Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and nondecreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.

Figure 1

Bounds on observable fluctuations for a two-level system. Transition rates are $W_{10}=\gamma$ and $W_{01}=\kappa$. We consider as observable $A$ the total number of $1\to 0$ jumps. In the stationary state $\langle a\rangle =\langle A\rangle /t=\gamma \kappa /(\gamma +\kappa )$. The average activity per unit time is $\langle k\rangle =2\langle a\rangle$. Panel (a) shows the rate function $\phi (A/t)$ [full (black)] for $\gamma =5$ and $\kappa =1.25$. The rate function is bounded from above everywhere by a CMP rate function, Eq. (17) [dashed (red)]. We also show for comparison a Poisson rate function with mean $\langle a\rangle$ [dotted (blue)]. Panel (b) shows the corresponding scaled cumulant generating function $\theta \left(s\right)=\frac{1}{2}\left[\sqrt{{(\gamma -\kappa )}^2+4\gamma \kappa e^{-s}}-(\gamma +\kappa )\right]$ [full (black)]. It is monotonic in $s$ since $A\ge 0$, and is bounded from below, Eq. (18), by ${\theta }_{*}\left(s\right)=\frac{2\gamma \kappa }{\gamma +\kappa }(e^{-s/2}-1)$ [dashed (red)]. For the case $\kappa =\gamma$ the bounds become exact in this simple model.

Figure 2

Bounds on first-passage time fluctuations for the two-level system of Fig. 1. The FPT $\tau$ is defined as the time when a total $A$ of up or down jumps $1\to 0$ is reached. In the stationary state $\langle \tau \rangle =A/\langle a\rangle =A(\gamma +\kappa )/\left(\gamma \kappa \right)$. Panel (a) shows the rate function $\varphi (\tau /A)$ [full (black)] for $\gamma =5$ and $\kappa =1.25$, and assuming the initial state is zero. The rate function is bounded from above everywhere by ${\varphi }_{*}(\tau /A)$, Eq. (37) [dashed (red)]. We also show for comparison the FPT rate function of a Poisson process with the same mean [dotted (blue)]. Panel (b) shows the FPT scaled cumulant generating function $g\left(\mu \right)=ln\left(\gamma \kappa \right)-ln\left[\right(\gamma +\mu \left)\right(\kappa +\mu \left)\right]$ [full (black)]. It is bounded from below, Eq. (35), by $g_{*}\left(\mu \right)$ [dashed (red)].

Figure 3

Bounds on rate functions for a three-level system. Transition rates are $W_{10}=W_{01}=\gamma$ and $W_{02}=W_{20}=\kappa$. We consider as observable $A$ the total number of $1\to 0$ jumps. Panel (a) shows the rate function $\phi (A/t)$ [full (black)] for $\gamma =1$ and $\kappa =0.5$. The rate function is bounded from above everywhere by a CMP rate function, Eq. (17) [dashed (red)]. We also show for comparison a Poisson rate function with mean $\langle a\rangle$ [dotted (blue)]. Panel (b) shows the FPT rate function $\varphi (\tau /A)$ [full (black)]. The rate function is bounded from above everywhere by ${\varphi }_{*}(\tau /A)$, Eq. (37) [dashed (red)]. The CMP bounds are not tight, due to the intermittent nature of the dynamics (i.e., events are correlated in time, or “bunched”) making the event statistics super-Poissonian.