# Mean First Passage Time for a Small Rotating Trap inside a Reflective Disk

@article{Tzou2015MeanFP,
title={Mean First Passage Time for a Small Rotating Trap inside a Reflective Disk},
author={J. C. Tzou and Theodore Kolokolnikov},
journal={Multiscale Model. Simul.},
year={2015},
volume={13},
pages={231-255}
}
• Published 9 May 2014
• Physics
• Multiscale Model. Simul.
We compute the mean first passage time (MFPT) for a Brownian particle inside a two-dimensional disk with reflective boundaries and a small interior trap that is rotating at a constant angular velocity. The inherent symmetry of the problem allows for a detailed analytic study of the situation. For a given angular velocity, we determine the optimal radius of rotation that minimizes the average MFPT over the disk. Several distinct regimes are observed, depending on the ratio between the angular…

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## References

SHOWING 1-10 OF 59 REFERENCES
An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere
• Mathematics
Multiscale Model. Simul.
• 2010
The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem, closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere.
Narrow Escape, Part II: The Circular Disk
• Mathematics
• 2004
We consider Brownian motion in a circular disk Ω, whose boundary $$\partial\Omega$$ is reflecting, except for a small arc, $$\partial\Omega_a$$, which is absorbing. As
Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps
• Mathematics
European Journal of Applied Mathematics
• 2005
An optimization problem for the fundamental eigenvalue $\lam_0$ of the Laplacian in a planar simply-connected domain that contains $N$ small identically-shaped holes, each of radius $\eps\ll 1$, is
First-passage times, mobile traps, and Hopf bifurcations.
• Mathematics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2014
A surprising relation between the oscillating trap problem and a moving-sink problem that describes reduced dynamics of a single spike in a certain regime of the Gray-Scott model is found and is used to prove the uniqueness of the Hopf bifurcation.
First passage time problems in time-dependent fields
• Mathematics
• 1988
This paper discusses the simplest first passage time problems for random walks and diffusion processes on a line segment. When a diffusing particle moves in a time-varying field, use of the adjoint
Diffusion on a Sphere with Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points
• Mathematics
SIAM J. Appl. Math.
• 2009
This work calculates asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere, in the presence of N partially absorbing traps of small radii.
Trapping reactions with randomly moving traps: exact asymptotic results for compact exploration.
• Mathematics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2002
It is shown that this remarkable behavior has a more general range of validity and holds for systems of an arbitrary dimension d, integer or fractal, provided that the traps are "compactly exploring" the space, i.e., the "fractal" dimension d(w) of traps' trajectories is greater than d.
First-passage quantities of Brownian motion in a bounded domain with multiple targets: a unified approach
• Mathematics
• 2011
In this paper, we introduce a general computation method to systematically determine the mean first-passage time (MFPT), the global mean first-passage time (GMFPT) and splitting probabilities for a
The Narrow Escape Problem—A Short Review of Recent Results
Recent results on the dependence of the absorption flux on the geometric properties of the domain are reviewed and thus geometrical features that can modulate the flux are revealed that indicate a possible way to code information physiologically.