Comment on"Inferring broken detailed balance in the absence of observable currents"

In this reply, we resolve the apparent discrepancy raised in the"Comment on Inferring broken detailed balance in the absence of observable currents"[arXiv:2112.08978v1]. We stress that the non-instantaneous transition paths originate from the choice of a decimation process non-local in time, in contrast to the decimation process described in our original work, which commutes with time-reversal. Therefore, the example using such a non-local decimation procedure, which indeed gives rise to spurious time-irreversibility, does not invalidate our main result, which was rigorously derived and proven mathematically. We hope this reply highlights the subtlety of the different choices for decimating time series data which is pivotal for a correct understanding of the thermodynamics of small systems.

In [1], Hartich and Godec (HG) present a counterexample that apparently refutes our results in [2]. While the content of their Comment is essentially correct, it does not invalidate our results, but rather raises an interesting question on the effect of coarse-graining on irreversibility, which we discuss in detail below. However, the way they write the Comment is slightly misleading. They highlight the fact that we never tested Eq. (4) in Ref. [2], implicitly suggesting that this equation is wrong. Eqs. (2)(3)(4) are an exact expression of the Kullbak-Leibler divergence (KLD) between a semi-Markov chain and its time reverse. There is no need of a "test" since it is an exact result, whose rigorous mathematical proof is given in the section Methods of Ref. [2]. Moreover, HG claim that our results and those of Wang and Qian [3] are "diametrically opposing". We want to stress that this is not true. Both our paper and [3] are fully correct and perfectly compatible.
The example in [1] has nothing to do with the validity of Eq. (4). Instead, it calls into question Eq. (1) in our paper [2], which asserts that the KLD between an observed trajectory and its time reversal is a lower bound to the physical entropy production. This claim is widely used in the literature and based on a well-known property of the KLD. The KLD between two stochastic processes measures our capacity to distinguish between them using data. If we remove information about these processes, by adding noise to the data or decimating the underlying network, it should be clear that this capacity decreases and, consequently, the KLD decreases. Since the entropy production is equal to the KLD between the microscopic trajectory and its time reversal [4][5][6], the KLD between a coarse-grained trajectory and its time reversal is a lower bound on the actual entropy production.
In more precise mathematical terms, this argument is based on the following property of the KLD between two * Electronic address: bisker@tauex.tau.ac.il † Electronic address: ignacio.martinez@ugent.be ‡ Electronic address: jmhorow@umich.edu § Electronic address: parrondo@ucm.es distributions p X and q X of a random variable X: where f (x) is an arbitrary (possibly random) function.
The equality holds if f is one-to-one and deterministic. The function f can represent a coarse-graining when several states {x 1 , . . . , x m } coalesce into a single state f (x 1 ) = · · · = f (x m ), or the removal of the information associated to a particular variable, when for instance The inequality (1) has a direct and easy interpretation: by manipulating the available data, that is, by transforming the data according to the function f , one cannot increase the distinguishability between the two probability distributions, p and q.
If X is a microscopic trajectory and ΘX is its time reversal, the entropy production ∆S verifies [4,5] ∆S = kD(p X ||p ΘX ). ( where k is Boltzmann constant. If we apply Eq. (1) to this expression, we obtain However, Eq. (1) in our work [2] slightly differs from this inequality. There, we assume that the observer has access to a coarse-grained trajectory f (X) and constructs the time reversal as Θf (X). Hence, to have from Eq. (3), a sufficient condition is that coarse-graining and time reversal commute, f (ΘX) = Θf (X). The example in [1] does not fulfill this commutation relation, as HG mention in their comment. In Fig. 1, we introduce a simplified version of the counterexample discussed in [1] that illustrates the origin of this issue. Here, the microscopic trajectory X is reversible, hence ΘX = X, and the commutation implies that f (X) is also reversible, as shown in the figure. On the other hand, the red dashed line results from the decimation procedure considered in [1], which is based on the arrivals at states A and B. This decimation does not commute with the timereversal operation, and the resulting trajectory is irreversible Θf (X) = f (X) = f (ΘX): in the red trajectory the system takes two units of time to jump down from B to C and one unit of time to jump up from B to A; whereas these waiting times are swapped when the trajectory is reversed. Notice that this decimation procedure is non local in time, i.e., the state of the decimated trajectory at time t is not a function of the micro-state at time t (open circles), but depends on the past. the states A, B, and C as jumps between states in the coarse-grained trajectory. In Fig. 1 (bottom), we plot a sketch of reversible microscopic trajectory (thin black lines) X, whereas the corresponding coarse-grained trajectory using this prescription (red dashed lines). Since X is reversible, X = ΘX, and the resulting trajectory is irreversible, hence Θf (X) = f (X) = f (ΘX). The irreversibility of the red trajectory is revealed by the waiting time distributions: in the forward trajectory, f (X), the system takes two units of time to jump down from B to C and one unit of time to jump up from B to A, whereas these waiting times are swapped when the trajectory is reversed. Notice that this decimation procedure is non local in time, i.e., the state of the decimated trajectory at time t is not a function of the micro-state at time t (open circles), but depends on the past.
We also plot in Fig. 1 (bottom), the trajectory resulting from lumping states B and b (blue dashed lines), which is reversible. This procedure is local in time, since the hidden state b is always mapped onto B, and consequently decimation and time reversal commute, f (ΘX) = Θf (X).
Summarizing, HG comment is interesting since it points out that a sufficient condition for Eq. (1) to be valid is that coarse-graining and time reversal commute. This condition was not mentioned in our paper because we implicitly assumed coarse-graining procedures that are local in time and consequently commute with the time-reversal operation, as implied in the description of the decimation process in the main text and in the Methods section in [2].