Xiangting Li
PHYS243M First passage time
Oct 24, 2023This series of notes is based on a comparison of stochastic processes in physics and mathematics, during my study of PHYS243M (Nonequilibrium Statistical Mechanics in Complex Systems) by Prof. Shenshen Wang at UCLA.
In this note, we discuss the calculation of mean first passage time to a region by a particle follows stochastic differential equation, or Langevin equation.
Derivation via survival probability
The general idea is that we track a system up to the moment at which the particle reaches the desired location. In mathematics, it is to “stop” a stochastic process $X_t$ with the first passage time $T$, giving rise to $X_{t\wedge T}$, $t\wedge T \equiv \min \{ t, T \}$. In this case, the Fokker-Planck equation of $X_{t\wedge T}$ behaves as if the target state is an absorbing state, i.e. absorbing boundary condition in the following.
The motion of a cloud of initial points satisfies the Fokker-Planck equation
\[\frac{\partial P(\vec{a}, t)}{\partial t}=-\nabla_{\vec{a}} \cdot(\vec{v}(\vec{a}) P)+\nabla_{\vec{a}} \cdot \vec{B} \cdot \nabla_{\vec{a}} P \equiv \mathcal{L} P\]- IC. $P(\vec{a}, \theta)=\delta\left(\vec{a}-\vec{a}_0\right), \quad \vec{a}_0 \in V$
- BC. $P(\vec{a}, t)=0, \quad \vec{a} \in \partial V$
Formal solution
\(P(\vec{a}, t \mid \vec{a}_0)=e^{\mathcal{L} t} \delta\left(\vec{a}-\vec{a}_0\right)\) with $P(\vec{a}, t \rightarrow \infty)=0$ due to absorbing B.C.
Survival probability
\[S\left(t, \vec{a}_0\right)=\int_V d \vec{a} P(\vec{a}, t)\]Dist. of first passage time (FPT)
\[\rho\left(t, \vec{a}_0\right)=-d S\left(t, \vec{a}_0\right) / d t\]MFPT (1st moment of $\rho$)
\[\tau\left(\vec{a}_0\right)=\int_0^{\infty} \mathrm{d} t t \underbrace{\rho\left(t, \vec{a}_0\right)}_{- \mathrm{d} S / \mathrm{d} t}=\int_0^{\infty} \mathrm{d} t S\left(t, \vec{a}_0\right)-\left.t S\right|_0 ^{\infty}\left(\textrm{using } S\left(\infty, \vec{a}_0\right)=0\right)\]The advantage of this approach is that we can get a complete characterization for the distribution of waiting time. The disadvantage is that we only get the waiting time distribution for a specific initial point.
Method of conditioning
This is an alternative method based on conditional probability or recurrence relation to calculate mean first passage time $\tau(\vec{a}_0)$ as a function of initial point $\vec{a}_0$. We formally write
\[\tau\left(\vec{a}_0\right)=\int_0^{\infty} \mathrm{d} t \int_V \mathrm{d} \vec{a} P\left(\vec{a}, t \mid \vec{a}_0\right)\]Let $\mathcal{L}^*$ represent the dual operator of the Fokker-Planck operator $\mathcal{L}$ with respect to the inner product $\langle u, \rho \rangle=\int u \rho \mathrm{d}x$, we claim that
\(\mathcal{L}^*_{(a_0)} P(\vec{a}, t \mid \vec{a}_0) = \mathcal{L}_{(a)}P(\vec{a},t \mid \vec{a}_0).\) Here, the subscripts $(a_0)$ and $(a)$ denote the variables the operator acts on.
See the note for the Kolmogorov backward equation for a proof on this relation.
Thus
\(\mathcal{L}^{*} P(\vec{a}, t \mid \vec{a}_0) = \partial_{t} P(\vec{a},t).\) We then observe that
\[\mathcal{L}^* \tau(\vec{a}_0) = \int_V \mathrm{d} \vec{a}\int_0^t \mathrm{d} t \partial_{t} P(\vec{a},t \mid \vec{a}_0) = \int_V \mathrm{d} \vec{a} \left( P(\vec{a}, \infty) - P(\vec{a}_0, 0) \right) = - 1.\]Therefore, we have the following description of mean first passage time.
\[\mathcal{L}^{*} \tau\left(\vec{a}_0\right)=-1, \quad \tau(\vec{a}_0)=0 ~ \forall \vec{a}_0 \in \partial_{} V.\]Note that a similar result holds for general continuous-time Markov chains.