The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology. The mathematical formulation is the following: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. When escape is even more stringent due to severe geometrical restrictions at the place of escape, the narrow escape problem becomes the dire strait problem. The narrow escape problem was proposed in the context of biology and biophysics by D. Holcman and Z. Schuss, and later on with A.Singer and led to the narrow escape theory in applied mathematics and computational biology.
Formulation The motion of a particle is described by the Smoluchowski limit of the Langevin equation:
d X t = 2 D d B t + 1 γ F ( x ) d t , {\displaystyle dX_{t}={\sqrt {2D}}\,dB_{t}+{\frac {1}{\gamma }}F(x)\,dt,}
where D {\displaystyle D} is the diffusion coefficient of the particle, γ {\displaystyle \gamma } is the friction coefficient per unit of mass, F ( x ) {\displaystyle F(x)} the force per unit of mass, and B t {\displaystyle B_{t}} is a Brownian motion.
Mean first passage time and the Fokker-Planck equation A common question is to estimate the mean sojourn time of a particle diffusing in a bounded domain Ω {\displaystyle \Omega } before it escapes through a small absorbing window ∂ Ω a {\displaystyle \partial \Omega _{a}} in its boundary ∂ Ω {\displaystyle \partial \Omega } . The time is estimated asymptotically in the limit ε = | ∂ Ω a | | ∂ Ω | ≪ 1 {\textstyle \varepsilon ={\frac {|\partial \Omega _{a}|}{|\partial \Omega |}}\ll 1}
The probability density function (pdf) p ε ( x , t ) {\displaystyle p_{\varepsilon }(x,t)} is the probability of finding the particle at position x {\displaystyle x} at time t {\displaystyle t} . The pdf satisfies the Fokker–Planck equation:
∂ ∂ t p ε ( x , t ) = D Δ p ε ( x , t ) − 1 γ ∇ ( p ε ( x , t ) F ( x ) ) {\displaystyle {\frac {\partial }{\partial t}}p_{\varepsilon }(x,t)=D\Delta p_{\varepsilon }(x,t)-{\frac {1}{\gamma }}\nabla (p_{\varepsilon }(x,t)F(x))}
with initial condition
p ε ( x , 0 ) = ρ 0 ( x ) {\displaystyle p_{\varepsilon }(x,0)=\rho _{0}(x)\,}
and mixed Dirichlet–Neumann boundary conditions ( t > 0 {\displaystyle t>0} )
p ε ( x , t ) = 0 for x ∈ ∂ Ω a {\displaystyle p_{\varepsilon }(x,t)=0{\text{ for }}x\in \partial \Omega _{a}}
D ∂ ∂ n p ε ( x , t ) − p ε ( x , t ) γ F ( x ) ⋅ n ( x ) = 0 for x ∈ ∂ Ω − ∂ Ω a {\displaystyle D{\frac {\partial }{\partial n}}p_{\varepsilon }(x,t)-{\frac {p_{\varepsilon }(x,t)}{\gamma }}F(x)\cdot n(x)=0{\text{ for }}x\in \partial \Omega -\partial \Omega _{a}}
The function
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