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Register a new userFull distribution of first exit times in the narrow escape problem
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In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small escape window in the otherwise impermeable boundary, once it arrives to this window and over-passes an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the containers boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.
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The time needed for a particle to exit a confining domain through a small window, called the narrow escape time (NET), is a limiting factor of various processes, such as some biochemical reactions in cells. Obtaining an estimate of the mean NET for a given geometric environment is therefore a requisite step to quantify the reaction rate constant of such processes, which has raised a growing interest in the last few years. In this Letter, we determine explicitly the scaling dependence of the mean NET on both the volume of the confining domain and the starting point to aperture distance. We show that this analytical approach is applicable to a very wide range of stochastic processes, including anomalous diffusion or diffusion in the presence of an external force field, which cover situations of biological relevance.
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.
We investigate the first-passage dynamics of symmetric and asymmetric Levy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Levy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Levy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.
In this short note, a correction is made to the recently proposed solution [1] to a 1D biased diffusion model for linear DNA translocation and a new analysis will be given to the data in [1]. It was pointed out [2] by us recently that this 1D linear translocation model is equivalent to the one that was considered by Schrodinger [3] for the Enrenhaft-Millikan measurements [4,5] on electron charge. Here we apply Schrodingers first-passage-time distribution formula to the data set in [1]. It is found that Schrodingers formula can be used to describe the time distribution of DNA translocation in solid-state nanopores. These fittings yield two useful parameters: drift velocity of DNA translocation and diffusion constant of DNA inside the nanopore. The results suggest two regimes of DNA translocation: (I) at low voltages, there are clear deviations from Smoluchowskis linear law of electrophoresis [6] which we attribute to the entropic barrier effects; (II) at high voltages, the translocation velocity is a linear function of the applied electric field. In regime II, the apparent diffusion constant exhibits a quadratic dependence on applied electric field, suggesting a mechanism of Taylor dispersion effect likely due the electro-osmotic flow field in the nanopore channel. This analysis yields a dispersion-free diffusion constant value for the segment of DNA inside the nanopore which is in agreement with Stokes-Einstein theory quantitatively. The implication of Schrodingers formula for DNA sequencing is discussed.
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.
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