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Register a new userFrom single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers
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We consider the first-passage problem for $N$ identical independent particles that are initially released uniformly in a finite domain $Omega$ and then diffuse toward a reactive area $Gamma$, which can be part of the outer boundary of $Omega$ or a reaction centre in the interior of $Omega$. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the $N$ particles reacts with $Gamma$. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fastest first-passage time with the particle number $N$, namely, a much stronger dependence ($1/N$ and $1/N^2$ for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.
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We study the problem of random search in finite networks with a tree topology, where it is expected that the distribution of the first-passage time F(t) decays exponentially. We show that the slope of the exponential tail is independent of the initial conditions of entering the tree in general, and scales exponentially or as a power law with the extent of the tree L, depending on the tendency p to jump toward the target node. It is unfeasible to uniquely determine L and p from measuring the tail slope or the mean first-passage time (MFPT) of an ordinary diffusion along the tree. To unravel the structure, we consider lazy random walkers that take steps with probability m when jumping on the nodes and return with probability q from the leaves. By deriving an exact analytical expression for the MFPT of the intermittent random walk, we verify that the structural information of the tree can be uniquely extracted by measuring the MFPT for two randomly chosen types of tracer particles with distinct experimental parameters m and q. We also address the applicability of our approach in the presence of disorder in the structure of the tree or statistical uncertainty in the experimental parameters.
We derive an approximate but fully explicit formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape in a general elongated domain in the plane. Our approximation combines conformal mapping, boundary homogenisation, and Fick-Jacobs equation to express the MFPT in terms of diffusivity and geometric parameters. A systematic comparison with a numerical solution of the original problem validates its accuracy when the starting point is not too close to the target. This is a practical tool for a rapid estimation of the MFPT for various applications in chemical physics and biology.
We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $dot{x}_t=sqrt{2 D_0 V(B_t)},xi_t$, where $xi_t$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V(B_t)$ is a stochastic diffusivity (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V(B_t) =Theta(B_t)$ (Model I), where $Theta(x)$ is the Heaviside theta function; a Geometric Brownian Motion $V(B_t)=exp(B_t)$ (Model II); and a case with $V(B_t)=B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the Levy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the Levy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.
We investigate the statistics of encounters of a diffusing particle with different subsets of the boundary of a confining domain. The encounters with each subset are characterized by the boundary local time on that subset. We extend a recently proposed approach to express the joint probability density of the particle position and of its multiple boundary local times via a multi-dimensional Laplace transform of the conventional propagator satisfying the diffusion equation with mixed Robin boundary conditions. In the particular cases of an interval, a circular annulus and a spherical shell, this representation can be explicitly inverted to access the statistics of two boundary local times. We provide the exact solutions and their probabilistic interpretation for the case of an interval and sketch their derivation for two other cases. We also obtain the distributions of various associated first-passage times and discuss their applications.
We study a stochastic process $X_t$ related to the Bessel and the Rayleigh processes, with various applications in physics, chemistry, biology, economics, finance and other fields. The stochastic differential equation is $dX_t = (nD/X_t) dt + sqrt{2D} dW_t$, where $W_t$ is the Wiener process. Due to the singularity of the drift term for $X_t = 0$, different natures of boundary at the origin arise depending on the real parameter $n$: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behaviour is observed in the case of a regular boundary.
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