Limits on Inferring the Past

Nature is irreversible. Any state of knowledge, regardless of how exact, will invariably and universally deteriorate into a entropy maximizing probability distribution. Our loss of information, forward in time, can be quantified by the entropy generation rate. Here we address the converse question of how well one can reconstruct a past state given exact information concerning the present. To this end, we invert Langevin dynamics to estimate the original starting point of a number of particles, given their exact positions at a later time. We define "reconstruction entropy" - a measure of how many good candidates there are for a past state, given the present state. We then evaluate reconstruction entropy for Langevin dynamics exactly for simple cases, explore general properties of reconstruction entropy, and make several conjectures about the behavior of Langevin reconstruction entropy as a function of time. Finally, we obtain a formula connecting reconstruction entropy, thermodynamic entropy, and system parameters.