Selective-Resetting Methods

• Selective-resetting methods are stochastic protocols that target specific states or regions to create tailored non-equilibrium steady states and control dynamic behaviors.
• They modify conventional reset frameworks by incorporating state- or space-dependent rates, linking renewal theory, information flow, and thermodynamic balances.
• Applications span optimized search strategies, controlled biochemical gating, and adaptive machine-learning protocols to manage work, entropy, and information costs.

Selective-resetting methods constitute a class of stochastic protocols in which reset events are triggered or applied not homogeneously, but in a way that depends on space, state, or other conditions within a system. Instead of indiscriminately initiating resets over all possible states, selective resetting targets specific regions, degrees of freedom, or process configurations, thereby enabling the design of tailored non-equilibrium steady states (NESS), enhanced search strategies, and efficient energetic or informational manipulations. This approach crucially links stochastic thermodynamics, renewal theory, and information theory—often underpinned by the need to balance trade-offs between speed, resource use, entropy production, and work cost.

1. Theoretical Framework of Selective Resetting

Selective resetting modifies the canonical stochastic resetting framework by introducing spatially, state-, or outcome-dependent resetting rates $r(x)$ or transition kernels. The canonical example in continuous space is an overdamped particle in a potential $V(x)$ satisfying the augmented Fokker–Planck equation,

$$\partial_t p(x) = -\partial_x j(x) - r(x)p(x) + \delta(x-x_0) \int dx'\, r(x')p(x'),$$

with $j(x) = F(x)p(x) - \partial_x p(x),\ F(x) = -dV/dx$. Here, $r(x)$ can be engineered to be nonzero solely in regions targeted for reset. This modifies the steady-state probability profile $p(x)$ and information content.

Within the renewal approach, selective resetting often results in an effective, modified survival probability $p_{\rm sel}(t)$ or a reset kernel. For observables depending only on the most recent renewal period, the Laplace-space solution generalizes as

$$\tilde{F}_r^{\rm sel}(s) = \frac{\tilde{F}_{\rm sel}(s)}{\tilde{p}_{\rm sel}(s)}.$$

This generality allows the same renewal-based methodology to apply, with appropriate modifications for the selectivity.

2. Entropy Production, Information Flow, and Thermodynamics

Selective resetting fundamentally alters the structure of entropy production and information flows. The total entropy balance in steady state becomes

$$\dot{S}_{\rm med} - \dot{S}_{\rm abs} - \dot{S}_{\rm ins} \equiv \dot{S}_{\rm tot} \geq 0,$$

where:

• $\dot{S}_{\rm med}$: entropy production in the medium (linked to heat dissipation $\dot{Q}$).
• $\dot{S}_{\rm abs} = \int dx\, r(x)p(x) \ln p(x)$: entropy absorbed as probability leaves resettable states.
• $\dot{S}_{\rm ins} = -\int dx\, r(x)p(x) \ln p(x_0)$: entropy insertion at the reset site.

Their sum yields the resetting entropy production rate,

$$\dot{S}_{\rm rst} = \int dx\, r(x)p(x) \ln\left[\frac{p(x)}{p(x_0)}\right].$$

Negative $\dot{S}_{\rm rst}$ corresponds to information erasure (compression, e.g., search restarts from a favored region), positive values to information creation. Landauer’s principle yields the bound on work required for resetting: $W \geq -\dot{Q} - \dot{S}_{\rm rst}.$ If selective resetting is designed to compress probability toward a high-probability target, work must be performed to satisfy the thermodynamic bound.

These thermodynamic identities enable bookkeeping of both energetic and informational balances for any selective-resetting protocol, as in kinetic proofreading or immune detection.

3. Dynamics, Renewal Structure, and Unified Approaches

In selective resetting, observables are functionals over the most recent renewal period—intervals between resets that are not identically distributed, as selective criteria may act only in certain regimes or states. The unified framework developed in (Jr. et al., 2019) provides, for the Laplace transforms of such observables,

$$\tilde{F}_r(s) = \frac{\tilde{F}(s)}{\tilde{p}(s)},$$

and the stationary value (if the mean reset interval $R$ is finite),

$F_r(\infty) = \frac{1}{R}\int_0^\infty F(t) dt.$

For intermittent, power-law resetting, clear asymptotic regimes exist: $$F_r(t) \sim \begin{cases} (r_0^{-\beta}K/(1+\gamma-\beta))\, t^{1+\gamma-\beta}, & 1<\beta<1+\gamma,\ \left[\Gamma(1+\gamma-\beta)/(\Gamma(1+\gamma)\Gamma(1-\beta))\right] K t^\gamma, & \beta<1. \end{cases}$$ By adjusting selectivity (through thresholds or spatial dependence), one can tailor not only stationary values but also the rates and scaling of observable growth.

Selective resetting gains further power in complex diffusive-decay settings, where the reset rate kernel becomes a function of microscopic disorder or other couplings. This permits engineered “control knobs,” e.g., confinement in disorder-rich regions, selective exploration, or altered transport spectra.

4. Practical Applications and Protocol Design

Selective-resetting protocols are pivotal for numerous applications:

• Optimized search strategies: By spatially varying $r(x)$ (or piecewise-constant rates, e.g., $r_+$ for $x>x_r$, $r_-$ for $x<x_r$), searchers prioritize domains more likely to contain a target. However, minimizing mean first-passage time can induce divergent fluctuations unless fluctuations are minimized alongside the mean (García-Valladares et al., 2023).
• Gated processes in biochemistry: Resetting is used to counterbalance gating (temporal selectivity of the target), with optimal reset rates $r^*$ determined by minimizing the completion time and controlled by phase transitions in the $(\lambda, p_r)$ plane (Biswas et al., 2023).
• Threshold and event-driven resetting: Instead of time-driven resets, the process is reset collectively when an internal system variable (e.g., position, failure, or energy) crosses a threshold. This yields correlated resets and non-trivial group statistics (Biswas et al., 18 Apr 2025).
• Selective resetting in information-processing: In photon frequency diffusion, resetting can be additive (frequency-independent) or multiplicative (frequency-scaled), controlling the occupation distribution and inducing deviations from Planck’s law in nonequilibrium optical media (Oliveira et al., 2022). In RNNs, selective-resetting through a composite loss allows internal “resets” at the output level for non-informative segments, eliminating the need for explicit state resets during continual inference (Yin et al., 20 Dec 2024).

Experimental systems—such as optical trap-based colloidal resetting with state-dependent traps or biological systems exploiting kinetic constraints—can implement these protocols via either controlled resetting rates or event-driven resets.

5. Statistical and Fluctuation-Based Criteria for Benefit

Selective resetting does not always offer improvement; its efficacy depends on detailed statistics of the process and outcomes:

• The classical criterion that resetting is beneficial if the unconditional coefficient of variation (CV) exceeds unity does not hold selectively. For a specific outcome (e.g., exit at a desired boundary), resetting expedites the outcome if

$$CV^\sigma > \Lambda^\sigma,\quad \text{with}\quad \Lambda^\sigma = \sqrt{\frac{\langle T_0\rangle^2}{2\langle T_0^\sigma\rangle^2}[1 + (CV)^2]}.$$

This threshold generalizes across dimensions and process types, permitting control over bias between outcomes (Pal et al., 13 Feb 2025).

• In cost-sensitive scenarios, e.g., where each reset incurs a distance-dependent cost, the functional form of the cost (linear, sub-linear, super-linear, or exponential) crucially alters optimal resetting rates, sometimes making any resetting prohibitively costly for heavy-tailed cost distributions (Sunil et al., 2023).

Mean-variance trade-offs and tail risk considerations dictate protocol selection, especially in spatially or outcome-selective resetting problems.

6. Numerical, Adaptive, and Machine-Learning Approaches

Recent advances incorporate numerical and adaptive frameworks for selective-resetting protocol optimization:

• The renewal approach can use measured propagators and empirically determined survival probabilities for prediction of steady states under arbitrary (possibly selective) resetting, without requiring full analytic description of the dynamics (Vatash et al., 14 Nov 2024).
• Adaptive resetting—where $r(\mathbf{X}, t)$ is a function of state and time—enables “informed” decisions leveraging trajectory or environmental knowledge. Key observables (mean first-passage time, propagator, NESS structure) can be computed from a single ensemble of reset-free trajectories using reweighting. The MFPT under adaptive resetting takes the form

$$\langle T_R \rangle = \frac{\langle \min(T, R)\rangle}{\Pr(T \leq R)},$$

enabling efficient evaluation over sweeping protocol sets (Keidar et al., 22 Sep 2024).

• Machine learning frameworks can optimize adaptive resetting rules for arbitrary objectives by parameterizing $r(\mathbf{X}, t)$ as a neural network and training using observables estimated via reweighting. Applications include acceleration of protein folding in molecular dynamics by optimizing resetting based on collective variables.

These computational strategies generalize selective-resetting optimization to high-dimensional and complex environments with minimal additional simulation overhead.

7. Outlook and Significance

Selective-resetting methods provide a versatile, thermodynamically and statistically principled framework for accelerating stochastic processes, steering information flows, and imposing outcome or spatial bias in both physical and computational systems. Their flexibility allows for:

• Space-, state-, and outcome-specific targeting in both theoretical and experimental contexts.
• Satisfying universal thermodynamic and fluctuation criteria while balancing work, entropy, and resource costs.
• Systematic design and optimization in complex, interacting, or disordered environments using numerical or machine-learning-based tools.

Whether applied to search theory, biological pattern formation, engineered control in soft matter or quantum optics, or continual representation in neural computation, selective resetting stands as a unifying principle enabling tailored manipulation of stochastic dynamics far from equilibrium.