Inference of time irreversibility from incomplete information: Linear systems and its pitfalls

Data from experiments and theoretical arguments are the two pillars sustaining the job of modeling physical systems through inference. In order to solve the inference problem, the data should satisfy certain conditions that depend also upon the particular questions addressed in a research. Here we focus on the characterization of systems in terms of a distance from equilibrium, typically the entropy production (time-reversal asymmetry) or the violation of the Kubo fluctuation-dissipation relation. We show how general, counterintuitive and negative for inference, is the problem of the impossibility to estimate the distance from equilibrium using a series of scalar data which have a Gaussian statistics. This impossibility occurs also when the data are correlated in time, and that is the most interesting case because it usually stems from a multi-dimensional linear Markovian system where there are many timescales associated to different variables and, possibly, thermal baths. Observing a single variable (or a linear combination of variables) results in a one-dimensional process which is always indistinguishable from an equilibrium one (unless a perturbation-response experiment is available). In a setting where only data analysis (and not new experiments) is allowed, we propose as a way out the combined use of different series of data acquired with different parameters. This strategy works when there is a sufficient knowledge of the connection between experimental parameters and model parameters. We also briefly discuss how such results emerge, similarly, in the context of Markov chains within certain coarse-graining schemes. Our conclusion is that the distance from equilibrium is related to quite a fine knowledge of the full phase space, and therefore typically hard to approximate in real experiments.

Figure 1

Average trajectory for Brownian gyrator (drift parameters $\alpha =\gamma =1, \lambda =\mu =-0.5$) for trajectories of duration $\tau =5$, from $x_i=(0,0)$ to $x_f=(1,2)$ and viceversa. The two panels represent different choices of temperatures $T_1,T_2$: in (a) the system satisfies detailed balance ($T_1=T_2=2$); in (b) detailed balance is broken ($T_1=10,T_2=1$). These are parametric plots, where the axis are the two average components of the bridge $L_i\left(t\right)={\langle x_i\left(t\right)\rangle }_{\tau }^{x_i\to x_f}$, while time $t$ is the parameter.

Figure 2

Single component of the average bridge trajectories of Fig. 1, as a function of $t$. The dotted lines correspond to $\pm \sqrt{Q_{11}}$ and give a better ideas of the fluctuations of the bridge trajectories. Same parameters of Fig. 1. Again, (a) is a case with detailed balance ($T_1=T_2$); (b) is the out-of equilibrium case ($T_1\ne T_2$).

Figure 3

In the case of a bridge from $x_i=(0,0)$ to $x_f=(0,0)$ after a time $\tau =5$, the values of the covariance matrix element $Q_{11}$ are shown as a function of time. For the equilibrium case $[T_1=T_2$, (a)] the curve is symmetric around $t=\tau /2$, and the direct curve corresponds exactly to the reverse bridge. On the contrary, out of equilibrium $[T_1\ne T_2$, (b)], direct and reverse path differ, and the curves are no more symmetric. The values of the paramters are the same as those in Fig. 1.

Figure 4

Average bridge shape for a stationary single component of a Brownian gyrator, with (a) and without (b) detailed balance. In each panel, the curves represent different duration values $\tau =1,2,...,10$, respectively, lines from bottom to top. The parameters of the gyrator are the same as those of Fig. 1.

Figure 5

(a) Autocorrelation function ${\mathcal{C}}_y\left(t\right)$ for the two systems $S_1$ (blue/dark gray) and $S_2$ (orange/light gray). (b) Response functions ${\mathcal{R}}_y\left(t\right)$ of $S_1$ (blue/dark gray) and $S_2$ (orange/light gray). The results have been obtained by means of numerical simulations. The analytical results are shown with dashed black lines.

Figure 6

Conditional average $\langle y\left(\tau \right)|y_0\rangle$ as a function of the time delay $\tau$ for the two systems $S_1$ (left) and $S_2$ (right). In both cases $y_0=0.7{\sigma }_y$, where ${\sigma }_y$ is the standard deviation. The results have been obtained by means of numerical simulations.

Figure 7

Distributions of the average entropy production rate $S$ for the two systems $S_1$ (a) and $S_2$ (b). The dotted lines correspond to the theoretical value of the average entropy production rate $S$.

Figure 8

Distributions of rescaled exit times ${\tau }_r=\frac{\tau -\langle \tau \rangle }{\sigma }$ (${\sigma }^2=\langle {\tau }^2\rangle -{\langle \tau \rangle }^2$) from macrostate 0 (containing many microstates) to macrostate 1 (pure state) for Markov chains on a ring. (a)–(c) Markov chain with $N=4$ states. (b)–(d) Markov chain with $N=5$ states. (a), (b) $\epsilon =0.135$ and $S=0.167$; (c), (d) $\epsilon =\frac{1-\alpha }{2}=0.45$ and $S=+\infty$.