Symmetric Inclusion Process Research Articles

Spectral gap of the symmetric inclusion process

Spectral gap of the symmetric inclusion process

Symmetric inclusion process with slow boundary: Hydrodynamics and hydrostatics

We study the hydrodynamic and hydrostatic limits of the one-dimensional open symmetric inclusion process with slow boundary. Depending on the value of the parameter tuning the interaction rate of the bulk of the system with the boundary, we obtain a linear heat equation with either Dirichlet, Robin or Neumann boundary conditions as hydrodynamic equation. In our approach, we combine duality and first-second class particle techniques to reduce the scaling limit of the inclusion process to the limiting behavior of a single, non-interacting, particle.

Condensation of SIP Particles and Sticky Brownian Motion

We study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP particles to be together at time t. This, combined with self-duality, allows us to obtain the explicit scaling for the variance of the fluctuation field.

Exact formulas for two interacting particles and applications in particle systems with duality

We consider two particles performing continuous-time nearest neighbor random walk on $\mathbb{Z}$ and interacting with each other when they are at neighboring positions. The interaction is either repulsive (partial exclusion process) or attractive (inclusion process). We provide an exact formula for the Laplace–Fourier transform of the transition probabilities of the two-particle dynamics. From this we derive a general scaling limit result, which shows that the possible scaling limits are coalescing Brownian motions, reflected Brownian motions and sticky Brownian motions. In particle systems with duality, the solution of the dynamics of two dual particles provides relevant information. We apply the exact formula to the the symmetric inclusion process, that is self-dual, in the condensation regime. We thus obtain two results. First, by computing the time-dependent covariance of the particle occupation number at two lattice sites we characterise the time-dependent coarsening in infinite volume when the process is started from a homogeneous product measure. Second, we identify the limiting variance of the density field in the diffusive scaling limit, relating it to the local time of sticky Brownian motion.

Generalized immediate exchange models and their symmetries

We reconsider the discrete dual of the immediate exchange model and define a more general class of models where mass is split, exchanged and merged. We relate the splitting process to the symmetric inclusion process via thermalization and from that obtain symmetries and self-duality for it and its generalization. We show that analogous properties hold for models where the splitting is related to the symmetric exclusion process or to independent random walkers.

Ergodic theory of the symmetric inclusion process

We prove the existence of a successful coupling for n particles in the symmetric inclusion process. As a consequence we characterise the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in the course of time to an invariant product measure.

Extreme current fluctuations of boundary-driven systems in the large-N limit

Current fluctuations in boundary-driven diffusive systems are, in many cases, studied using hydrodynamic theories. Their predictions are then expected to be valid for currents which scale inversely with the system size. To study this question in detail, we introduce a class of large -N models of one-dimensional boundary-driven diffusive systems, whose current large deviation functions are exactly derivable for any finite number of sites. Surprisingly, we find that for some systems the predictions of the hydrodynamic theory may hold well beyond their naive regime of validity. Specifically, we show that, while a symmetric partial exclusion process exhibits non-hydrodynamic behaviors sufficiently far beyond the naive hydrodynamic regime, a symmetric inclusion process is well described by the hydrodynamic theory for arbitrarily large currents. We conjecture, and verify for zero-range processes, that the hydrodynamic theory captures the statistics of arbitrarily large currents for all models where the mobility coefficient as a function of density is unbounded from above. In addition, for the large-N models, we prove the additivity principle under the assumption that the large deviation function has no discontinuous transitions.

Asymmetric Stochastic Transport Models with $${\mathscr {U}}_q(\mathfrak {su}(1,1))$$ U q ( su ( 1 , 1 ) ) Symmetry

By using the algebraic construction outlined in \cite{CGRS}, we introduce several Markov processes related to the ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the Brownian Energy Process and which turns out to have the symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.

Duality and Stationary Distributions of the “Immediate Exchange Model” and Its Generalizations

We prove that the "Immediate Exchange Model" of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and prove full SU(1,1) for both the continuous and discrete model.

Coupling and Hydrodynamic Limit for the Inclusion Process

We show propagation of local equilibrium for the symmetric inclusion process (SIP) after diffusive rescaling of space and time, as well as the local equilibrium property of the non-equilibrium steady state in the boundary driven SIP. The main tool is self-duality and a coupling between \(n\) SIP particles and \(n\) independent random walkers.

Weakly nonequilibrium properties of a symmetric inclusion process with open boundaries.

We study close-to-equilibrium properties of a one-dimensional symmetric inclusion process (SIP) with finite size by coupling it to two particle reservoirs at the two boundaries with slightly different chemical potentials. The boundaries introduce irreversibility and induce a weak particle current in the system. We calculate the McLennan ensemble for the SIP, which corresponds to the entropy production, and the first-order nonequilibrium correction for the stationary state. We find that the first-order correction is a product measure and is exactly consistent with the local equilibrium measure corresponding to the steady-state density profile in the finite-size SIP, without the need to be in the thermodynamic limit. This provides a novel example for microscopic extensions of the McLennan formula and the interpretation of first-order nonequilibrium correction as entropy production.

Dynamics of condensation in the symmetric inclusion process

The inclusion process is a stochastic lattice gas, which is a natural bosonic counterpart of the well-studied exclusion process and has strong connections to models of heat conduction and applications in population genetics. Like the zero-range process, due to attractive interaction between the particles, the inclusion process can exhibit a condensation transition. In this paper we present first rigorous results on the dynamics of the condensate formation for this class of models. We study the symmetric inclusion process on a finite set $S$ with total number of particles $N$ in the regime of strong interaction, i.e. with independent diffusion rate $m=m_N \to 0$. For the case $N m_N\to\infty$ we show that on the time scale $1/m_N$ condensates emerge from general homogeneous initial conditions, and we precisely characterize their limiting dynamics. In the simplest case of two sites or a fully connected underlying random walk kernel, there is a single condensate hopping over $S$ as a continuous-time random walk. In the non fully connected case several condensates can coexist and exchange mass via intermediate sites in an interesting coarsening process, which consists of a mixture of a diffusive motion and a jump process, until a single condensate is formed. Our result is based on a general two-scale form of the generator, with a fast-scale neutral Wright-Fisher diffusion and a slow-scale deterministic motion. The motion of the condensates is described in terms of the generator of the deterministic motion and the harmonic projection corresponding to the absorbing states of the Wright-Fisher diffusion.

Correlation Inequalities for Interacting Particle Systems with Duality

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other—a process which we call here the symmetric inclusion process (SIP)—or repel each other—a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction,—the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.