How rare are diffusive rare events?

We study the time until first occurrence, the first-passage time, of rare density fluctuations in diffusive systems. We approach the problem using a model consisting of many independent random walkers on a lattice. The existence of spatial correlations makes this problem analytically intractable. However, for a mean-field approximation in which the walkers can jump anywhere in the system, we obtain a simple asymptotic form for the mean first-passage time to have a given number k of particles at a distinguished site. We show numerically, and argue heuristically, that for large enough k, the mean-field results give a good approximation for firstpassage times for systems with nearest-neighbour dynamics, especially for two and higher spatial dimensions. Finally, we show how the results change when density fluctuations anywhere in the system, rather than at a specific distinguished site, are considered. Introduction. – Rare events control the kinetics of many physical systems. They are frequently associated with activated processes, corresponding to the crossing of a high free-energy barrier, as happens, for example, in nucleation processes [1]. On the other hand, in diffusive systems, where particles undergo Brownian motion, there are no such energy barriers. Here, questions arise such as: what is the time required for a particle to first reach a particular region of space, or for a pair of particles to meet for the first time? Depending on the circumstances, such events can be very rare, in which case they may be the limiting step in the dynamics of the system. It is then crucial to understand when such events will occur. Diffusive systems exhibit universality, in the sense that their behaviour is often independent of the microscopic details. It is then convenient to study simple models, in the hope that the results will extend to more complicated systems. A common choice is that of particles undergoing a random walk on a lattice [2]. For a single-particle random walk, first-passage times to a given site in an infinite system have long been studied [3], and much is now known about their statistics [2,4]. Recent progress in this (a)E-mail: [email protected]. Current address: Departamento de F́ısica, Facultad de Ciencias, Universidad Nacional Autónoma de México – Circuito Exterior, Ciudad Universitaria, 04510 México D.F., Mexico. direction was made by Condamin et al., who studied firstpassage times and distributions for a single particle diffusing from a source site to a destination site in a confined region [5–7]. Other properties, however, are relevant only when many walkers are present, for example the sequence of times for each to arrive at a given point [8], and the territory covered after a given number of steps [9, 10]. In this Letter, we study the following first-passage problem for systems containing many random walkers on a periodic lattice: how long does it take for k of the walkers to accumulate at a given site? For k much larger (or much smaller) than the mean density, this corresponds to a large local density fluctuation, and is thus a rare event. A related quantity was studied numerically in [11], as a model of particles traversing a membrane pore. Special cases have also been studied in relation to Ritort’s backgammon model [12,13], and another related model was recently used in a study of cooperativity in chemical kinetics [14]. The general problem has, however, received scant attention, despite its relevance for many physical systems. Indeed, the problem pertains to any systems whose dynamics may be affected by the accumulation of diffusing particles in a given region. A particular physical example that we have in mind is a fluid of hard discs exhibiting glassy behaviour. When the density of discs is close to the