Decision Making in the Arrow of Time

We show that the steady-state entropy production rate of a stochastic process is inversely proportional to the minimal time needed to decide on the direction of the arrow of time. Here we apply Wald’s sequential probability ratio test to optimally decide on the direction of time’s arrow in stationary Markov processes. Furthermore, the steady-state entropy production rate can be estimated using mean first-passage times of suitable physical variables. We derive a first-passage time fluctuation theorem which implies that the decision time distributions for correct and wrong decisions are equal. Our results are illustrated by numerical simulations of two simple examples of nonequilibrium processes.

Figure 1

Log-likelihood ratio of the sequential probability ratio test in the arrow of time as a function of time in a drift-diffusion process with diffusion coefficient $D=0.52\mu {\mathrm{m}}^2/\mathrm{s}$ and drift velocity $v=65\mu \mathrm{m}/\mathrm{s}$. The simulation time step is $\mathrm{\Delta }\mathrm{t}=0.1\mathrm{ms}$. The thresholds of the test are shown as horizontal lines for symmetric error probabilities equal to 20% (thin and red) and 1% (thick and blue). The thresholds $L_0$ and $L_1$ correspond to the choice of one of the two hypotheses: the sequence runs forward ($L_1$) or backward ($L_0$) in time. With 20% error probability, the decision is made faster (red circle and vertical thin red dashed line) than for 1% error probability (blue circle and vertical thick blue dashed line).

Figure 2

Estimation of the steady-state entropy production rate of a drift-diffusion process with periodic boundary conditions (e.g., particle in a ring, see inset) by a sequential probability ratio test in the arrow of time. The estimator $L(1-2\alpha )/⟨{\tau }_{\mathrm{dec}}⟩$ is shown as a function of the error probability $\alpha$ for different simulation time steps $\mathrm{\Delta }t/{\tau }_c$ and normalized by ${\tau }_c=1/[⟨d{S}_{\mathrm{tot}}/dt⟩/k]=D/v^2$. For the vertical axis we use the empirical mean of ${\tau }_{\mathrm{dec}}$, and $L$ is the threshold for the decision. The data are obtained from 1000 numerical simulations with drift velocity $v=65\mu \mathrm{m}/\mathrm{s}$ and diffusion coefficient $D=0.52\mu {\mathrm{m}}^2/\mathrm{s}$. The horizontal line corresponds to the steady-state entropy production rate.

Figure 3

Conditional distributions of the decision time $P({\tau }_{\mathrm{dec}}|\to )$ and $P({\tau }_{\mathrm{dec}}|\leftarrow )$ obtained from ${10}^6$ numerical simulations of a drift-diffusion process with diffusion $D=0.52\mu {\mathrm{m}}^2/\mathrm{s}$ and drift $v=65\mu \mathrm{m}/\mathrm{s}$. The simulation time step is $\mathrm{\Delta }\mathrm{t}=0.1\mathrm{ms}$ and the error probability $\alpha =0.01$.

Figure 4

Estimation of the entropy production rate using the SPRT in the arrow of time in a flashing ratchet model using the right-hand side in Eq. (8) (blue open circles) and the right-hand side in Eq. (6) (blue open diamonds) as a function of the reliability of the test. Red open squares are given by the ratio between $D[\to \parallel \leftarrow ]$ and the mean first-passage time $⟨\tau ⟩$ of the position of the particle in Eq. (9). The results were obtained from 1000 numerical simulations with time step $\mathrm{\Delta }\mathrm{t}=1\mu \mathrm{s}$, diffusion coefficient $D=1\mu {\mathrm{m}}^2/\mathrm{ms}$, $V_0=10\text{kT}$, $a=1/3\mu \mathrm{m}$, and $\omega =10\mathrm{kHz}$. The characteristic time ${\tau }_c=0.07\mathrm{ms}$ is the numerical estimate of $k/⟨d{S}_{\mathrm{tot}}/dt⟩$ obtained from a single stationary trajectory of ${10}^7$ data points. Inset: Correction term in Eq. (6) given by $[⟨\mathrm{\Delta }{S}_{\mathrm{ex}}{⟩}_{\mathrm{dec}}/k⟨{\tau }_{\mathrm{dec}}⟩]{\tau }_c$ as a function of $1-\alpha$ (blue triangles). The solid line is a linear fit of the data.

Figure 5

Conditional distributions of the decision time $P({\tau }_{\mathrm{dec}}|\to )$ and $P({\tau }_{\mathrm{dec}}|\leftarrow )$ obtained from ${10}^6$ numerical simulations of the flashing ratchet with the same simulation parameters as in Fig. 4. The two figures show the distributions for two different error probabilities $\alpha$.