Self-Organization as Phase Transition in Decentralized Groups of Robots: A Study Based on Boltzmann Entropy Gianluca Baldassarre 1 Introduction An important goal of collective robotics (Dudek et al., 1996; Cao et al., 1997; Dorigo and Sahin, 2004) is the development of multi-robot systems capable of accomplish- ing collective tasks without centralized coordination (Kube and Zhang, 1993; Hol- land and Melhuish, 1999; Ijspeert et al., 2001; Quinn et al., 2003). From an engineer- ing point of view, decentralized multi-robot systems have several advantages vs. centralized ones, at least in some tasks. For example, they are more robust with re- spect to the failure of some of their composing robots, do not require a control sys- tem or robot with sophisticated computational capabilities to manage the centralized control (Kube and Bonabeau, 2000), have a high scalability with respect to the whole system’s size (Baldassarre et al., 2006; Baldassarre et al., in press a), and tend to require simpler robots due to the low requirements of communication as they often can rely upon implicit coordination (Beckers et al., 1994; Trianni et al., 2006). Decentralized coordination is often based on self-organizing principles. Very of- ten research on decentralized multi-robot systems makes a general claim on the pres- ence of these principles behind the success of the studied systems, but it does not conduct a detailed analysis of which specific principles are at work, nor it attempts to measure their effects in terms of the evolution of the system’s organization in time or to analyze the robustness of its operation versus noise (cf. Holland and Melhuish, 1999; Krieger et al., 2000; Kube and Bonabeau, 2000; Quinn et al., 2003). This pa- per studies some of these issues in a multi-robot system presented in detail elsewhere (Baldassarre et al., 2003; Baldassarre et al., 2006; Baldassarre et al., in press; Baldas- sarre et al., in press a). This system is formed by robots that are physically connected between them and have to coordinate their direction of motion to explore an open arena without relying on a centralized coordination. The robots are controlled by an identical neural network whose weights are evolved through a genetic algorithm.