Mapping of uncertainty relations between continuous and discrete time

Lower bounds on fluctuations of thermodynamic currents depend on the nature of time, discrete or continuous. To understand the physical reason, we compare current fluctuations in discrete-time Markov chains and continuous-time master equations. We prove that current fluctuations in the master equations are always more likely, due to random timings of transitions. This comparison leads to a mapping of the moments of a current between discrete and continuous time. We exploit this mapping to obtain uncertainty bounds. Our results reduce the quests for uncertainty bounds in discrete and continuous time to a single problem.

Figure 1

Biased random walk. (a) Transition network. (b) Sample trajectory for the Markov chain. (1). (c) Sample trajectory for the master equation (2). Notice the fluctuations in the jump times in (c), which are absent in (b).

Figure 2

Comparison of the rate functions for the discrete and continuous cases [see Eq. (19)] for the biased random walk. The four panels correspond to different choices of the bias $a$, as shown in the legend. The rate functions are computed analytically for a three-state random walk with periodic boundary conditions and $\tau =1$. Note that the comparison is possible only in the interval shown in the figure, since $j\in [-1,1]$ in the discrete case.

Figure 3

Uncertainty bounds in the scaling form of Eq. (26). (a) Discrete-time bounds. The blue dot-dashed line and the red dashed line represent the mapped Barato-Gingrich bound (23) and the Proesmans–Van den Broeck bound (24), respectively. (b) Continuous-time bounds. The blue dot-dashed line and the red dashed line are the Barato-Gingrich bound (21) and the mapped Proesmans–Van den Broeck bound (25), respectively.

Figure 4

Comparison of the bound (25) with examples of a biased random walk and a fully connected four-state network. In both cases, we set $\tau =1$. For the biased random walk, the current is $j=J/t$, where $J$ is given by Eq. (3) and entropy production is tuned by varying the bias parameter $a$. For the four-state network, the parameters $j_{kl}$ defining the generalized current were independently drawn from a Gaussian distribution with zero average and unit variance. We then numerically minimize ${\tilde{\sigma }}_{j,c}^2/{\langle j\rangle }_c^2$ as a function of ${\mathrm{\Sigma }}_c$, with the constraint $W_{ll}\ge -1$ for all $l$. The moments of the currents are computed with the method of Ref. [26]. The minimization is carried out with the matlab patternsearch algorithm. To avoid local minima, for each value of ${\mathrm{\Sigma }}_c$, we perform $N=25$ different runs of patternsearch starting from different initial conditions. The smallest value among these realizations is plot in the figure. We verified that performing the minimization after a different random choice of the coefficients $j_{kl}$ leads to very similar results. Notice how the two systems are very close to the mapped Proesmans–Van den Broeck bound (25) (blue solid line) for a broad range of ${\mathrm{\Sigma }}_c$ values.