Brownian Dynamics at Boundaries and Interfaces

The basic concepts in the axiomatic definition of the one-dimensional Brownian motion as a mathematical object are a space of events Ω, whose elementary events are real-valued continuous functions ω = ω( ⋅) on the positive axis \(\mathbb{R}_{+}\).

A discrete computer simulation of an Itô SDE is a necessary computational tool for the study of the behavior of diffusing particles in situations in which the FPE does not provide the needed information or when its analytical or numerical solutions are not feasible. This is the situation, for example, in the study of interacting particles, in the study of diffusion through narrow passages, in the simulation of ions in a small volume in a large continuum, and many more situations.

It was argued in Sect. 2.6 that mathematical Brownian motion (MBM) is the overdamped limit of the Langevin displacement process. It is tempting, therefore, to coarse-grain the two-dimensional phase space simulation of the overdamped one-dimensional Langevin equation into the one-dimensional configuration space of an MBM. This simplification, however, comes at a price: it inherits the artifacts of the MBM, such as the infinite rate of level crossing, with fatal consequences. An MBM that crosses a boundary (a point, a curve, or a surface) recrosses it infinitely many times in any time interval. Therefore, the number of recrossings of a boundary increases indefinitely as the step size of the simulation is decreased. Consequently, it becomes impossible to determine when a simulated trajectory is on one side of the boundary or the other. This phenomenon shows up, for example, in the simulation of ions in a given (small) volume in solution: simulated ionic trajectories have to enter and leave the simulation domain an unbounded number of times as the step size of the simulation decreases, leaving no room for determining the convergence of the simulation. Quoting Einstein (1956) in this context, “the movements of one and the same particle after different intervals of time must be considered as mutually independent processes, so long as we think of these intervals of time as being chosen not too small.” This means that the MBM idealization should be taken with a grain of salt; the time step in a simulation cannot be refined beyond a certain limit. This limit has to be determined from the more refined Langevin model of the Brownian movement in the limit of large damping.

This chapter relates the first passage time (FPT) from a point to the boundary of a given domain to total population, flux, rate, mean time spent at a point, eigenvalues, and other quantities

Biological microstructures such as synapses, dendritic spines, subcellular domains, sensor cells, and many other structures are regulated by chemical reactions that involve only a small number of molecules, that is, between a few and up to thousands of molecules. Traditional chemical kinetics theory may provide an inadequate description of chemical reactions in such microdomains. Models with a small number of diffusers can be used to describe noise due to gating of ionic channels by random binding and unbinding of ligands in biological sensor cells, such as olfactory cilia, photoreceptors, and hair cells in the cochlea. A chemical reaction that involves only 10–100 proteins can cause a qualitative transition in the physiological behavior of a given part of a cell. Large fluctuations should be expected in a reaction if so few molecules are involved, both in transient and persistent binding and unbinding reactions. In the latter case, large fluctuations in the number of bound molecules can force the physiological state to change all the time, unless there is a specific mechanism that prevents the switch and stabilizes the physiological state. Therefore, a theory of chemical kinetics of such reactions is needed to predict the threshold at which switches occur and to explain how the physiological function is regulated in molecular terms at a subcellular level.

This chapter introduces the concept of the stochastic separatrix and elaborates its application in clarifying the notion of transitions between relatively long-lived states and short-lived transition states. These may be noise-induced transitions over high barriers or the squeezing of Brownian motion through narrow necks connecting relatively large confining compartments. The stochastic separatrix plays a role in determining the dependence of the first nonzero eigenvalue of the Fokker–Planck operator (FPO) on the geometry of the drift field and on the geometry of the domain.

The narrow escape problem in diffusion theory, which goes back to Lord Rayleigh (in the context of the theory of sound), is to calculate the mean first passage time of Brownian motion to a small absorbing window on the otherwise reflecting boundary of a bounded domain (see Fig. 7.1). The MFPT in this problem is also called the narrow escape time (NET).

The NET problem in three dimensions is more complicated than that in two dimension, primarily because the singularity of Neumann’s function for a regular domain is more complicated than (7.1).