From single-particle to collective effective temperatures in an active fluid of self-propelled particles

We present a comprehensive analysis of effective temperatures based on fluctuationdissipation relations in a model of an active fluid composed of self-propelled hard disks. We first investigate the relevance of effective temperatures in the dilute and moderately dense fluids. We find that a unique effective temperature does not in general characterize the non-equilibrium dynamics of the active fluid over this broad range of densities, because fluctuation-dissipation relations yield a lengthscale-dependent effective temperature. By contrast, we find that the approach to a non-equilibrium glass transition at very large densities is accompanied by the emergence of a unique effective temperature shared by fluctuations at all lengthscales. This suggests that an effective thermal dynamics generically emerges at long times in very dense suspensions of active particles due to the collective freezing occurring at non-equilibrium glass transitions. Copyright c © EPLA, 2015 Introduction. – Statistical mechanics provides a unified theoretical description of systems at thermal equilibrium in terms of the probability distribution over phase space, from which thermodynamic quantities such as temperature can be defined [1]. A similar framework is lacking for out-of-equilibrium systems for which the definition of a temperature remains an open issue [2]. Active matter formed by assemblies of living cells [3], bacteria [4], self-propelled colloids [5–8] or grains [9,10], is a coherent class of non-equilibrium systems receiving increasing attention, because they raise fundamental issues and for potential applications in soft matter and biophysics [11,12]. A number of recent studies have addressed the question of whether “effective” thermodynamic concepts can be fruitfully applied to describe the phase behaviour and microscopic dynamics of active matter. This question is natural because if some mapping to an equilibrium situation exists, then the whole arsenal of equilibrium statistical mechanics becomes available for further theoretical treatment. In particular the definitions of a non-equilibrium temperature [13–18], of an active pressure [8,19,20], of activity-induced interactions [8,21] and non-equilibrium free energies [22,23] have been investigated for self-propelled particles. An effective temperature Teff defined as the parameter replacing the thermal bath temperature in fluctuation-dissipation relations was extensively studied in slowly relaxing materials, such as spin and structural glasses [24–27]. In these systems, the effective temperature becomes a meaningful thermodynamic concept [28]. It can be used to quantify heat flows and can be defined through a microcanonical construction over a restricted region of the phase space. Physically, this approach is possible because there is a strong decoupling of timescales [2] between thermal motion at short times, which essentially follows an equilibrium statistics, and an “effective” thermal dynamics at long times, which is an emerging collective property characterizing slow dynamical events leading to structural relaxation in driven and aging glasses. In active matter, the possibility to define a single-particle effective temperature in the dilute regime where particles do not interact has been discussed [14,16,17,29,30]. The simple fluid state has been analysed mostly by numerical simulations determining fluctuation-dissipation relations for different observables in various models of self-propelled particles [13,15,18]. Finally, in the limit of very large densities where selfpropelled particles may undergo a non-equilibrium glass transition, the driven glassy dynamics should resemble the one of slowly driven glasses and a collective effective temperature was predicted to emerge from the analysis of a mean-field active glass model [31], but this prediction