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We present a comprehensive study about the relationship between the way Detailed Balance is broken in non-equilibrium systems and the resulting violations of the Fluctuation-Dissipation Theorem. Starting from stochastic dynamics with both odd and even variables under Time-Reversal, we exploit the relation between entropy production and the breakdown of Detailed Balance to establish general constraints on the non-equilibrium steady-states (NESS), which relate the non-equilibrium character of the dynamics with symmetry properties of the NESS distribution. This provides a direct route to derive extended Fluctuation-Dissipation Relations, expressing the linear response function in terms of NESS correlations. Such framework provides a unified way to understand the departure from equilibrium of active systems and its linear response. We then consider two paradigmatic models of interacting selfpropelled particles, namely Active Brownian Particles (ABP) and Active Ornstein-Uhlenbeck Particles (AOUP). We analyze the non-equilibrium character of these systems (also within a Markov and a Chapman-Enskog approximation) and derive extended Fluctuation-Dissipation Relations for them, clarifying which features of these active model systems are genuinely non-equilibrium.

We explore the phase diagram of a two-dimensional model of active glass. Using local order parameters and studying their relaxation dynamics, we establish the presence of four distinct regimes. A Motility-Induced Phase Separation (MIPS) region where the system segregates into a high and low-density phase, a re-entrant liquid/hexatic transition, and, for large enough packing fractions and small enough activity, a disordered glassy phase. Although the dense phase of MIPS is an assembly of geometrically frustrated particles, we do not observe any glassy feature in such a regime. For gaining insight into the difference between dense active matter at high and low activity, we investigate the structural properties of the inherent and instantaneous configurations. We find that the energy spectrum of the instantaneous vibrational modes sheds new light on the fundamental differences between equilibrium and active glasses. From this analysis, indeed, it is possible to make a connection between the breakdown of the concept of effective temperature (defined from the spectrum of vibrations) in active matter at high densities and the mechanical stability of the glass state. Finally, we show that such breakdown of the effective temperature marks a crossover between a low-activity regime, where the system shares many similarities with its equilibrium counterpart, and a high-activity regime, which drives the system towards MIPS.

Motion of active particles, such as catalytic micro-and nano-motors, is usually characterized via either dynamic light scattering or optical microscopy. In both cases, speed of particles is obtained from the calculus of the mean square displacement (MSD) and typically, the theoretical formula of the MSD is derived from the motion equations of an active Brownian particle. One of the most commonly reported parameters is the speed of the particle, usually attributed to its propulsion, and is widely used to compare the motion efficiency of catalytic motors. However, it is common to find different methods to compute this parameter, which are not equivalent approximations and do not possess the same physical meaning. Here, we review the standard methods of speed analysis and focus on the errors that arise when analyzing the MSD of self-propelled particles. We analyze the errors from the computation of the instantaneous speed, as well as the propulsive speed and diffusion coefficient through fittings to parabolic equations, and we propose a revised formula for the motion analysis of catalytic particles moving with constant speed that can improve the accuracy and the amount of information obtained from the MSD. Moreover, we emphasize the importance of spotting the presence of different motion dynamics, such as particles with active angular speed or that move under the presence of drift, and how the breaking of ergodicity can completely change the analysis by considering particles with an exponentially decaying speed. In all cases, real data from enzymatically propelled micro-motors and simulations are used to back up the theories. Finally, we propose several analytical approaches and analyze limiting cases that will help to deal with these scenarios while still obtaining accurate results.