Physica A 390 (2011) 3086–3094 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Accuracy of coarse grained Markovian dynamics Karl Heinz Hoffmann a,∗ , Peter Salamon b a Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany b Department of Mathematical Sciences, San Diego State University, CA 92182-7720, San Diego, USA article info Article history: Received 23 December 2010 Received in revised form 19 April 2011 Available online 1 May 2011 Keywords: Irreversible thermodynamics Markov processes Markov chain Lumping techniques Coarse grained Markovian dynamics abstract Markov chain models on a mesoscopic level are a widely used description for complex systems. They are based on the assumption that certain sets of microstates can be coarse grained as their internal dynamics is faster than the time scales considered in the modeling. Here we analyze quantitatively the errors made by using lumping techniques and present the first rigorous proof for bounds on such errors. Our bounds express the deviations from a full microscopic description for all subsequent time steps in terms of the deviations in the first time step. © 2011 Elsevier B.V. All rights reserved. 1. Introduction In physics, Markov chains provide a paradigm dynamics for irreversible processes. The applications of Markov chains are typically via the master equation [1,2], with the transition probability matrix containing the specifics of the system under investigation. But Markov chains also occur in other fields of science. They are an often used modeling tool with applications ranging from business [3,4] and sociology [5] to chemistry [6], biology [7], and computer science [8]. Most applications of Markov chains model the states of the chain already as some kind of average over many factors which cannot be individually controlled. In physical theories based on the master equation, this comes about through coarse graining often implicitly by defining the states of the Markov chain to be aggregates or, in the mathematical language, lumps of microstates. Our goal is to analyze the accuracy of such coarse grained models as compared to the exact microscopic behavior, i.e., to bound the error in the dynamics as predicted by a coarse grained description. Apart from the mathematical facts [9] we present application areas as well as examples showing the strength of our bounds. We present the first rigorous proof of a fact that mesoscopic models take for granted. In these approaches, certain sets of microstates are not resolved in the modeling, but are rather treated as mesostates. One assumes that the mesoscopic dynamics appropriately models the behavior of the original system, at least on time scales that are large compared to the time scale of relaxation within one mesostate. While this assumption seems reasonable, it remains unclear to what extent the dynamics quantitatively mirrors the underlying system dynamics. The rationale behind the mesoscopic approach is to facilitate calculations with a simplified model as a full microscopic approach is usually beyond technical feasibility. Our theorem below bounds the error that results from the mesoscopic modeling in terms of the error in one step. It thereby gives a precise criterion for when and to what extent such an approach is justified. ∗ Corresponding author. E-mail address: hoffmann@physik.tu-chemnitz.de (K.H. Hoffmann). 0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.04.027