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At finite concentrations of reacting molecules, kinetics of diffusion-controlled reactions is affected by intra-reactant interactions. As a result, multi-particle reaction statistics cannot be deduced from single-particle results. Here we briefly review a recent progress in overcoming this fundamental difficulty. We show that the fluctuating hydrodynamics and macroscopic fluctuation theory provide a simple, general and versatile framework for studying a whole class of problems of survival, absorption and escape of interacting diffusing particles.
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The narrow escape problem deals with the calculation of the mean escape time (MET) of a Brownian particle from a bounded domain through a small hole on the domains boundary. Here we develop a formalism that allows us to evaluate the emph{non-escape probability} of a gas of diffusing particles that may interact with each other. In some cases the non-escape probability allows us to evaluate the MET of the first particle. The formalism is based on the fluctuating hydrodynamics and the recently developed macroscopic fluctuation theory. We also uncover an unexpected connection between the narrow escape of interacting particles and thermal runaway in chemical reactors.
Suppose that a $d$-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density $n_0$. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability ${mathcal P}$ that no particles are absorbed during a long time $T$. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time $T$. As a result, ${mathcal P}$ decays exponentially with $T$ for a whole class of interacting diffusive gases in any dimension. For $d=1$ the stationary gas density profile and ${mathcal P}$ can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that $-ln {mathcal P}simeq D_0TL^{d-2} ,s(n_0)$, where $D_0$ is the gas diffusivity, and $L$ is the linear size of the system. We calculate the rescaled action $s(n_0)$ for $d=1$, for rectangular domains in $d=2$, and for spherical domains. Near close packing of the SSEP $s(n_0)$ can be found analytically for domains of any shape and in any dimension.
We study fluctuations of particle absorption by a three-dimensional domain with multiple absorbing patches. The domain is in contact with a gas of interacting diffusing particles. This problem is motivated by living cell sensing via multiple receptors distributed over the cell surface. Employing the macroscopic fluctuation theory, we calculate the covariance matrix of the particle absorption by different patches, extending previous works which addressed fluctuations of a single current. We find a condition when the sign of correlations between different patches is fully determined by the transport coefficients of the gas and is independent of the problems geometry. We show that the fluctuating particle flux field typically develops vorticity. We establish a simple connection between the statistics of particle absorption by all the patches combined and the statistics of current in a non-equilibrium steady state in one dimension. We also discuss connections between the absorption statistics and (i) statistics of electric currents in multi-terminal diffusive conductors and (ii) statistics of wave transmission through disordered media with multiple absorbers.
The time which a diffusing particle spends in a certain region of space is known as the occupation time, or the residence time. Recently the joint occupation time statistics of an ensemble of non-interacting particles was addressed using the single-particle statistics. Here we employ the Macroscopic Fluctuation Theory (MFT) to study the occupation time statistics of many emph{interacting} particles. We find that interactions can significantly change the statistics and, in some models, even cause a singularity of the large-deviation function describing these statistics. This singularity can be interpreted as a dynamical phase transition. We also point out to a close relation between the MFT description of the occupation-time statistics of non-interacting particles and the level 2 large deviation formalism which describes the occupation-time statistics of a single particle.
We investigate how confinement may drastically change both the probability density of the first-encounter time and the related survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case when the diffusivities are not equal, and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as translational invariance of such systems is broken.
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