arXiv subjects

Zeev Schuss

Publications and source records attributed to Zeev Schuss.

15 recordsLinked to original sources

arXiv2018

Extreme Narrow Escape: shortest paths for the first particles to escape through a small window

What is the path associated with the fastest Brownian particle that reaches a narrow window located on the boundary of a domain? Although the distribution of the fastest arrival times has been well studied in dimension 1, much less is known in higher dimensions. Based on the Wiener path-integral, we derive a variational principle for the path associated with the fastest arrival particle. Specifically, we show that in a large ensemble of independent Brownian trajectories, the first moment of the shortest arrival time is associated with the minimization of the energy-action and the optimal trajectories are geodesics. Escape trajectories concentrate along these geodesics, as confirmed by stochastic simulations when an obstacle is positioned in front of the narrow window. To conclude paths in stochastic dynamics and their time scale can differ significantly from the mean properties, usually at the basis of the Smoluchowski's theory of chemical reactions.

cond-mat.stat-mech↗

arXiv2017

The case of escape probability as linear in short time

We derive rigorously the short-time escape probability of a quantum particle from its compactly supported initial state, which has a discontinuous derivative at the boundary of the support. We show that this probability is liner in time, which seems to be a new result. The novelty of our calculation is the inclusion of the boundary layer of the propagated wave function formed outside the initial support. This result has applications to the decay law of the particle, to the Zeno behavior, quantum absorption, time of arrival, quantum measurements, and more, as will be discussed separately.

quant-ph↗

arXiv2017

Do cells sense time by number of divisions?

Do biological cells sense time by the number of their divisions, a process that ends at senescence? We consider the question "can the cell's perception of time be expressed through the length of the shortest telomere?" The answer is that the absolute time before senescence cannot be expressed by the telomere's length and that a cell can survive many more divisions than intuitively expected. This apparent paradox is due to shortening and elongation of the telomere, which suggests a random walk model of the telomere's length. The model indicates two phases, first, a determinist drift of the length toward a quasi-equilibrium state, and second, persistence of the length near an attracting state for the majority of divisions prior to senescence. The measure of stability of the latter phase is the expected number of divisions at the attractor ("lifetime") prior to crossing a threshold to senescence. The telomerase regulates stability by creating an effective potential barrier that separates statistically the shortest lifetime from the next shortest. The random walk has to overcome the barrier in order to extend the range of the first regime. The model explains how random telomere dynamics underlies the extension of cell survival time.

q-bio.SC↗

arXiv2017

Asymptotics of extreme statistics of escape time in 1,2 and 3-dimensional diffusions

The first of $N$ identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target, sets the time scale of activation, which in general is much faster than the arrival to the target of only a single trajectory. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number of i.i.d. Brownian trajectories to a small target in 1,2, and 3 dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular transduction.

q-bio.SC↗

arXiv2017

Geometrical effects on nonlinear electrodiffusion in cell physiology

We report here new electrical laws, derived from nonlinear electro-diffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nano-pipettes and help to understand the local voltage changes inside dendrites and axons with heterogenous local geometry.

q-bio.SC↗

arXiv2017

Electrostatics of non-neutral biological microdomains

Voltage and charge distributions in cellular microdomains regulate communications, excitability, and signal transduction. We report here new electrical laws in a cell, which follow from a nonlinear electro-diffusion model. These newly discovered relations derive from the geometrical cell-membrane properties, such as membrane curvature, volume, and surface area. These electro-diffusion laws can now be used to predict and interpret voltage distribution in cellular microdomains.

physics.bio-ph↗

arXiv2016

100 years after Smoluchowski: stochastic processes in cell biology

100 years after Smoluchowski introduces his approach to stochastic processes, they are now at the basis of mathematical and physical modeling in cellular biology: they are used for example to analyse and to extract features from large number (tens of thousands) of single molecular trajectories or to study the diffusive motion of molecules, proteins or receptors. Stochastic modeling is a new step in large data analysis that serves extracting cell biology concepts. We review here the Smoluchowski's approach to stochastic processes and provide several applications for coarse-graining diffusion, studying polymer models for understanding nuclear organization and finally, we discuss the stochastic jump dynamics of telomeres across cell division and stochastic gene regulation.

physics.data-an↗

arXiv2016

On the spatial coordinate measurement of two identical particles

Theoretically, the coordinate measurement of two identical particles at a point by two narrowly separated narrow detectors, is interpreted in the limit of shrinking width and separation, as the detection of two particles by a single narrow detector. { Ordinarily, the ratio between probabilities of point measurements is independent of the width of the narrow detectors.} We show here that not only this is not the case, but that in some scenarios the results depend on the way the dimensions shrink to zero. The ratio between the width and the separation determines the detection result. { In particular, it is shown that the bunching parameter of bosons is not a well-defined physical property. Moreover, it may suggests that } there is a difficulty in quantum measurement theory in the interpretation of coordinate measurement of two particles.

quant-ph↗

arXiv2014

Stochastic model of a pension plan

Structuring a viable pension plan is a problem that arises in the study of financial contracts pricing and bears special importance these days. Deterministic pension models often rely on projections that are based on several assumptions concerning the "average" long-time behavior of the stock market. Our aim here is to examine some of the popular "average" assumptions in a more realistic setting of a stochastic model. Thus, we examine the contention that investment in the stock market is similar to gambling in a casino, while purchasing companies, after due diligence, is safer under the premise that acting as a holding company that wholly owns other companies avoids some of the stock market risks. We show that the stock market index faithfully reflects its companies' profits at the time of their publication. We compare the shifted historical dynamics of the S\&P500's aggregated financial earnings to its value, and find a high degree of correlation. We conclude that there is no benefit to a pension fund in wholly owning a super trust. We verify, by examining historical data, that stock earnings follow an exponential (geometric) Brownian motion and estimate its parameters. The robustness of this model is examined by an estimate of a pensioner's accumulated assets over a saving period. We also estimate the survival probability and mean survival time of the accumulated individual fund with pension consumption over the residual life of the pensioner.

q-fin.RM↗

arXiv2014

Oscillatory survival probability and eigenvalues of the non-self adjoint Fokker-Planck operator

We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics $d\x_\eps=\mb{a}(\x_\eps)\, dt +\sqrt{2\eps}\,\mb{b}(\x_\eps)\,d\w$, which is activated by small noise over the boundary of the domain of attraction $D$ of a stable focus of the drift $\mb{a}(\x)$. The boundary $\p D$ of the domain is an unstable limit cycle of $\mb{a}(\x)$. The oscillations are explained by a singular perturbation expansion of the spectrum of the Dirichlet problem for the non-self adjoint Fokker-Planck operator in $D$ \[L_\eps u(\x)=\,\eps\sum_{i,j=1}^2 \frac{\p ^2\left[ σ^{i,j}\left(\x\right) u(\x) \right]}{\p x^i\p x^j}-\sum_{i=1}^2\frac {\p \left[ a^i\left(\x\right) u(\x)\right]} {\p x^i} =-λ_\eps u(\x),\] with $\mbσ(\x)=\mb{b}(\x)\mb{b}^T(\x)$. We calculate the leading-order asymptotic expansion of all eigenvalues $λ_\eps$ for small $\eps$. The principal eigenvalue is known to decay exponentially fast as $\eps\to0$. We find that for small $\eps$ the higher-order eigenvalues are given by $λ_{m,n}=nω_1+miω_2+O(\eps)$ for $n=1,2,\ldots,\,m=\pm1,\ldots$, where $ω_1$ and $ω_2$ are explicitly computed constants. We also find the asymptotic structure of the eigenfunctions of $L_\eps$ and of its adjoint $L^*_\eps$. We illustrate the oscillatory decay with a model of synaptic depression of neuronal network in neurobiology.

math.AP↗

arXiv2012

Brownian Motion in Dire Straits

The passage of Brownian motion through a bottleneck in a bounded domain is a rare event and the mean time for such passage increases indefinitely as the bottleneck's radius shrinks to zero. Its calculation reveals the effect of geometry and smoothness on the flux through the bottleneck. We find new behavior of the narrow escape time through bottlenecks in planar and spatial domains and on a surface. Some applications are discussed.

math-ph↗

arXiv2010

A subluminous Schroedinger equation

The standard derivation of Schroedinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of Schroedinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit, which produces an intermediate model between non-relativistic and relativistic quantum mechanics: it produces Schroedinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. These boundary conditions relieve the Schroedinger equation of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), the EPR paradox, and so on. The practical implications of the present formulation are yet to be seen.

quant-ph↗

arXiv2009

Stochastic resonance with applied and induced fields: the case of voltage-gated ion channels

We consider a charged Brownian particle in an asymmetric bistable electrostatic potential biased by an externally applied or induced time periodic electric field. While the amplitude of the applied field is independent of frequency, that of the one induced by a magnetic field is. Borrowing from protein channel terminology, we define the open probability as the relative time the Brownian particle spends on a prescribed side of the potential barrier. We show that while there is no peak in the open probability as the frequency of the applied field and the bias (depolarization) of the potential are varied, there is a narrow range of low frequencies of the induced field and a narrow range of the low bias of the potential where the open probability peaks. This manifestation of stochastic resonance is consistent with experimental results on the voltage gated Iks and KCNQ1 potassium channels of biological membranes and on cardiac myocytes.

q-bio.BM↗

arXiv2008

The exit problem in optimal non-causal extimation

We study the phenomenon of loss of lock in the optimal non-causal phase estimation problem, a benchmark problem in nonlinear estimation. Our method is based on the computation of the asymptotic distribution of the optimal estimation error in case the number of trajectories in the optimization problem is finite. The computation is based directly on the minimum noise energy optimality criterion rather than on state equations of the error, as is the usual case in the literature. The results include an asymptotic computation of the mean time to lose lock (MTLL) in the optimal smoother. We show that the MTLL in the first and second order smoothers is significantly longer than that in the causal extended Kalman filter.

math.OC↗

arXiv2001

Total Absorption in Finite Time in an $iδ$ Potential

We consider the evolution of Green's function of the one-dimensional Schrödinger equation in the presence of the complex potential $-ikδ(x)$. Our result is the construction of an explicit time-dependent solution which we use to calculate the time-dependent survival probability of a quantum particle. The survival probability decays to zero in finite time, which means that the complex delta potential well is a total absorber for quantum particles. This potential can be interpreted as a killing measure with infinite killing rate concentrated at the origin.

quant-ph↗