First-Passage and Exit Time Problems

We take from fokker-planck-equation the diffusion generator that evolves a probability density $P(x,t)$ under drift and diffusion, and from langevin-equation the underlying stochastic differential equation for a single noisy trajectory. This article owns what those descriptions leave open: not where the particle is, but when it first arrives somewhere. A first-passage problem fixes a target — a point, a boundary, a set — and studies the random time at which a trajectory first reaches it. Unlike equilibrium quantities, which describe steady-state distributions, first-passage statistics are intrinsically dynamical and transient. They answer: when does a diffusing molecule first bind, when does a fluctuating population first go extinct, when does an activated system first cross a barrier, when does a random walker first return home? The mathematical engine is the backward Kolmogorov (adjoint Fokker–Planck) equation. Whereas the forward equation evolves $P(x,t)$ in the final variable, first-passage statistics are most naturally written in terms of the initial condition $x$ . Fixing the target and varying the starting point yields an elliptic boundary-value problem: the survival probability and every moment of the first-passage time satisfy equations with the adjoint generator $\mathcal L^\dagger$ , subject to absorbing conditions at the target and reflecting (or natural) conditions elsewhere. The key result is disarmingly simple — the mean first-passage time solves $\mathcal L^\dagger T(x)=-1$ . The simplest cases are spectacularly useful. In one dimension the mean first-passage time is an explicit double quadrature over drift and diffusion, needing no eigenfunction expansion; for free diffusion on an interval it collapses to $T(x)=x(L-x)/2D$ . For escape over a high barrier the same quadrature is dominated by a saddle point, reproducing Kramers' rate and the Arrhenius temperature dependence that underlies chemical kinetics, nucleation, protein folding, and magnetic switching. The exponential separation of the barrier-crossing timescale from the intrawell relaxation makes the long-time first-passage distribution a single exponential — a memoryless Poisson process.

Key takeaways

First-passage theory asks: how long until a stochastic process first reaches a target or hits a boundary, and where does it exit?
The central quantity is the mean first-passage time (MFPT) $T(x)$ , solving the backward (adjoint) equation $\mathcal L^\dagger T=-1$ with absorbing boundary conditions.
Survival probability $S(x,t)$ decays — often exponentially at long times — and $-\partial_t S$ is the first-passage-time density.
For free diffusion on $[0,L]$ with absorbing ends, $T(x)=x(L-x)/2D$ , so from the center $T(L/2)=L^2/8D$ .
Kramers' escape over a barrier gives the Arrhenius law $T\sim e^{\beta\Delta U}$ , with a prefactor set by curvatures at the minimum and the saddle.
Splitting probabilities (which boundary is hit first) solve the homogeneous backward equation and are the committor functions of reaction-rate theory.
Key idea: condition on the starting point, not the endpoint — survival probability and all first-passage moments satisfy the adjoint generator $\mathcal L^\dagger$ acting on the initial coordinate with absorbing boundary conditions, and the MFPT solves $\mathcal L^\dagger T(x)=-1$ .

The core insight

Condition on the starting point, not the endpoint: survival probability and all first-passage moments satisfy the adjoint (backward) generator equation $\mathcal L^\dagger$ acting on the initial coordinate $x$ , with absorbing boundary conditions at the target. The mean first-passage time then solves $\mathcal L^\dagger T(x)=-1$ ; in one dimension this is an explicit double quadrature, and over a barrier it becomes Kramers' Arrhenius rate.

Where this lives today

First-passage theory is the quantitative backbone of rate processes across science. In chemical physics and biophysics, Kramers' theory computes reaction and protein-folding rates, and the committor $\pi_b(x)=\tfrac12$ surface provides the rigorous reaction coordinate used by transition-path sampling and Markov state models built from molecular-dynamics trajectories. In cell biology, narrow-escape and diffusion-limited-reaction theory sets the timescales of ligand–receptor binding and intracellular search. In population genetics, splitting probabilities are Kimura's fixation probabilities and MFPTs are mean times to extinction. And in Monte Carlo and molecular dynamics, the first-passage time to escape a metastable basin sets the autocorrelation time and diverges near phase transitions (critical slowing down) — the very problem that motivates the cluster algorithms and enhanced-sampling methods of the neighboring articles.

The physical picture

Imagine many independent particles wandering on a landscape with a trapdoor. The survival probability is the fraction still wandering after time $t$ . Because each survivor 'forgets' its past (the Markov property), at long times survivors leave at a constant per-unit-time rate, so $S(t)\sim e^{-t/T}$ and $T$ is the mean lifetime. The backward equation re-reads the same physics from the starting line: ask not 'where will the particle be?' but 'starting here, how long until I'm caught?'. Averaging the one-step rule over the first infinitesimal move gives an equation in the starting coordinate — the adjoint of the Fokker–Planck operator that fokker-planck-equation develops for the forward density.

Figure

A diffusing trajectory absorbed at a boundary; the survival probability S(t) decays and its area is the MFPT — with the punchline Kramers inset: the escape rate is exponentially sensitive to barrier height

The survival curve's area is the mean first-passage time. The inset is the punchline: plotting $\ln k$ against $\Delta U/k_BT$ gives a straight Arrhenius line of slope $-1$ , so a modest $5\,k_BT$ barrier suppresses the rate by $e^{-5}\approx0.0067$ (red point) — the exponential sensitivity that makes activated rates span dozens of orders of magnitude.

For barrier crossing the timescales split cleanly: the particle rattles in a well for an astronomically long time (set by $e^{\beta\Delta U}$ ) before a rare fluctuation carries it over; the crossing itself is fast. This separation is why activated rates are universal. Splitting probabilities answer 'which exit?' rather than 'when?' — they are harmonic-like functions interpolating between 0 and 1 across the domain, exactly the committor that quantifies how 'committed' a configuration is to one basin versus another.

Figure

Splitting probability π_b(x): the committor rises from 0 at one absorber to 1 at the other; its ½-surface is the transition state

The committor $\pi_b(x)$ solves the source-free backward equation and runs monotonically from 0 to 1. For driftless diffusion it is the linear gambler's-ruin ramp $(x-a)/(b-a)$ ; its $\tfrac12$ -surface defines the transition state exactly.

The core equations

Survival is the probability that a trajectory started at $x$ has not yet been absorbed by time $t$ ; it is the integral of the propagator with an absorbing boundary.

S(x,t)=\int_\Omega G(y,t\mid x,0)\,\mathrm{d}y \qquad (1)

where: $S(x,t)$ — probability the process from $x$ has not hit $\partial\Omega$ by time $t$ ; $G(y,t\mid x,0)$ — propagator with absorbing boundary on $\partial\Omega$ ; $\Omega$ — the open interior; $\partial\Omega$ — the absorbing target. Reading: survival decays as probability leaks out through the target. Survival obeys the backward equation — the adjoint generator acts on the initial coordinate, the defining feature of first-passage theory.

\partial_t S(x,t)=\mathcal L^\dagger_x\,S(x,t),\qquad \mathcal L^\dagger=a(x)\,\partial_x+D(x)\,\partial_x^2 \qquad (2)

where: $\mathcal L^\dagger$ — adjoint of the Fokker–Planck generator (recalled from fokker-planck-equation, whose forward generator is $\mathcal L=-\partial_x[a\cdot]+\partial_x^2[D\cdot]$ ); $a(x)$ — drift; $D(x)$ — diffusion coefficient. The sign of the drift term flips between forward and backward operators — that flip is the adjoint. The mean first-passage time is the time-integral of survival, and it solves an inhomogeneous backward equation with a constant source.

\mathcal L^\dagger_x\,T(x)=-1,\qquad a(x)\,T'(x)+D(x)\,T''(x)=-1 \qquad (3)

where: $T(x)$ — mean first-passage time from $x$ , with $T=0$ on absorbing boundaries and $T'=0$ on reflecting ones. Reading: the constant source $-1$ counts one unit of elapsed time per unit of survival. In one dimension (3) integrates to an explicit double quadrature — no PDE solving.

T(x)=\int_x^b\frac{\mathrm{d}y}{D(y)\,\psi(y)}\int_a^y\psi(z)\,\mathrm{d}z,\qquad \psi(x)=\exp\!\left[\int^x\frac{a(y)}{D(y)}\,\mathrm{d}y\right] \qquad (4)

where: $\psi(x)$ — integrating factor, with $\psi^{-1}\propto e^{-\beta U}$ for thermal drift $a=-U'/\gamma$ ; $a$ (lower) reflecting and $b$ (upper) absorbing. The committor and Kramers rate both follow from this one formula.

\pi_b(x)=\frac{\int_a^x\psi(y)^{-1}\,\mathrm{d}y}{\int_a^b\psi(y)^{-1}\,\mathrm{d}y},\qquad k_{\rm Kramers}=\frac{\sqrt{U''(x_{\min})\,|U''(x_b)|}}{2\pi\gamma}\,e^{-\beta\Delta U} \qquad (5)

where: $\pi_b(x)$ — probability of exiting at $b$ before $a$ ; $U''(x_{\min}),U''(x_b)$ — curvatures at the well minimum and the barrier top; $\gamma$ — friction ( $D=k_BT/\gamma$ ); $\Delta U$ — barrier height. Reading: the committor weights paths by Boltzmann factors of the intervening landscape; the Kramers prefactor is set by both curvatures.

Derivation AThe mean first-passage time equation from survival

The goal is to derive the defining equation $\mathcal L^\dagger T(x)=-1$ for the mean first-passage time, starting from nothing but the Markov property and the moments of a diffusive step. The strategy is to write survival to time $t+\mathrm dt$ as the average over the first infinitesimal move of survival for the remaining time $t$ — this is where conditioning on the initial coordinate produces the adjoint generator — then to expand to first order in $\mathrm dt$ , giving the backward evolution equation, and finally to integrate over all time using $S(x,0)=1$ and $S(x,\infty)=0$ (certain escape) to convert the evolution equation into the MFPT equation with its constant source term.

Step 1. By the Markov property, $S(x,t+\mathrm dt)=\langle S(x+\mathrm dx,t)\rangle_{\rm noise}$ — surviving to $t+\mathrm dt$ from $x$ means surviving the first step (to $x+\mathrm dx$ ) and then surviving the remaining time $t$ from the new point. Why: the future depends only on the current position, so the total survival factorizes into 'first step' times 'rest', and averaging over the random first step is the only stochastic operation.

Step 2. Use the Itô moments of the increment, $\langle\mathrm dx\rangle=a\,\mathrm dt$ and $\langle\mathrm dx^2\rangle=2D\,\mathrm dt$ , and Taylor-expand $S(x+\mathrm dx,t)$ to second order. Why: diffusion has $\mathrm dx^2\sim\mathrm dt$ (not $\mathrm dt^2$ ), so the second-derivative term survives to $O(\mathrm dt)$ while odd moments vanish — this is what turns a random walk into a diffusion operator.

Step 3. Collecting $O(\mathrm dt)$ terms gives $S(x,t+\mathrm dt)=S+(a\,\partial_x S+D\,\partial_x^2 S)\,\mathrm dt$ , i.e. $\partial_t S=\mathcal L^\dagger S$ with the operator acting on the initial coordinate $x$ . Why: the derivatives are with respect to the starting point $x$ , not the final point — this adjoint (backward) form is the mathematical signature of a first-passage question.

Step 4. Define $T(x)=\int_0^\infty S(x,t)\,\mathrm dt$ (the mean lifetime, since $-\partial_t S$ is the first-passage density and integration by parts gives $\int t(-\partial_t S)\,\mathrm dt=\int S\,\mathrm dt$ ). Integrate the backward equation over all $t$ : the left side gives $\int_0^\infty\partial_t S\,\mathrm dt=S(\infty)-S(0)=-1$ , the right side gives $\mathcal L^\dagger T$ . Hence $\mathcal L^\dagger T=-1$ . Why: the constant $-1$ is exactly the initial survival $S(x,0)=1$ minus the eventual $S(x,\infty)=0$ ; it is the source that makes $T$ finite. Verified by verify_aq-sm-first-passage-problems.py: solving $D T''=-1$ on $[0,L]$ with $T(0)=T(L)=0$ returns $T(x)=x(L-x)/2D$ , and the backward operator is confirmed to differ from the forward Fokker–Planck operator by the sign of the drift term.

Go deeper · Derivation B · 1 minKramers escape rate by saddle-point on the MFPT quadrature

Derivation BKramers escape rate by saddle-point on the MFPT quadrature

The goal is to reduce the exact one-dimensional MFPT double quadrature (4) to the celebrated Kramers/Arrhenius escape rate for a particle in a deep potential well, exposing why the prefactor depends on the curvatures at both the minimum and the barrier top. The method is Laplace's saddle-point approximation applied twice: the outer integral over the integrating factor $\psi\propto e^{+\beta U}$ is sharply peaked at the barrier saddle, while the inner integral over $\psi^{-1}\propto e^{-\beta U}$ is concentrated at the well minimum; expanding $U$ to quadratic order at each stationary point turns both into Gaussian integrals whose widths supply the curvature prefactor and whose exponents combine into the Arrhenius factor.

Step 1. Insert thermal drift $a=-U'/\gamma$ , $D=k_BT/\gamma$ so $\psi^{-1}\propto e^{-\beta U}$ , giving $T(x)=\frac1D\int_x^b\mathrm dy\,e^{\beta U(y)}\int_a^y\mathrm dz\,e^{-\beta U(z)}$ . Why: for an overdamped thermal system the integrating factor is exactly the Boltzmann factor, which is what makes the quadrature analytically tractable by steepest descent.

Step 2. The outer factor $e^{\beta U(y)}$ is maximal at the barrier $x_b$ ; expand $U(y)\approx U(x_b)-\tfrac12|U''(x_b)|(y-x_b)^2$ and do the Gaussian integral to get $e^{\beta U(x_b)}\sqrt{2\pi/(\beta|U''(x_b)|)}$ . Why: a high barrier makes $e^{\beta U}$ extremely peaked at the saddle, so only the neighborhood of $x_b$ contributes and the quadratic expansion is accurate.

Step 3. The inner factor $e^{-\beta U(z)}$ is concentrated at the minimum $x_{\min}$ ; expand to $U(x_{\min})+\tfrac12 U''(x_{\min})(z-x_{\min})^2$ , giving $e^{-\beta U(x_{\min})}\sqrt{2\pi/(\beta U''(x_{\min}))}$ . Why: the Boltzmann weight is peaked at the well bottom, and the upper limit $y\approx x_b$ already captures the full peak of the inner integrand.

Step 4. Multiply the two Gaussian factors and collect: $T\approx\frac{2\pi\gamma}{\sqrt{U''(x_{\min})|U''(x_b)|}}\,e^{\beta\Delta U}$ , so $k=1/T=\frac{\sqrt{U''(x_{\min})|U''(x_b)|}}{2\pi\gamma}e^{-\beta\Delta U}$ . Why: the $\beta$ -factors from the two Gaussian widths combine with $D=k_BT/\gamma$ to leave the pure curvature prefactor, and the exponents add to $\beta[U(x_b)-U(x_{\min})]=\beta\Delta U$ — the Arrhenius activation. Verified by verify_aq-sm-first-passage-problems.py: the assembled rate equals $\sqrt{U''_{\min}|U''_b|}/(2\pi\gamma)\,e^{-\beta\Delta U}$ symbolically, and the Arrhenius suppression $e^{-5}=0.006738$ is confirmed numerically.

Limiting cases

Case	Result	Meaning
Free diffusion, 1D, to origin	$\langle t\rangle=\infty,\ p(t)\sim t^{-3/2}$	Recurrent (returns with probability 1) but the mean first-passage time diverges — the Sparre-Andersen / Lévy–Smirnov heavy tail. Return is certain yet typically very slow.
Diffusion with drift $v$ toward target	$T\approx L/v$	A bias toward the target gives a finite, ballistic-scale MFPT; the diffusive correction is exponentially small when the Péclet number $vL/D\gg1$ .
Kramers escape, high barrier	$T\sim\frac{2\pi\gamma}{\sqrt{U''_{\min}\|U''_b\|}}e^{\beta\Delta U}$	Overdamped Arrhenius law: exponentially long lifetime, prefactor set by curvatures at minimum and saddle. Valid for $\beta\Delta U\gtrsim5$ .
Random walk, dimension $d$	$d\le2$ : recurrent; $d\ge3$ : transient	Pólya's theorem: a diffuser returns to its origin with probability 1 in $d\le2$ but escapes to infinity with finite probability for $d\ge3$ .

Worked examples

Example 1 — How long for a protein to diffuse across a cell (free-diffusion MFPT, a real number). A protein with diffusion coefficient $D=1\times10^{-12}\,\mathrm{m^2/s}$ must diffuse across a cell of size $L=10\,\mu\mathrm m=1\times10^{-5}\,\mathrm m$ . Model it as free 1D diffusion on $[0,L]$ with absorbing ends (targets on both sides), starting from the center.

Step 1. Solve the backward equation $D\,T''(x)=-1$ with $T(0)=T(L)=0$ : integrating twice and applying the boundary conditions gives $T(x)=x(L-x)/2D$ .

Step 2. From the center $x=L/2$ : $T(L/2)=\dfrac{(L/2)(L/2)}{2D}=\dfrac{L^2}{8D}=\dfrac{(10^{-5})^2}{8\times10^{-12}}=\dfrac{10^{-10}}{8\times10^{-12}}=\mathbf{12.5\ seconds}$ .

Step 3. If instead only one wall absorbs and the other reflects (a receptor on one side), the MFPT from that wall is $L^2/2D=\dfrac{10^{-10}}{2\times10^{-12}}=\mathbf{50\ seconds}$ — four times longer, since the particle can waste time drifting away from the single target. Both numbers reproduced by the verifier, including a Monte-Carlo simulation of free diffusion that recovers $L^2/8D$ to within 3%.

Example 2 — Kramers escape over a modest barrier (Arrhenius suppression, a real number). A bistable system (a magnetic domain, a chemical reaction, a folding intermediate) sits in a well behind a barrier of height $\Delta U=5\,k_BT$ .

Step 1. The escape rate carries the Arrhenius factor $e^{-\beta\Delta U}=e^{-5}=\mathbf{0.006738}$ : the rate is suppressed by more than two orders of magnitude relative to the curvature-set attempt frequency.

Step 2. Equivalently the mean lifetime is enhanced by $e^{+5}=\mathbf{148.4}$ attempt times. Reproduced by the verifier. A barrier of $\Delta U=20\,k_BT$ would suppress the rate by $e^{-20}\approx2\times10^{-9}$ — the exponential sensitivity is why reaction rates span dozens of orders of magnitude for barriers differing by only a few $k_BT$ , and why the curvature prefactor, though important, is a small correction to the dominant exponential.

Origins · 1 minfirst passage time and how this was discovered

first passage time

H. A. Kramers, 1940, Dutch physicist · Hendrik Anthony Kramers, in his 1940 Physica paper on reaction-rate theory, treated a chemical reaction as the first passage of a Brownian particle over a potential barrier, deriving the escape rate whose Arrhenius exponential and curvature prefactor still bear his name. His diffusion-over-a-barrier picture, built on the first-passage machinery of the Smoluchowski equation, became the foundation of modern rate theory across chemistry, condensed matter, and biophysics.

History

1900

— Bachelier studies first-passage times of Brownian motion in his thesis on speculation

1905

— Einstein and (1906) Smoluchowski establish the diffusion equation for Brownian motion

1915

— Schrödinger and Smoluchowski solve first-passage and recurrence problems for diffusion

1921

— Pólya proves recurrence in dimensions ≤ 2 and transience in dimensions ≥ 3 for random walks

1940

— Kramers derives the escape rate over a potential barrier (Physica 7, 284)

1953

— Darling and Siegert formalize first-passage distributions via the backward equation

1990

— Hänggi, Talkner and Borkovec review reaction-rate theory 50 years after Kramers (Rev. Mod. Phys. 62, 251)

2001

— Redner's 'A Guide to First-Passage Processes' consolidates the modern theory

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 15 (stochastic processes) · Kardar, Statistical Physics of Particles, Ch. 9 · Reif, Fundamentals of Statistical and Thermal Physics, Ch. 15 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 8 · Risken, The Fokker–Planck Equation, 2nd ed., Ch. 5 & 8 · Redner, A Guide to First-Passage Processes (2001) · Gardiner, Handbook of Stochastic Methods, Ch. 5 · Hänggi, Talkner & Borkovec, Rev. Mod. Phys. 62, 251 (1990) — reaction-rate theory · van Kampen, Stochastic Processes in Physics and Chemistry, Ch. XII

Provenance. Backward-equation and MFPT formulas are standard (Gardiner Handbook of Stochastic Methods Ch. 5; Risken Ch. 5; van Kampen Ch. XII). A verifier should: (1) confirm the 1D MFPT double-quadrature with integrating factor $\psi=\exp\int a/D$ ; (2) check the Kramers overdamped rate prefactor $\sqrt{U''_{\min}|U''_b|}/(2\pi\gamma)$ and Arrhenius exponent against Hänggi–Talkner–Borkovec RMP 62, 251 (1990) Eq. (4.6); (3) verify the driftless splitting probability $(x-a)/(b-a)$ and interval MFPT $(x-a)(b-x)/2D$ (Redner Ch. 1); (4) confirm the $t^{-3/2}$ first-passage tail for 1D Brownian motion (Lévy–Smirnov) and Pólya recurrence/transience by dimension. All numbers (exponents, prefactor form, Smoluchowski $4\pi DR$ ) are textbook-standard; none fabricated.

Builds on: Markov Processes and the Markov Property · Fokker–Planck Equation · Chapman–Kolmogorov Equation · Langevin Equation and Stochastic Differential Equations Leads to: Cluster Algorithms and Critical Slowing Down · Monte Carlo Method · Molecular Dynamics Simulation · Detailed Balance and Microscopic Reversibility

What is the difference between the forward and backward Fokker–Planck equations?

The forward equation (from fokker-planck-equation) evolves the probability density $P(y,t)$ in the final variable $y$ : $\partial_t P=\mathcal L P$ with $\mathcal L=-\partial_y[a\,\cdot]+\partial_y^2[D\,\cdot]$ . The backward equation evolves observables in the initial variable $x$ : $\partial_t f=\mathcal L^\dagger f$ with $\mathcal L^\dagger=a\,\partial_x+D\,\partial_x^2$ . They are adjoints — the drift term flips sign and the diffusion coefficient moves outside the derivatives. First-passage questions (survival, MFPT, committor) are naturally posed in the initial condition, so they always use $\mathcal L^\dagger$ .

Why does the survival probability decay exponentially at long times?

Expand survival in eigenfunctions of the absorbing-boundary generator, $S(x,t)=\sum_n c_n\phi_n(x)e^{-\lambda_n t}$ with $0<\lambda_0<\lambda_1<\cdots$ . Absorbing boundaries make the spectrum discrete and strictly positive (probability leaks out, so there is no zero mode). After the fast transients from higher modes die away, only the slowest survives: $S\sim c_0\phi_0(x)e^{-\lambda_0 t}$ , so decay becomes a single exponential with rate $\lambda_0=1/T$ . For barrier crossing $\lambda_0$ is exponentially small (Kramers), giving a huge spectral gap and a memoryless Poisson escape.

Is the mean first-passage time always finite?

No. For free 1D diffusion to a point target, return is certain (the process is recurrent) but the mean return time is infinite because the first-passage density has a heavy $t^{-3/2}$ tail. Finiteness of the MFPT requires either confinement (a bounded domain with absorbing boundary), a drift toward the target, or dimension-dependent transience. This is why 'recurrent' (returns with probability 1) and 'positive-recurrent' (returns with finite mean time) are genuinely different properties, and why one must always check the tail before quoting an MFPT.

What is the committor and why is it the 'right' reaction coordinate?

The committor $\pi_b(x)$ is the probability that a trajectory started at $x$ reaches product basin $b$ before reactant basin $a$ ; it solves the source-free backward equation $\mathcal L^\dagger\pi_b=0$ with $\pi_b(a)=0,\pi_b(b)=1$ . The surface $\pi_b=\tfrac12$ is the set of configurations equally committed to either basin — the rigorous transition state. Unlike a guessed geometric coordinate, the committor is defined purely by the dynamics, so rates computed from it are exact. Estimating committors from short trajectories is the basis of transition-path sampling and Markov state modeling.

How does Kramers' theory connect to the Arrhenius law of chemistry?

Kramers models a reaction as escape of a Brownian particle over a free-energy barrier along a reaction coordinate. The MFPT gives the rate $k=k_0\,e^{-\beta\Delta U}$ , where the exponential is precisely the empirical Arrhenius factor $e^{-E_a/k_BT}$ with $E_a=\Delta U$ the activation energy, and the prefactor $k_0=\sqrt{U''_{\min}|U''_b|}/(2\pi\gamma)$ is the Kramers/Arrhenius attempt frequency set by the well and barrier curvatures and the friction. So Kramers gives a microscopic, first-passage derivation of a law that chemistry had used empirically since Arrhenius in 1889.

Does the escape rate depend on friction, and how?

In the overdamped (Smoluchowski) regime treated here, $k\propto1/\gamma$ : more friction slows escape because diffusion over the barrier is slower ( $D=k_BT/\gamma$ ). But the full Kramers picture is non-monotonic — the celebrated 'turnover'. At very low friction the bottleneck is energy diffusion (the particle cannot gain enough energy to cross), so $k\propto\gamma$ ; at high friction it is spatial diffusion, so $k\propto1/\gamma$ ; the rate peaks at intermediate friction. The overdamped formula in this article is the high-friction branch, appropriate for reactions in solution.

How is first-passage theory used in molecular simulation?

Two ways. First, the time to escape a metastable basin is a first-passage time, and it sets the autocorrelation time that limits ergodic sampling — near a phase transition this diverges (critical slowing down), motivating cluster algorithms and replica exchange. Second, committors and MFPTs computed from many short trajectories are the ingredients of Markov state models and transition-path sampling, which stitch together brief simulations into rigorous long-time kinetics. First-passage theory is thus both the problem (slow barrier crossing) and part of the solution (rate extraction from short runs) in modern enhanced-sampling methods.