Diffusion-mediated surface phenomena are crucial for human life and industry, with examples ranging from oxygen capture by lung alveolar surface to heterogeneous catalysis, gene regulation, membrane ...permeation, and filtration processes. Their current description via diffusion equations with mixed boundary conditions is limited to simple surface reactions with infinite or constant reactivity. In this Letter, we propose a probabilistic approach based on the concept of boundary local time to investigate the intricate dynamics of diffusing particles near a reactive surface. Reformulating surface-particle interactions in terms of stopping conditions, we obtain in a unified way major diffusion-reaction characteristics such as the propagator, the survival probability, the first-passage time distribution, and the reaction rate. This general formalism allows us to describe new surface reaction mechanisms such as for instance surface reactivity depending on the number of encounters with the diffusing particle that can model the effects of catalyst fooling or membrane degradation. The disentanglement of the geometric structure of the medium from surface reactivity opens far-reaching perspectives for modeling, optimization, and control of diffusion-mediated surface phenomena.
Abstract
We develop an encounter-based approach for describing restricted diffusion with a gradient drift toward a partially reactive boundary. For this purpose, we introduce an extension of the ...Dirichlet-to-Neumann operator and use its eigenbasis to derive a spectral decomposition for the full propagator, i.e. the joint probability density function for the particle position and its boundary local time. This is the central quantity that determines various characteristics of diffusion-influenced reactions such as conventional propagators, survival probability, first-passage time distribution, boundary local time distribution, and reaction rate. As an illustration, we investigate the impact of a constant drift onto the boundary local time for restricted diffusion on an interval. More generally, this approach accesses how external forces may influence the statistics of encounters of a diffusing particle with the reactive boundary.
We derive a general exact formula for the mean first passage time (MFPT) from a fixed point inside a planar domain to an escape region on its boundary. The underlying mixed Dirichlet-Neumann boundary ...value problem is conformally mapped onto the unit disk, solved exactly, and mapped back. The resulting formula for the MFPT is valid for an arbitrary space-dependent diffusion coefficient, while the leading logarithmic term is explicit, simple, and remarkably universal. In contrast to earlier works, we show that the natural small parameter of the problem is the harmonic measure of the escape region, not its perimeter. The conventional scaling of the MFPT with the area of the domain is altered when diffusing particles are released near the escape region. These findings change the current view of escape problems and related chemical or biochemical kinetics in complex, multiscale, porous or fractal domains, while the fundamental relation to the harmonic measure opens new ways of computing and interpreting MFPTs.
Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly ...change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule.
We consider a particle diffusing outside a compact planar set and investigate its boundary local time ℓt, i.e., the rescaled number of encounters between the particle and the boundary up to time t. ...In the case of a disk, this is also the (rescaled) number of encounters of two diffusing circular particles in the plane. For that case, we derive explicit integral representations for the probability density of the boundary local time ℓt and for the probability density of the first-crossing time of a given threshold by ℓt. The latter density is shown to exhibit a very slow long-time decay due to extremely long diffusive excursions between encounters. We briefly discuss some practical consequences of this behavior for applications in chemical physics and biology.
We propose a unifying theoretical framework for the analysis of first-passage time distributions in two important classes of stochastic processes in which the diffusivity of a particle evolves ...randomly in time. In the first class of 'diffusing diffusivity' models, the diffusivity changes continuously via a prescribed stochastic equation. In turn, the diffusivity switches randomly between discrete values in the second class of 'switching diffusion' models. For both cases, we derive exact formulas for the probability density function of the first-passage time and quantify the impact of the diffusivity dynamics via the moment-generating function of the integrated diffusivity.
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