Capturing the local entropy production by data compression

We introduce a universal protocol to estimate local entropy production by data compression. The Kullback-Leibler divergence (KLD), or relative entropy, between the probability of observing a trajectory with time running forward and its time reversal quantitatively characterizes the breakdown of time-reversal symmetry and gives the entropy production of non-equilibrium steady states. Here we use a cross-parsing compression algorithm (Ziv-Merhav) to estimate KLD between two individual sequences. By compressing the forward time sequence of states using its time reversed, we obtain a close estimation of the entropy production for finite state systems. A natural spatial decomposition of entropy production can then be defined by applying the algorithms on time series of local representation of states, which gives a spatial distribution of the local entropy production. We validate this method on motility-induced phase separation of active Brownian particles systems. The local entropy production distributions show good agreement among MD simulation, lattice model and active field theory. Experimental systems such as bacterias driven by funnel structures are also analyzed.