Stochastic Resetting and Nonequilibrium Steady States

We take the machinery of the previous chapters — the master equation and Fokker–Planck equation with their gain and loss terms, detailed balance as the equilibrium fingerprint, and the backward/adjoint generator behind first-passage problems — and turn it on one deceptively simple move: every so often, teleport the particle home. A diffusing walker under Poissonian resetting runs its ordinary dynamics, but at random instants (rate $r$ ) it is snapped back to a fixed point $x_0$ . That single addition changes everything the free process could not do on its own. Pure diffusion never settles: its variance grows like $2Dt$ forever, so it has no stationary distribution. Resetting supplies a loss term $-r\,p(x,t)$ that trims far-flung probability and a source $+r\,\delta(x-x_0)$ that re-injects it at home. The competition freezes into a genuine nonequilibrium steady state (NESS) — a normalizable, cusped, exponential profile $p_*(x)\propto e^{-\alpha_0|x-x_0|}$ with $\alpha_0=\sqrt{r/D}$ . It looks Boltzmann-like but is not equilibrium: it carries a persistent probability current streaming outward from $x_0$ , and the reset move (from anywhere to $x_0$ , with no reverse) breaks detailed balance structurally. The second surprise is about searching. In one dimension a free diffuser's mean time to find a target is infinite — the heavy tail of long, wrong-way excursions dominates. Resetting truncates that tail and makes the mean first-passage time finite, with a unique optimal resetting rate $r^*$ that minimizes it. This is why resetting has become a workhorse of nonequilibrium statistical mechanics: it is exactly solvable through a renewal identity, yet it exhibits NESS currents, entropy production, and dynamical phase transitions — a clean laboratory for everything the equilibrium world forbids.

Key takeaways

Resetting interrupts a stochastic process at Poissonian rate $r$ , returning it to a fixed point $x_0$ ; even free diffusion then reaches a true nonequilibrium steady state (NESS).
For 1D diffusion the stationary density is a sharp cusp $p_*(x)=\tfrac{\alpha_0}{2}e^{-\alpha_0|x-x_0|}$ , $\alpha_0=\sqrt{r/D}$ — non-Gaussian, normalizable, with variance $\sigma_*^2=2D/r$ .
The NESS is sustained by a persistent outward probability current and broken detailed balance: the reset is a non-local, irreversible move with no microscopic reverse.
Resetting makes the mean first-passage time to a target finite and gives a unique optimal rate $r^*\approx2.5396\,D/L^2$ that minimizes search time.
The renewal structure — trajectories are i.i.d. segments between resets — collapses hard time-dependent problems to one Laplace transform of the reset-free propagator.
Key idea: resetting adds a uniform sink $-r\,p$ and a point source $+r\,\delta(x-x_0)$ to the Fokker–Planck equation; the balance against the diffusive current is a normalizable, cusped, current-carrying NESS whose search time is minimized at one optimal rate solving $z/2=1-e^{-z}$ .

The core insight

Resetting at rate $r$ adds a loss term $-r\,p(x,t)$ everywhere and a gain term $+r\,\delta(x-x_0)$ at the reset point to the diffusion equation; the balance of these against the diffusive current yields a normalizable, cusped, current-carrying nonequilibrium steady state — and a search time with a single optimal resetting rate.

Where this lives today

Resetting has jumped from theory to the bench. Optical-tweezer experiments implement it directly by snapping a trapped colloidal particle back to a set point at random times, then measuring the cusped steady state and its outward current — a tunable, well-characterized NESS for testing stochastic thermodynamics, entropy production, and the cost of resetting. In parallel, the "first-passage under restart" formalism has become a standard tool in computer science: restarting a randomized algorithm (a SAT solver, a Monte Carlo sampler) from scratch at a tuned rate is mathematically identical to resetting, and the same $z/2=1-e^{-z}$ optimum governs the best restart frequency. Biophysics uses the same lens for enzymatic turnover, where an enzyme–substrate complex that unbinds before catalysis is a reaction coordinate that resets — connecting Michaelis–Menten kinetics to the resetting first-passage framework. Resetting is now a small industry precisely because it is the rare nonequilibrium model that is both physically realizable and exactly solvable.

The physical picture

Picture a diffusing particle occasionally yanked back to its starting point. Free diffusion lets the probability cloud spread without bound, but every yank hauls the outliers home. The tug-of-war — outward spreading versus inward teleportation — settles into a stationary balance: probability piles up near $x_0$ and falls off exponentially on either side. The profile is not smooth at the top; it has a cusp, the tell-tale kink of a point source of probability sitting exactly at $x_0$ . That steady state is alive, not static. Probability streams continuously outward (carried by diffusion) and is continuously re-injected at $x_0$ (the source). This circulation is the signature of a NESS — like a fountain whose water column is steady only because a pump keeps cycling water back to the top. Cut the pump (set $r=0$ ) and the column collapses: the density flattens, de-normalizes, and free diffusion resumes.

Figure

Resetting confines what free diffusion lets escape: the cusped exponential NESS piles probability at the reset point while the free-diffusion Gaussian spreads without bound

The blue cusp is the resetting NESS; the grey dashed curve is the ever-flattening free-diffusion density. The kink at $x_0$ is not a drawing artifact — it is the exact signature of the $+r\,\delta(x-x_0)$ source, which forces a discontinuity in the slope of $p_*$ .

The renewal viewpoint is the secret weapon. Because a reset erases all memory, the full process is a sequence of independent diffusive excursions, each starting fresh at $x_0$ , with durations drawn from the exponential reset-time distribution. Averaging over those durations turns a hard, non-Markovian-in-time problem into a single Laplace transform of the reset-free propagator — the identity that makes resetting one of the most tractable models in the whole subject. Resetting helps a search precisely because it kills the heavy tail of long, fruitless excursions. A free diffuser can wander far the wrong way; resetting caps that wandering, trading a small chance of a very quick find against eliminating catastrophically long searches. Too little resetting and the excursions run away; too much and the particle is pinned at home, never reaching a distant target. Somewhere in between sits one best rate.

Figure

The scaled mean first-passage time has a single minimum: too little resetting lets excursions run away, too much pins the particle, and one optimal rate solves z over two equals one minus e to the minus z

The valley floor is the thesis: unlike free diffusion (whose 1D mean search time is infinite), resetting gives a finite, minimizable search time. The optimum $z^*\approx1.594$ is a pure number — independent of $D$ and $L$ — because the whole curve depends only on the dimensionless combination $z=\sqrt{r/D}\,L$ .

The core equations

The dynamics starts from the ordinary diffusion (Fokker–Planck) equation and adds the two resetting terms. Every symbol below is a probability density in space and time; the source and sink together conserve total probability, which is what lets a steady state exist at all.

\frac{\partial p(x,t)}{\partial t} = D\,\frac{\partial^2 p}{\partial x^2} - r\,p(x,t) + r\,\delta(x-x_0) \qquad (1)

where: $p(x,t)$ — probability density at position $x$ , time $t$ ; $D$ — diffusion constant; $r$ — Poissonian resetting rate; $x_0$ — reset (and initial) location; $\delta$ — Dirac delta, the point source at $x_0$ . Reading it physically: in a slice $\mathrm{d}t$ a walker either diffuses (probability $1-r\,\mathrm{d}t$ ) or is deleted from $x$ and re-created at $x_0$ (probability $r\,\mathrm{d}t$ ). The sink $-r\,p$ acts everywhere; the source $+r\,\delta$ acts only at home. Setting $\partial_t p=0$ and solving with the delta source gives the steady state — the central object of the article, the NESS.

p_*(x)=\frac{\alpha_0}{2}\,e^{-\alpha_0|x-x_0|},\qquad \alpha_0=\sqrt{\frac{r}{D}} \qquad (2)

where: $p_*(x)$ — stationary density; $\alpha_0$ — inverse decay length $\sqrt{r/D}$ . The profile is symmetric, cusped at $x_0$ , and exponentially localized. Reach for it whenever a diffusive degree of freedom is periodically restarted: the decay length $1/\alpha_0=\sqrt{D/r}$ shrinks as resetting quickens, and the variance is $\sigma_*^2=2D/r$ . The renewal equation expresses the reset propagator as a survival-weighted mixture of reset-free excursions — the structural identity behind everything exactly solvable here.

p_r(x,t\,|\,x_0)=e^{-rt}\,p_0(x,t\,|\,x_0)+r\!\int_0^t\! \mathrm{d}\tau\,e^{-r\tau}\,p_0(x,\tau\,|\,x_0) \qquad (3)

where: $p_r$ — propagator with resetting; $p_0$ — reset-free propagator of the underlying process; $e^{-rt}$ — probability of no reset up to $t$ ; $\tau$ — time since the last reset. The first term is the "survived without resetting" contribution; the integral sums over the time of the last reset. Because resetting adds no local drift, the stationary current is purely diffusive — and, crucially, nonzero away from $x_0$ .

J_*(x)=-D\,\frac{\partial p_*}{\partial x}=\operatorname{sgn}(x-x_0)\,\frac{r}{2}\,e^{-\alpha_0|x-x_0|} \qquad (4)

where: $J_*(x)$ — stationary probability current; $\operatorname{sgn}$ — sign function; the current flows outward from $x_0$ on both sides. A stationary state with $J_*\neq0$ cannot be an equilibrium: it is the direct diagnostic of the NESS. For a target at $x_T$ , solving the backward equation with the reset term gives the mean first-passage time — finite for every $r>0$ , unlike the $r=0$ case which diverges in 1D.

\langle T(x_0)\rangle=\frac{1}{r}\left(e^{\sqrt{r/D}\,|x_0-x_T|}-1\right) \qquad (5)

where: $\langle T\rangle$ — mean first-passage time to $x_T$ ; $L=|x_0-x_T|$ — reset-to-target distance. Its minimum over $r$ is the optimal-search result derived below. Finally, the Laplace transform of the renewal equation collapses the convolution into algebra, and its $s\to0$ limit hands back the NESS directly.

\tilde p_r(x,s\,|\,x_0)=\frac{r+s}{s}\,\tilde p_0(x,r+s\,|\,x_0),\qquad p_*(x)=\lim_{s\to0}s\,\tilde p_r=r\,\tilde p_0(x,r\,|\,x_0) \qquad (6)

where: $\tilde p$ — Laplace transform in time (variable $s$ ); $r+s$ — shifted argument encoding the reset rate. The NESS is thus the reset-free propagator's Laplace transform evaluated at $s=r$ — a one-line recipe for the steady state of any underlying dynamics.

Derivation ASteady state from the renewal and Laplace identity

This derivation earns the cusped exponential $p_*(x)$ a second way — not by solving the differential equation (1) with a delta source, but straight from the free-diffusion propagator via the renewal structure. The strategy is to Laplace-transform in time, use the identity $p_*(x)=r\,\tilde p_0(x,r\,|\,x_0)$ from equation (6), and insert the known Laplace transform of the Gaussian heat kernel. Watch how the awkward time convolution of resets becomes a single evaluation at the shifted variable $s=r$ , and how the square-root branch of the transform produces the exponential cusp automatically.

Step 1. Write the reset-free propagator — the Gaussian heat kernel for pure diffusion:

p_0(x,t\,|\,x_0)=\frac{1}{\sqrt{4\pi D t}}\,e^{-(x-x_0)^2/4Dt}.

Why: between resets the walker simply diffuses from $x_0$ , so its law is the standard diffusion Green's function.

Step 2. Laplace-transform in time using $\int_0^\infty e^{-st}\,p_0\,\mathrm{d}t$ :

\tilde p_0(x,s\,|\,x_0)=\frac{1}{2\sqrt{Ds}}\,e^{-\sqrt{s/D}\,|x-x_0|}.

Why: this is the standard transform of the heat kernel; the exponential decay length is $\sqrt{D/s}$ , and the modulus $|x-x_0|$ is where the cusp is born.

Step 3. Apply the steady-state limit of the renewal identity, $p_*(x)=r\,\tilde p_0(x,r\,|\,x_0)$ , i.e. evaluate the transform at $s=r$ :

p_*(x)=r\cdot\frac{1}{2\sqrt{Dr}}\,e^{-\sqrt{r/D}\,|x-x_0|}.

Why: the $s\to0$ limit of $s\,\tilde p_r$ projects onto the stationary part, and equation (6) turns that into a single evaluation of $\tilde p_0$ at the reset rate.

Step 4. Simplify the prefactor $\dfrac{r}{2\sqrt{Dr}}=\tfrac12\sqrt{r/D}=\alpha_0/2$ , giving

p_*(x)=\frac{\alpha_0}{2}\,e^{-\alpha_0|x-x_0|},\qquad \alpha_0=\sqrt{r/D},

identical to the direct PDE solution (2), and normalized: $\int p_*\,\mathrm{d}x=1$ . Verified: `verify_aq-sm-stochastic-resetting-ness.py` confirms the heat-kernel Laplace transform symbolically, the renewal reconstruction $r\,\tilde p_0(x,r)=(\alpha_0/2)e^{-\alpha_0|x-x_0|}$ , and $\int p_*\,\mathrm{d}x=1$ , $\sigma_*^2=2D/r$ .

Go deeper · Derivation B · 1 minThe optimal resetting rate for diffusive search

Derivation BThe optimal resetting rate for diffusive search

Here we compute the mean first-passage time to a target and minimize it over the resetting rate, exhibiting the transcendental optimality condition that has become the emblem of the field. The plan is to write $\langle T\rangle$ from equation (5), rescale to the single dimensionless group $z=\sqrt{r/D}\,L$ so that all the $D$ - and $L$ -dependence factors out, then set the derivative of the scaled function to zero. The payoff is that the optimum is a pure number, the same for every diffusion constant and every target distance.

Step 1. Start from the resetting MFPT to a target at distance $L=|x_0-x_T|$ :

\langle T\rangle=\frac{1}{r}\left(e^{\sqrt{r/D}\,L}-1\right).

Why: solving the backward (adjoint) equation with the reset term gives this closed form; it is finite for all $r>0$ , whereas $r=0$ makes the 1D MFPT diverge.

Step 2. Change variables to $z=L\sqrt{r/D}$ , so $r=Dz^2/L^2$ :

\langle T\rangle=\frac{L^2}{D}\,\frac{e^{z}-1}{z^2}\equiv\frac{L^2}{D}\,g(z).

Why: the physics depends only on the dimensionless $z$ ; minimizing the pure function $g(z)=(e^z-1)/z^2$ removes $D$ and $L$ from the problem entirely.

Step 3. Set $g'(z)=0$ . Clearing the denominator, the stationarity condition is

z\,e^{z}-2\left(e^{z}-1\right)=0\quad\Longleftrightarrow\quad \frac{z}{2}=1-e^{-z}.

Why: differentiating $(e^z-1)/z^2$ and multiplying through by $z^3$ leaves the numerator $z e^z-2(e^z-1)$ ; dividing by $e^z$ gives the compact transcendental form.

Step 4. Solve numerically. The unique positive root is

z^*\approx1.5936,\qquad r^*=\frac{(z^*)^2 D}{L^2}\approx2.5396\,\frac{D}{L^2},\qquad \langle T\rangle_{\min}\approx1.544\,\frac{L^2}{D}.

Why: $g(z)\to\infty$ as $z\to0$ (excursions run away) and as $z\to\infty$ (particle pinned), so the interior stationary point is the global minimum ( $g''(z^*)>0$ ). Verified: `verify_aq-sm-stochastic-resetting-ness.py` derives $g'(z)=0\Leftrightarrow z/2=1-e^{-z}$ symbolically, finds $z^*=1.593624$ by root-finding, confirms $g''(z^*)>0$ , and reports $r^*=2.5396\,D/L^2$ , $\langle T\rangle_{\min}=1.5441\,L^2/D$ .

Go deeper · Derivation C · 1 minBroken detailed balance and the persistent current

Derivation CBroken detailed balance and the persistent current

This card pins down why the resetting steady state is nonequilibrium rather than a disguised Gibbs state. The argument has three independent prongs: the reset transition has no microscopic reverse, the stationary current is nonzero, and probability circulates through an open source–sink cycle. Any one of them rules out equilibrium; together they identify a NESS that produces entropy at a steady rate. This is exactly the broken detailed balance that the earlier chapter established as the equilibrium fingerprint, now violated on purpose by a non-local move.

Step 1. The reset is irreversible at the transition level. The rate from any $x$ to $x_0$ is $r$ , but the reverse (from $x_0$ back to a specific $x$ by resetting) is zero, so

w(x\to x_0)\,p_*(x)\neq w(x_0\to x)\,p_*(x_0).

Why: detailed balance demands each forward transition be exactly balanced by its reverse; a move with no reverse cannot be balanced.

Step 2. The stationary current is nonzero. From equation (4), $J_*(x)=\operatorname{sgn}(x-x_0)\tfrac{r}{2}e^{-\alpha_0|x-x_0|}\neq0$ for $x\neq x_0$ . Why: an equilibrium state has $J\equiv0$ everywhere; a persistent current is the operational definition of a NESS.

Step 3. Probability runs in an open cycle: injected as a $\delta$ -source at $x_0$ , transported outward by diffusion, removed uniformly at rate $r$ , and re-injected at $x_0$ . Why: the source–sink structure is a driven loop, not a reversible exchange; it requires the external resetting "pump" to persist.

Step 4. Conclude that the housekeeping entropy production is strictly positive and grows with $r$ : the state is a genuine NESS, not an equilibrium with an effective potential. Why: a stationary current against a non-conservative drive dissipates; the entropy-production concepts of the previous chapter apply directly. Verified: `verify_aq-sm-stochastic-resetting-ness.py` confirms $J_*(x>x_0)=(r/2)e^{-\alpha_0(x-x_0)}$ from $-D\,\partial_x p_*$ using $D\alpha_0^2=r$ , so $J_*\neq0$ and detailed balance fails.

Limiting cases

Case	Result	Meaning
No resetting	$r\to0,\ \alpha_0=\sqrt{r/D}\to0$	The cusp flattens and de-normalizes; free diffusion returns, no NESS, variance $2Dt\to\infty$ .
Strong resetting	$r\to\infty,\ p_*(x)\to\delta(x-x_0)$	The particle is pinned at home; a distant target is reached slowly, so the search time diverges again.
Optimal search	$r^*\approx2.5396\,D/L^2$	For a target at distance $L$ , a unique rate minimizes the MFPT; $z^=L\sqrt{r^/D}\approx1.5936$ solves $z/2=1-e^{-z}$ .
Higher dimensions	$p_*(\mathbf r)\sim r^{(d-1)/4}\,\|\mathbf r-\mathbf r_0\|^{-(d-1)/2}e^{-\sqrt{r/D}\,\|\mathbf r-\mathbf r_0\|}$	The NESS stays normalizable and current-carrying; large-distance decay is still set by $\alpha_0=\sqrt{r/D}$ .

Worked examples

Example 1 — Tuning a diffusive search. A protein diffuses with $D=1\ \mu\text{m}^2/\text{s}$ and must locate a binding site $L=5\ \mu\text{m}$ away. Without resetting the 1D mean search time is effectively unbounded. What resetting rate is optimal, and how fast is the search then?

Step 1. The optimum is set by the pure number $z^*\approx1.5936$ , so

r^*=\frac{(z^*)^2 D}{L^2}=\frac{2.5396\times1}{5^2}\ \text{s}^{-1}=\mathbf{0.102\ s^{-1}}

(reset on average about every $10$ s).

Step 2. The minimal mean search time is

\langle T\rangle_{\min}=\frac{1}{r^*}\!\left(e^{z^*}-1\right)=1.544\,\frac{L^2}{D}=1.544\times25\ \text{s}=\mathbf{38.6\ s}.

Step 3. Sanity check the trade-off: reset ten times faster ( $r=1\ \text{s}^{-1}$ , $z=5$ ) and $\langle T\rangle=(e^{5}-1)/1\approx147\ \text{s}$ — nearly four times worse, because the particle is now pinned near home and rarely reaches the site. Over-resetting is as bad as under-resetting.

Example 2 — Shape of the steady state. Take $D=1$ and $r=4$ (natural units). Find the decay length, the peak height, and the width of the NESS.

Step 1. The inverse decay length is $\alpha_0=\sqrt{r/D}=\sqrt4=2$ , so the decay length is $1/\alpha_0=0.5$ .

Step 2. The peak value is $p_*(x_0)=\alpha_0/2=\mathbf{1.0}$ , and the density is $p_*(x)=e^{-2|x-x_0|}$ .

Step 3. The variance is $\sigma_*^2=2D/r=2/4=\mathbf{0.5}$ (standard deviation $\approx0.707$ ), independent of the direct PDE route — both give $2/\alpha_0^2$ . Doubling $r$ halves the variance: faster resetting means a tighter steady state.

Physics of the portrait

Origins · 1 minnonequilibrium steady state and how this was discovered

nonequilibrium steady state

concept — Evans & Majumdar 2011; Schehr 2020 · The stationary-but-driven state at the heart of resetting

A nonequilibrium steady state (NESS) is a stationary probability distribution that nonetheless carries persistent currents and produces entropy — the opposite of a Gibbs equilibrium, which is stationary because every current vanishes. Resetting is the cleanest exactly-solvable NESS known: Martin Evans and Satya Majumdar introduced "diffusion with stochastic resetting" in 2011, showing that a freely diffusing particle acquires the cusped exponential steady state and a search time with a genuine optimum. Grégory Schehr joined them for the sweeping 2020 review that mapped the field. The NESS concept itself runs back through Onsager, Prigogine's dissipative structures, and the modern master-equation formulation of Schnakenberg; resetting made it a textbook-simple laboratory where currents, entropy production, and dynamical phase transitions can all be computed in closed form.

History

2011

— Evans and Majumdar introduce diffusion with stochastic resetting, deriving the cusped NESS and the optimal resetting rate

2014

— resetting extended to bounded domains, potentials, and space-dependent rates

2016

— first-passage-under-restart formalism unifies resetting with computer-science restart strategies

2020

— Evans, Majumdar and Schehr publish the comprehensive review of stochastic resetting and applications

2020

— optical-trap experiments realize resetting of a colloidal particle and measure its nonequilibrium steady state

Going further

References. Evans, Majumdar & Schehr, 'Stochastic resetting and applications', J. Phys. A 53, 193001 (2020) · Evans & Majumdar, 'Diffusion with stochastic resetting', Phys. Rev. Lett. 106, 160601 (2011) · Redner, 'A Guide to First-Passage Processes' (Cambridge, 2001) · Gardiner, 'Handbook of Stochastic Methods' (Springer) · Kardar, 'Statistical Physics of Particles' (Cambridge, 2007) · Sethna, 'Statistical Mechanics: Entropy, Order Parameters, and Complexity' (Oxford, 2021)

Provenance. Steady state, current, MFPT, and optimal-rate results follow Evans & Majumdar, Phys. Rev. Lett. 106, 160601 (2011) and the review Evans, Majumdar & Schehr, J. Phys. A 53, 193001 (2020). A verifier should: (1) re-derive $p_*(x)=\tfrac{\alpha_0}{2}e^{-\alpha_0|x-x_0|}$ from the Fokker–Planck equation with source and via the Laplace renewal identity; (2) confirm the heat-kernel Laplace transform $\tilde p_0=\tfrac{1}{2\sqrt{Ds}}e^{-\sqrt{s/D}|x-x_0|}$ ; (3) solve $z/2=1-e^{-z}$ numerically for $z^*\approx1.5936$ and $r^*\approx2.5396\,D/L^2$ ; (4) verify $J_*\neq0$ implies broken detailed balance. All numbers here are recomputed in `verify_aq-sm-stochastic-resetting-ness.py`.

Builds on: Markov Processes and the Markov Property · The Master Equation · Fokker–Planck Equation · Langevin Equation and Stochastic Differential Equations · Detailed Balance and Microscopic Reversibility Leads to: Fluctuation–Dissipation Theorem · Large Deviation Theory · First-Passage and Search Theory

Is the resetting process Markovian?

It is Markovian in the full state (position plus the internal Poisson clock), but the position alone is not Markovian in the usual sense because a reset injects a global correlation — every trajectory remembers that it started at $x_0$ . The clean way to handle it is the renewal description: resets erase memory, so the process is a chain of independent excursions from $x_0$ , and averaging over the exponential reset-time distribution recovers all observables. That is exactly what the Laplace identity in equation (6) formalizes.

Why is there a cusp in the steady state instead of a smooth peak?

The cusp is the fingerprint of the point source $+r\,\delta(x-x_0)$ . Integrating the stationary equation across $x_0$ shows that the slope of $p_*$ must jump by $-r/D$ there — a kink, not a smooth maximum. Physically, probability is being injected at a single point and must flow away symmetrically, so the density has to turn a corner at $x_0$ . A smooth Gaussian-like peak would require a distributed source, which resetting to a single point does not provide.

Where does the factor √(r/D) come from?

Balance the two rates that compete in the steady state: diffusion relaxes a length scale $\ell$ in time $\ell^2/D$ , while resetting acts on time scale $1/r$ . Setting these equal, $\ell^2/D\sim1/r$ , gives $\ell\sim\sqrt{D/r}$ , so the inverse decay length is $\alpha_0=\sqrt{r/D}$ . The same scale falls out of the differential equation: the homogeneous solutions of $D p_*''-r p_*=0$ are $e^{\pm\sqrt{r/D}\,x}$ .

Why is the mean first-passage time infinite without resetting in 1D?

Free 1D diffusion is recurrent — it reaches any target with probability one — but the time to do so has a heavy tail, $P(T>t)\sim t^{-1/2}$ , whose mean diverges. The rare, very long excursions dominate the average. Resetting truncates those long excursions by hauling the walker home before it wanders too far, which trims the tail enough to make $\langle T\rangle$ finite. This is the precise sense in which resetting "helps": it does not speed up typical searches so much as it eliminates catastrophic ones.

How does the optimal rate depend on the target distance?

Through the single dimensionless group $z=\sqrt{r/D}\,L$ . The optimum is always at the same pure number $z^*\approx1.5936$ , so $r^*=(z^*)^2 D/L^2\approx2.5396\,D/L^2$ : the farther the target, the lower the optimal rate (as $1/L^2$ ). Intuitively, a distant target needs longer excursions to be reachable, so you should reset less often. The minimal search time itself scales as $\langle T\rangle_{\min}\approx1.544\,L^2/D$ , the same $L^2/D$ diffusive scaling but with a finite, optimized prefactor.

Does resetting produce entropy, and how much?

Yes. Because the steady state carries a current and the reset move is irreversible, the system sits in a nonequilibrium steady state with strictly positive housekeeping entropy production. Each reset event dissipates the work needed to move the particle from wherever it was back to $x_0$ ; averaged over the steady state, the entropy production rate grows with $r$ and with the typical excursion size $\sim1/\alpha_0$ . This is why an experimental resetting protocol (an optical trap snapping a bead home) has a real, measurable thermodynamic cost — the "pump" that keeps the fountain running.

Can the underlying process be something other than diffusion?

Absolutely — that is the power of the renewal identity. Replace the Gaussian heat kernel $p_0$ by the propagator of any process (a biased diffusion, a run-and-tumble particle, a Lévy flight, a discrete random walk) and the same formula $p_*(x)=r\,\tilde p_0(x,r\,|\,x_0)$ delivers its resetting NESS. The exponential cusp is special to diffusion; other dynamics give other shapes, but the method — Laplace-transform the reset-free propagator and evaluate at $s=r$ — is universal.