Mean First-Reaction Time (MFRT)

• MFRT is a stochastic measure defining the average time until a reaction event occurs, often equated with first-passage time under perfect reactivity.
• It quantifies the impact of partial reactivity, spatial geometry, and non-uniform transition metrics, distinguishing between first encounter and successful reaction.
• MFRT analysis leverages backward equations, absorbing Markov chain formulations, and spectral methods to address diverse systems from chemical kinetics to transport phenomena.

Mean first-reaction time (MFRT) is the expected time until a specified reaction event first occurs. In many stochastic models, the reaction event is represented as first arrival to an absorbing target, event state, or reactive boundary, so MFRT coincides with mean first-passage time (MFPT); this identification is exact for perfectly reactive first-encounter models and becomes a generalized first-reaction problem under partial reactivity, repeated unsuccessful encounters, or nonuniform physical time per transition (Saglam et al., 2014). Across absorbing Markov chains, confined diffusion, stochastic chemical reaction networks, and persistent-transport models, MFRT is therefore a transport-to-event observable, but its interpretation depends strongly on geometry, reactivity, memory, and on whether the full first-reaction-time distribution is narrow or strongly defocused (Grebenkov et al., 2019).

1. Definition and relation to adjacent first-event observables

At the most general level, MFRT is the expectation of a stopping time $\tau$ defined by the first occurrence of a designated reaction event. In the simplest case, the event is first encounter with a perfectly reactive target. For surface-mediated diffusion in a disk or sphere, the target is a reactive patch on the confining boundary and reaction occurs instantaneously upon first hitting that target, so the MFPT to the target equals the MFRT under the paper’s assumptions of perfect reaction at first encounter (Bénichou et al., 2010). The same identification is used for intramolecular first-contact kinetics, where the first time two reactive monomers satisfy $|\mathbf R(t)|<a$ is simultaneously a first-contact time and the relevant mean first-reaction time (Dolgushev et al., 2015).

The distinction between MFPT and MFRT becomes explicit when the target is only partially reactive. For diffusion to a reactive boundary $\Gamma$, the mean first-reaction time $U(\mathbf x)$ solves

$$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$

with mixed Robin–Neumann conditions

$$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$

so arrival at the boundary and successful reaction are no longer the same event (Grebenkov et al., 22 Sep 2025). In a boundary-local-time formulation, the first-reaction time is

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$

where the reaction threshold $\hat\ell$ is exponential with

$$\mathbb P\{\hat\ell>\ell\}=e^{-q\ell}, \qquad q=\frac{\kappa}{D},$$

which makes explicit that multiple encounters may precede the first successful reaction (Ye et al., 18 Dec 2025).

A further refinement is required when one transition step does not correspond to one unit of physical time. In absorbing Markov-chain language, MFPT is the special case of mean first-passage value (MFPV) obtained when every transition contributes one discrete time unit. If transition $i\to j$ carries a value $|\mathbf R(t)|<a$0 interpreted as physical time, energy, dose, distance, reward, or cost, then the correct first-reaction observable is the corresponding MFPV rather than step count (Saglam et al., 2014). This distinction is central whenever “time to first reaction” is not the same as “number of transitions to reaction.”

2. State-space, backward-equation, and spectral formulations

One common formalization of MFRT is the absorbing Markov-chain setting. A discrete-time stochastic system with finite state set $|\mathbf R(t)|<a$1 is evolved by a stochastic transition matrix $|\mathbf R(t)|<a$2, and the reaction event is represented by a halt state, taken without loss of generality to be $|\mathbf R(t)|<a$3, made absorbing. The state-specific first-passage times satisfy the first-step recurrence

$|\mathbf R(t)|<a$4

or, in block form,

$|\mathbf R(t)|<a$5

with $|\mathbf R(t)|<a$6 the transient block of the chain (Saglam et al., 2014). This gives a fully state-resolved MFRT when the reaction is encoded as absorption.

For metastable systems, the same paper identifies a system-wide mean through the dominant transient eigenvalue. If $|\mathbf R(t)|<a$7 is the Perron root of the transient block, then the quasi-stationary or metastable distribution $|\mathbf R(t)|<a$8 is built from the associated nonnegative eigenvector, and the first-passage law from $|\mathbf R(t)|<a$9 is geometric. The resulting system-wide mean is

$\Gamma$0

which transfers directly to MFRT when the reaction is the absorbing event (Saglam et al., 2014). This representation is especially useful when the chain forgets its initial condition rapidly relative to the absorption time.

In continuum settings, MFRT is usually obtained from backward equations. For partially reactive diffusion in a spherical shell $\Gamma$1, the exact Laplace transform $\Gamma$2 of the reaction-time density satisfies

$\Gamma$3

with reflecting outer boundary and radiation condition

$\Gamma$4

at the inner target (Grebenkov et al., 2019). For partially reactive patches on a bounded 3D domain, the mean itself solves the Robin–Neumann boundary-value problem described above (Grebenkov et al., 22 Sep 2025).

Persistent transport requires an enlarged state space. For velocity-jump processes, the mean first passage time $\Gamma$5 depends on both position and velocity and satisfies the exact backward kinetic equation

$\Gamma$6

with absorbing boundary conditions on outgoing states at the target boundary (Hillen et al., 2024). This is the natural MFRT backbone for reaction/search processes in which transport has persistence or directional structure.

3. Distributional structure and the limits of the mean

The mean is often analytically tractable, but the literature repeatedly shows that MFRT alone can be misleading. In the metastable absorbing-chain regime, if the process starts from the metastable distribution, the first-passage law is geometric:

$\Gamma$7

so

$\Gamma$8

However, the same model gives

$\Gamma$9

which means that for metastable systems with $U(\mathbf x)$0, the standard deviation is close to the mean. The corresponding confidence bounds are

$U(\mathbf x)$1

and in the metastable limit the probability that the first-passage time exceeds its mean is only about $U(\mathbf x)$2 (Saglam et al., 2014). Thus the mean is neither a median nor a high-probability guarantee.

A more detailed example is the partially reactive concentric-sphere model, in which the exact reaction-time density has four regimes delimited by three characteristic time scales $U(\mathbf x)$3, $U(\mathbf x)$4, and $U(\mathbf x)$5. The most probable time is approximately

$U(\mathbf x)$6

the confinement crossover is

$U(\mathbf x)$7

and the exact MFRT is

$U(\mathbf x)$8

The paper’s main point is that the mean can be dominated by late events in the plateau and long-time tail, while the most probable reaction time remains much smaller; the mean and mode can differ by orders of magnitude (Grebenkov et al., 2019).

Non-Markovian recrossing provides a different failure mode for a single mean. In a bistable generalized Langevin system, the mean first-to-first passage time $U(\mathbf x)$9 and the mean all-to-first passage time $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$0 are equivalent in the Markovian limit, but separate strongly once memory induces rapid state recrossings. Their distributions are linked by

$$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$1

so the all-to-first statistic weights long events more heavily than the waiting-time statistic (Zhou et al., 2024). This makes “the MFRT” operationally ambiguous unless the event definition and estimator are stated explicitly.

A related lesson emerges in nonlinear chemical reaction networks. For the first firing of a designated reaction, the exact object is the survival probability $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$2, from which

$$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$3

is only a derived summary. The exact framework for monomolecular networks coupled to a single $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$4-type target reaction is designed precisely to recover the full first-reaction-time law for arbitrary discrete initial conditions, not merely its mean (Rao et al., 6 Mar 2025).

4. Representative model classes and exact or asymptotic results

Surface-mediated diffusion in confinement provides one of the clearest MFRT models. In a disk or sphere, a particle alternates between surface diffusion with coefficient $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$5 and bulk diffusion with coefficient $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$6, desorbing from the surface at rate $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$7 and being radially ejected a distance $$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$8 into the bulk. For the 2D point-target case, the exact mean reaction time for a particle initially uniform on the boundary is

$$\Delta U = -\frac{1}{D}, \qquad \mathbf x\in \Omega,$$9

with $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$0 and $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$1 (Bénichou et al., 2010). Exact integral-equation formulations for 2D and 3D spherical domains, together with tractable approximation schemes, are given in a companion treatment (Bénichou et al., 2011). Both works show that the reaction time can be minimized as a function of the desorption rate.

For partially reactive boundaries, the asymptotic structure changes qualitatively. In a sphere whose boundary is mostly reflecting except for $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$2 small partially reactive patches, the volume-averaged dimensionless MFRT satisfies

$$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$3

A central scaling comparison follows from this expansion: for finite intrinsic reactivity, the MFRT scales like patch area, $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$4, whereas for perfectly absorbing patches it scales like patch diameter/capacitance, $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$5 (Grebenkov et al., 22 Sep 2025). The limits $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$6 and $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$7 therefore do not commute.

In the exactly solvable spherical-shell model with partial reactivity, the MFRT has the additive Collins–Kimball-like form

$$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$8

where the first term is the perfectly reactive MFPT and the second is the penalty due to imperfect reaction, independent of starting radius $$D\partial_n U + \kappa_i^{\rm dim} U = 0 \quad \text{on } \partial\Omega_i, \qquad \partial_n U = 0 \quad \text{on } \partial\Omega_r,$$9 (Grebenkov et al., 2019). By contrast, a boundary-local-time treatment expresses the same quantity as

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$0

and yields the weak-reactivity asymptotic

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$1

which is universal at leading order and independent of the starting point (Ye et al., 18 Dec 2025).

MFRT also appears beyond single-particle boundary hitting. In fractal macromolecules, the mean first-contact time between two reactive monomers is the relevant intramolecular MFRT. For large equilibrium separation $\tau = \inf\{t>0:\ell_t>\hat\ell\},$2, the paper finds the universal scaling

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$3

while for small capture radius $\tau = \inf\{t>0:\ell_t>\hat\ell\},$4,

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$5

with substantial quantitative corrections from non-Markovian reactive conformations (Dolgushev et al., 2015). In lattice-based reaction–diffusion models for trimolecular reactions, the mean reaction time decomposes as

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$6

and in the fine-lattice limit $\tau = \inf\{t>0:\ell_t>\hat\ell\},$7,

$\tau = \inf\{t>0:\ell_t>\hat\ell\},$8

so the mean reaction time diverges logarithmically because three-body coincidence becomes rare (Li et al., 2016).

5. Optimization, control, perturbation, and computation

Because MFRT is a first-event functional, it is naturally a control objective. In the absorbing-chain framework, many hybrid systems can be meshed into finite-state, finite-action, finite-randomness models, producing a policy-induced absorbing Markov chain to which the MFPT/MFPV analysis applies. The objective may be either minimizing or maximizing the value until escape, depending on whether the event represents success or failure. Computationally, the same framework emphasizes two routes: solve a linear system for state-specific first-passage times, or compute the dominant system-wide mean from the leading transient eigenvalue, which is especially attractive for very large state spaces (Saglam et al., 2014).

A complementary result is a universal rare-perturbation response theory. If a perturbation is activated at an exponential time with small rate $\tau = \inf\{t>0:\ell_t>\hat\ell\},$9, then

$\hat\ell$0

with

$\hat\ell$1

The sign criterion

$\hat\ell$2

shows that large unperturbed fluctuations can make a rare perturbation accelerate completion even when the post-activation residual mean $\hat\ell$3 is not especially small (Keidar et al., 2024). This applies directly to first-reaction problems whenever the reaction time is the relevant completion time.

Optimization by direct MFPT minimization also appears algorithmically in enhanced sampling. In a discretized Markov state model with rate matrix $\hat\ell$4, the bias is chosen to minimize the MFPT from a starting state $\hat\ell$5 to a target state $\hat\ell$6,

$\hat\ell$7

using relations between biased and unbiased kinetics obtained from dynamic histogram analysis (Wei et al., 2024). The same logic is visible in stochastic resetting. For an active fluctuating membrane whose local height first hits a threshold, the MFPT under resetting rate $\hat\ell$8 obeys

$\hat\ell$9

and the paper shows that there is an optimal resetting rate, with the optimal rate much smaller under active noise than under thermal noise (Singha, 2023).

6. Scope, terminology, and recurrent misconceptions

The first recurrent misconception is that MFRT is always identical to MFPT. It is identical only when the reaction occurs immediately and certainly at first encounter with the target. This is the case for the perfectly reactive surface targets treated in the spherical surface-mediated diffusion models (Bénichou et al., 2010). It is not the case for partially reactive targets, repeated failed encounters, or finite intrinsic reaction rates, where one must distinguish first arrival from first successful reaction (Ye et al., 18 Dec 2025).

The second misconception is that one global mean always captures the kinetics. In metastable absorbing chains, a system-wide mean based on the second eigenvalue is most meaningful when the chain forgets its initial condition quickly; the relevant indicator is the “memory constant”

$$\mathbb P\{\hat\ell>\ell\}=e^{-q\ell}, \qquad q=\frac{\kappa}{D},$$0

so large initial-condition dependence requires the full statewise vector rather than a single scalar mean (Saglam et al., 2014). In few-molecule partially reactive systems, the mean is often dominated by plateau and tail contributions rather than by the most probable events (Grebenkov et al., 2019). In non-Markovian barrier crossing, different operational definitions of first-passage time can imply different reaction kinetics altogether (Zhou et al., 2024).

The third misconception is that elapsed time is the only meaningful first-reaction observable. In many applications the appropriate quantity is cumulative physical time, energy, dose, distance, or cost accrued until the first event. The mean first-passage value formalism makes this explicit by treating MFPT as only the special case in which every transition contributes one unit of time (Saglam et al., 2014). This suggests that “MFRT” is best viewed not as a single formula but as a class of first-event expectations whose precise meaning is fixed by the event definition, the transport model, and the reaction mechanism.

Taken together, these results support a unified but technically strict interpretation: MFRT is the expectation of a first-event stopping time, often reducible to a first-passage problem, but only under model-specific assumptions about absorption, reactivity, memory, and the physical meaning of a transition.