011)]. The solution to this relation…
We determine the $N$-particle stationary states of a staggered stochastic hopping model with reflective boundaries. It is shown that the stationary states are in fact so-called optimum ground states. Recursion relations in the particle number for any $l$-point density correlation function will be derived. Furthermore, the connection between reflective boundaries and the occurrence of optimum ground states is examined. An explicit counterexample shows that reflective boundaries do not enforce the stationary state to be an optimum ground state.
We study the large deviation functional of the current for the Weakly Asymmetric Simple Exclusion Process in contact with two reservoirs.We compare this functional in the large drift limit to the one of the Totally Asymmetric Simple Exclusion Process, in particular to the Jensen-Varadhan functional. Conjectures for generalizing the Jensen-Varadhan functional to open systems are also stated.
We study the large deviations of time-integrated observables of Markov diffusions that have perfectly reflecting boundaries. We discuss how the standard spectral approach to dynamical large deviations must be modified to account for such boundaries by imposing zero-current conditions, leading to Neumann or Robin boundary conditions, and how these conditions affect the driven process, which describes how large deviations arise in the long-time limit. The results are illustrated with the drifted Brownian motion and the Ornstein-Uhlenbeck process reflected at the origin. Other types of boundaries…
We obtain the exact large deviation functions of the density profile and of the current, in the non-equilibrium steady state of a one dimensional symmetric simple exclusion process coupled to boundary reservoirs with slow rates. Compared to earlier results, where rates at the boundaries are comparable to the bulk ones, we show how macroscopic fluctuations are modified when the boundary rates are slower by an order of inverse of the system length.
Galerkin Methods for Boltzmann-Poisson transport with reflection conditions on rough boundaries
TU Wien - Institute for Analysis and Scientific Computing（维也纳工业大学分析与科学计算研究所）；The University of Texas at Austin - Institute for Computational Engineering and Sciences & Department of Mathematics（德克萨斯大学奥斯汀分校计算工程与科学学院及数学系）
In this thesis, we consider one of the most popular models of non-equilibrium statistical physics: the Asymmetric Simple Exclusion Process, in which particles jump stochastically on a one-dimensional lattice, between two reservoirs at fixed densities, with the constraint that each site can hold at most one particle at a given time. This model has the mathematical property of being integrable, which makes it a good candidate for exact calculations. What interests us in particular is the current of particles that flows through the system (which is a sign of it being out of equilibrium), and how …
We consider the one dimensional asymmetric exclusion process with particle injection and extraction at two boundaries. The model is known to exhibit four distinct phases in its stationary state. We analyze the current statistics at the first site in the low and high density phases. In the limit of infinite system size, we conjecture an exact expression for the current large deviation function.
We consider lattice gas diffusive dynamics with creation-annihilation in the bulk and maintained out of equilibrium by two reservoirs at the boundaries. This stochastic particle system can be viewed as a toy model for granular gases where the energy is injected at the boundary and dissipated in the bulk. The large deviation functional for the particle currents flowing through the system is computed and some physical consequences are discussed: the mechanism for local current fluctuations, dynamical phase transitions, the fluctuation-relation.
We study the totally asymmetric exclusion process (TASEP) on a finite one-dimensional lattice with open boundaries, i.e., in contact with two reservoirs at different potentials. The total (time-integrated) current through the system is a random variable that scales linearly with time in the long time limit. We give a parametric representation for the generating function of the cumulants of the current, which is related to the large deviation function by Laplace transform. This formula is valid for all system sizes and for all values of the boundary coupling parameters.
We study rare events in systems of diffusive fields driven out of equilibrium by the boundaries. We present a numerical technique and use it to calculate the probabilities of rare events in one and two dimensions. Using this technique, we show that the probability density of a slowly varying configuration can be captured with a small number of long wave-length modes. For a configuration which varies rapidly in space this description can be complemented by a local equilibrium assumption.
We study the probability of arbitrary density profiles in conserving diffusive fields which are driven by the boundaries. We demonstrate the existence of singularities in the large-deviation functional, the direct analog of the free-energy in non-equilibrium systems. These singularities are unique to non-equilibrium systems and are a direct consequence of the breaking of time-reversal symmetry. This is demonstrated in an exactly-solvable model and also in numerical simulations on a boundary-driven Ising model. We argue that this singular behavior is expected to occur in models where the compre…
The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in non--equilibrium, namely for non reversible systems. In this paper we consider a simple example of a non--equilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so called quasi potential,…
We describe how to solve the problem of Taylor dispersion in the presence of absorbing boundaries using an exact stochastic formulation. In addition to providing a clear stochastic picture of Taylor dispersion, our method leads to closed-form expressions for all the moments of the convective displacement of the dispersing particles in terms of the transverse diffusion eigenmodes. We also find that the cumulants grow asymptotically linearly with time, ensuring a Gaussian distribution in the long-time limit. As a demonstration of the technique, the first two longitudinal cumulants (yielding resp…
We study an open-boundary version of the on-off zero-range process introduced in Hirschberg et al. [Phys. Rev. Lett. 103, 090602 (2009)]. This model includes temporal correlations which can promote the condensation of particles, a situation observed in real-world dynamics. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices, and also explore the large-deviation properties of the particle current. Analytical and numerical calculations show that, although the particle distribution is well described by…
Dynamical phase transitions are crucial features of the fluctuations of statistical systems, corresponding to boundaries between qualitatively different mechanisms of maintaining unlikely values of dynamical observables over long periods of time. They manifest themselves in the form of non-analyticities in the large deviation function of those observables. In this paper, we look at bulk-driven exclusion processes with open boundaries. It is known that the standard asymmetric simple exclusion process exhibits a dynamical phase transition in the large deviations of the current of particles flowi…
Fractional Brownian motion (FBM), a non-Markovian self-similar Gaussian stochastic process with long-ranged correlations, represents a widely applied, paradigmatic mathematical model of anomalous diffusion. We report the results of large-scale computer simulations of FBM in one, two, and three dimensions in the presence of reflecting boundaries that confine the motion to finite regions in space. Generalizing earlier results for finite and semi-infinite one-dimensional intervals, we observe that the interplay between the long-time correlations of FBM and the reflecting boundaries leads to strik…
We have considered a one-dimensional coagulation-decoagulation system of classical particles on a finite lattice with reflecting boundaries. It is known that the system undergoes a phase transition from a high-density to a low-density phase. Using a matrix product approach we have obtained an exact expression for the average entropy production rate of the system in the thermodynamic limit. We have also performed a large deviation analysis for fluctuations of entropy production rate and particle current. It turns out that the characteristics of the kink in the large deviation function can be us…
Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude $1/L$ in an interval of length $L$ found for reflected normal Brownian motion. While for superdiffusion, cor…
We study spectral gaps of the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries in the maximal current phase. Earlier results for the model with periodic boundaries suggest that the gaps contributing to the universal KPZ regime may be understood as points on an infinite genus Riemann surface built from a parametric representation of the cumulant generating function of the current. We perform explicit analytic continuations from the known large deviations of the current for open TASEP, and confirm the results for the gaps by an exact Bethe ansatz calculati…
We study the statistical properties of the long-time dynamics of the rule 54 reversible cellular automaton (CA), driven stochastically at its boundaries. This CA can be considered as a discrete-time and deterministic version of the Fredrickson-Andersen kinetically constrained model (KCM). By means of a matrix product ansatz, we compute the exact large deviation cumulant generating functions for a wide range of time-extensive observables of the dynamics, together with their associated rate functions and conditioned long-time distributions over configurations. We show that for all instances of b…
The presence of a confining boundary can modify the local structure of a liquid markedly. In addition, small samples of finite size are known to exhibit systematic deviations of thermodynamic quantities relative to their bulk values. Here, we consider the static structure factor of a liquid sample in slab geometry with open boundaries at the surfaces, which can be thought of as virtually cutting out the sample from a macroscopically large, homogeneous fluid. This situation is a relevant limit for the interpretation of grazing-incidence diffraction experiments at liquid interfaces and films. We…
We consider the fluctuations of generalized currents in stochastic Markovian dynamics. The large deviations of current fluctuations are shown to obey a Gallavotti-Cohen (GC) type symmetry in systems with a finite state space. However, this symmetry is not guaranteed to hold in systems with an infinite state space. A simple example of such a case is the Zero-Range Process (ZRP). Here we discuss in more detail the already reported breakdown of the GC symmetry in the context of the ZRP with open boundaries and we give a physical interpretation of the phases that appear. Furthermore, the earlier a…
