Boundary driven Brownian gas

Abstract

We consider a gas of independent Brownian particles on a bounded interval in contact with two particle reservoirs at the endpoints. Due to the Brownian nature of the particles, infinitely many particles enter and leave the system in each time interval. Nonetheless, the dynamics can be constructed as a Markov process with continuous paths on a suitable space. If λ 0 and λ 1 are the chemical potentials of the boundary reservoirs, the stationary distribution (reversible if and only if λ 0 = λ 1 ) is a Poisson point process with intensity given by the linear interpolation between λ 0 and λ 1 . We then analyze the empirical flow that it is defined by counting, in a time interval [0, t], the net number of particles crossing a given point x. In the stationary regime we identify its statistics and show that it is given, apart an x dependent correction that is bounded for large t, by the difference of two independent Poisson processes with parameters λ 0 and λ 1 .

Boundary driven Markov gas: duality and scaling limits

“…In the interval [0, 1], the naive idea would be to study a system of independent Brownian motions that are absorbed at the boundaries 0 and 1, with additional creation of particles at 0 and 1. However, as was noticed in [3], this approach does not work, because in the continuum particles put at the boundary would immediately leave via that same boundary. Therefore, the problem of modeling reservoirs in the continuum is more involved than in the discrete setting.…”

“…Therefore, the problem of modeling reservoirs in the continuum is more involved than in the discrete setting. In [3] the boundary-driven Brownian gas on [0, 1] has been defined as the sum of two independent processes: one process modeling the evolution of the particles initially present in the system and moving as independent Brownian motions absorbed at 0 and at 1; and another Poisson point process adding particles on (0, 1) with well-chosen intensity. The creation of particles no longer takes place at the boundaries only, rather particles are created everywhere in (0, 1) with an intensity that guarantees the prescribed densities of the reservoirs.…”

“…The creation of particles no longer takes place at the boundaries only, rather particles are created everywhere in (0, 1) with an intensity that guarantees the prescribed densities of the reservoirs. The authors in [3] then proceed by proving that this process is Markov.…”

“…Next, a second aim is to generalize this duality to the abstract setting of general boundary driven systems of independent particles in the continuum. For this we will need to generalize the construction of Bertini and Posta [3] first to systems of independent diffusion processes evolving on regular domain D ⊂ R d and second to systems of general independent Markov processes which are allowed to jump and which thus can leave D without hitting its boundary. As a by-product of such general construction and our duality relations two results will follow.…”

“…Section 4 contains the main result of this paper regarding boundary driven systems. We start by recalling the definition of the boundary driven Brownian gas on [0, 1], introduced in [3]. We then generalize this construction to general independent diffusion processes moving on regular domains D ⊂ R d and finally to general independent Markov processes which can make jumps and thus can exit D without hitting its boundary.…”