Reaction Kinetics in the Few-Encounter Limit

Hartich, David; Godec, Aljaž

doi:10.1142/9781786347015_0011

Cited by 2 publications

(4 citation statements)

References 50 publications

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“…( 54). A similar interlacing for absorption and relaxation rates has been found for the first passage problem of classical random walks [46]. Now we consider τ > 0 but small.…”

Section: The Zeno Limit In the Electrostatic Formalism A Non-hermitia...supporting

confidence: 74%

“…Eqs. (45,46) are nicely demonstrated in Figs. 3(2) and 4(4), where the total detection probability and the mean show constant and linear behavior, respectively.…”

Section: Discrete Energy Spectra and The Electrostatic Formalismmentioning

confidence: 59%

“…The quantity of interest is the (random) first detection time T , the time of the first successful detection attempt. This is a generalization of the time-of-arrival problem [32][33][34][35][36][37][38][39][40][41] and the quantum analogue to the important first passage problem of statistical mechanics [42][43][44][45][46]. In the context of quantum information, it can also be seen as a protocol for quantum search [10,[47][48][49][50][51][52] or state transfer [53].…”

Section: Introductionmentioning

confidence: 99%

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Non-Hermitian and Zeno limit of quantum systems under rapid measurements

Thiel,

Kessler

2020

Preprint

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We investigate in depth the relation between the first detection time of an isolated quantum system that is repeatedly perturbed by strong local measurements with a large fixed frequency 1/τ , determining whether it is in some given state |ψ d , and the time of absorption to the same state of the same system with the added imaginary potential 2i |ψ d ψ d | /τ . As opposed to previous works, we compare directly the solutions of both problems in the small τ , i.e., Zeno, limit. We find a scaling collapse in F (t) with respect to τ and compute the total detection probability as well as the moments of the first detection time probability density F (t) in the Zeno limit. We show that both solutions approach the same result in this small τ limit, as long as the initial state |ψin is not parallel to the detection state, i.e. as long as | ψ d |ψin | < 1. However, when this condition is violated, the small probability density to detect the state on time scales much larger than τ is precisely a factor of four different for all such times. We express the solution of the Zeno limit of both problems formally in terms of an electrostatic analogy. Our results are corroborated with numerical simulations.

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“…( 54). A similar interlacing for absorption and relaxation rates has been found for the first passage problem of classical random walks [46]. Now we consider τ > 0 but small.…”

Section: The Zeno Limit In the Electrostatic Formalism A Non-hermitia...supporting

confidence: 74%

“…Eqs. (45,46) are nicely demonstrated in Figs. 3(2) and 4(4), where the total detection probability and the mean show constant and linear behavior, respectively.…”

Section: Discrete Energy Spectra and The Electrostatic Formalismmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

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Non-Hermitian and Zeno limit of quantum systems under rapid measurements

Thiel,

Kessler

2020

Preprint

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show abstract

“…Many authors have investigated extreme FPTs of diffusive searchers [25,34,36,50,[54][55][56][57][58][59][60]] (see also [61][62][63][64][65] for interesting related work). Most of these prior works assume that the searchers all start at some xed location, and they study the large N asymptotics of T N .…”

Section: Parameter Regimementioning

confidence: 99%

Competition between slow and fast regimes for extreme first passage times of diffusion

Madrid

Lawley

2020

J. Phys. A: Math. Theor.

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Many physical, chemical, and biological systems depend on the first passage time (FPT) of a diffusive searcher to a target. Typically, this FPT is much slower than the characteristic diffusion timescale. For example, this is the case if the target is small (the narrow escape problem) or if the searcher must escape a potential well. However, many systems depend on the first time a searcher finds the target out of a large group of searchers, which is the so-called extreme FPT. Since this extreme FPT vanishes in the limit of many searchers, the prohibitively slow FPTs of diffusive search can be negated by deploying enough searchers. However, the notion of ‘enough searchers’ is poorly understood. How can one determine if a system is in the slow regime (dominated by small targets or a deep potential, for example) or the fast regime (dominated by many searchers)? How can one estimate the extreme FPT in these different regimes? In this paper, we answer these questions by deriving conditions which ensure that a system is in either regime and finding approximations of the full distribution and all the moments of the extreme FPT in these regimes. Our analysis reveals the critical effect that initial searcher distribution and target reactivity can have on extreme FPTs.

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Reaction Kinetics in the Few-Encounter Limit

Cited by 2 publications

References 50 publications

Non-Hermitian and Zeno limit of quantum systems under rapid measurements

Non-Hermitian and Zeno limit of quantum systems under rapid measurements

Competition between slow and fast regimes for extreme first passage times of diffusion

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