First-passage times in renewal and nonrenewal systems

Krzysztof Ptaszyński

Phys. Rev. E 97, 012127 – Published 22 January 2018

Abstract

Fluctuations in stochastic systems are usually characterized by full counting statistics, which analyzes the distribution of the number of events taking place in the fixed time interval. In an alternative approach, the distribution of the first-passage times, i.e., the time delays after which the counting variable reaches a certain threshold value, is studied. This paper presents the approach to calculate the first-passage time distribution in systems in which the analyzed current is associated with an arbitrary set of transitions within the Markovian network. Using this approach, it is shown that when the subsequent first-passage times are uncorrelated, there exist strict relations between the cumulants of the full counting statistics and the first-passage time distribution. On the other hand, when the correlations of the first-passage times are present, their distribution may provide additional information about the internal dynamics of the system in comparison to the full counting statistics; for example, it may reveal the switching between different dynamical states of the system. Additionally, I show that breaking of the fluctuation theorem for first-passage times may reveal the multicyclic nature of the Markovian network.

Received 18 September 2017
Revised 14 November 2017

DOI:https://doi.org/10.1103/PhysRevE.97.012127

Physics Subject Headings (PhySH)

Fluctuations & noise

Quantum dots

Markovian processes

Statistical Physics & Thermodynamics

Authors & Affiliations

Krzysztof Ptaszyński^*

Institute of Molecular Physics, Polish Academy of Sciences, ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland

^*krzysztof.ptaszynski@ifmpan.poznan.pl

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Vol. 97, Iss. 1 — January 2018

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Images

Figure 1
Plot of an exemplary function $g (z)$ illustrating the properties of the functions $z_{+} (s)$ and $z_{-} (s)$ , i.e., the larger and the smaller root of the equation $g (z) = s$ [cf. the discussion below Eq. (22)].
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Figure 2
(a) Scheme of the system of two capacitively coupled quantum dots, each attached to two leads. Orbital energy in the upper dot is equal to $ε$ . The electrochemical potentials and temperatures of the leads attached to the upper dot are equal to $μ_{α}$ and $T_{α}$ , respectively, with $α \in {L, R}$ . The transport in the lower dot is unidirectional, occurring from the left to the right lead, due to the high voltage bias. Interdot Coulomb interaction is equal to $U$ . Tunneling rates in the upper and lower dot, respectively, are denoted as $Γ_{α}$ and $γ_{α}$ . (b) Four-state model of the dynamics of the system. Terms $f_{α}$ and $f_{α}^{U}$ are Fermi distribution functions defined as $f_{α} = f [(ε - μ_{α}) / k_{B} T_{α}]$ and $f_{α}^{U} = f [(ε + U - μ_{α}) / k_{B} T_{α}]$ . Red arrows with bullet tails denote the forward transitions, and blue arrows with square tails denote the backward transitions.
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Figure 3
Randomness parameter $R_{N}$ multiplied by $N$ in the double-dot system as a function of $N$ for different values of $γ / Γ$ (with $γ = γ_{L} = γ_{R}$ and $Γ = Γ_{L} = Γ_{R}$ ). All results for $ε = 0$ , $μ_{L} = 1$ , $μ_{R} = - 1$ , $k_{B} T_{L} = k_{B} T_{R} = 1$ , and $U = 3$ .
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Figure 4
The Fano factor (FF) (red solid line) and the randomness parameter $R_{1}$ (black dots) in the double-dot system as a function of (a) $U$ for $k_{B} T_{L} = k_{B} T_{R} = 1$ , and (b) $k_{B} T (= k_{B} T_{L} = k_{B} T_{R})$ for $U = 2$ . All results for $ε = 0$ , $μ_{L} = 1$ , $μ_{R} = - 1$ , $Γ_{L} = Γ_{R} = 1$ , and $γ_{L} = γ_{R} = 0.05$ .
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Figure 5
(a) Scheme of the transport through a single quantum dot. The cases when a single level $A$ or two levels $A$ and $B$ are available for transport are considered. The energies of the dot levels $A$ and $B$ are equal to $ε_{A}$ and $ε_{B}$ . The dot is attached to two leads with electrochemical potentials $μ_{α}$ and temperatures $T_{α}$ with $α \in {L, R}$ . Tunneling rates between the leads and specific levels are denoted as $Γ_{α}^{i}$ with $i \in {A, B}$ . (b) Three-state Markovian model of the dynamics of the system. Terms $f_{α}^{i}$ are the Fermi distribution functions defined as $f_{α}^{i} = f [(ε_{i} - μ_{α}) / k_{B} T_{α}]$ . Solid (dashed) lines correspond to the tunneling through the level $A$ ( $B$ ). Red arrows with bullet tails denote the forward transitions, while blue arrows with square tails denote the backward transitions.
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Figure 6
First-passage time distribution $F (1 | τ)$ (orange solid line) and the normalized first-passage time distribution $\tilde{F} (- 1 | τ) = F (- 1 | τ) / [\int_{0}^{\infty} F (- 1 | τ) d τ]$ (black dashed line) for (a) a single-level dot with $ε_{A} = 0$ and $Γ_{L}^{A} = Γ_{R}^{A} = Γ$ , and (b) a two-level dot with $ε_{A} = 0$ , $ε_{B} = - 2$ , and $Γ_{L}^{A} = Γ_{R}^{A} = Γ_{L}^{B} = Γ_{L}^{B} = Γ$ . All results for $μ_{L} = 1$ , $μ_{R} = - 1$ , $k_{B} T_{L} = 1$ , and $k_{B} T_{R} = 2$ .
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