Active systems are composed of units able to transform non-thermal energy into motion in the presence of dissipation. Examples abound in biology, ranging across scales from animal groups, insect swarms, suspensions of bacteria and microtubules driven by molecular motors, among many others; synthetic realizations are nowadays also engineered using, for instance, catalytic colloidal particles. When many active particles are put together, they are capable of fascinating collective behavior, think of the shapes that a flock of birds takes in the sky. Moreover, active systems can be seen as a broad, but sufficiently well-defined class, of non-equilibrium systems, so that we can hope for a relatively coherent theoretical description. Indeed, in the last 20 years, a lot of work from statistical and soft-matter physicists has tried to understand and predict their emerging collective phenomena. In these lectures, I will give a far-from-comprehensive overview of the microscopic and continuum models used to describe them.
Large ensembles of coupled oscillators are systems out of equilibrium, which can demonstrate transition to synchrony as a non-equilibrium phase transition. The Kuramoto system of globally coupled phase oscillators is a paradigmatic model for such a transition. In this lecture I will show how one can reduce the dynamics of deterministic and noisy oscillators to a finite-dimensional dynamical system for a few complex order parameters. This will be a generalization of the famous Ott-Antonsen ansatz.
Tuesday, February 12, 2019, 14:00
Collective organization of networked phase oscillators: explosive synchronization, dynamically interdependent networks and Bellerophon states
I will try to discuss the spontaneous emergence of some complex collective dynamics in networked phase oscillators. As a first step, I will discuss how synchronization may emerge in the graph. Synchronization is a process in which dynamical systems adjust some properties of their trajectories (due to their interactions, or to a driving force) so that they eventually operate in a macroscopically coherent way. A common result is that the vast majority of transitions to synchronization are of the second-order type, continuous and reversible. However, as soon as networked units with complex architectures of interaction are taken into consideration, abrupt and irreversible phenomena may emerge, namely explosive synchronization, which rather remind first-order like transitions.
In the second part of my talk, I will concentrate on a recently unveiled coherent state, the Bellerophon state, which is generically observed in the proximity of explosive synchronization at intermediate values of the coupling strength. Bellerophon states are multi-clustered states emerging in symmetric pairs. In these states, oscillators belonging to a given cluster are not locked in their instantaneous phases or frequencies, rather they display the same long-time average frequency (a sort of effective global frequency). Moreover, Bellerophon states feature quantum traits, in that such average frequencies are all odd multiples of a fundamental frequency. Finally, if there is sufficient time, I will try to show a generalization of the concept of interdependence of graphs when dynamical systems are considered to be the constituents of the networks, and in relationship to the setting of collective dynamics.
When the network is reconstructed, two types of errors can occur: false positive and false negative errors about the presence or absence of links. In this talk, the influence of these two errors on the vertex degree distribution is analytically analysed. Moreover, an analytic formula of the density of the biased vertex degree distribution is found. In the inverse problem, we find a reliable procedure to reconstruct analytically the density of the vertex degree distribution of any network based on the inferred network and estimates for the false positive and false negative errors.
They are expressed in terms of the statistics of fluctuating thermodynamic quantities (heat and work) in a non-equilibrium situation, and are a reflection of an underlying symmetry of the micro-dynamics (typically time-reversal symmetry). Close to equilibrium they imply the well known results of linear response theory (notably the fluctuation dissipation theorem, Kubo relations, Onsager Casimir relations etc), but hold regardless of how far a system is driven away from equilibrium. They can also be understood as a refinement of the second law of thermodynamics. The theory of fluctuation relations has been developed and checked experimentally both for classical and quantum systems, and as well in various non-equilibrium scenarios, namely driven system (closed or open), systems subject to non-equilibrium boundary conditions, or both (heat engines), and even in presence of invasive quantum measurements. I will provide an overview on the subject for the non-specialist.
Collective behavior in biological systems is a complex topic, to say the least. It runs wildly across scales in both space and time, involving taxonomically vastly different organisms, from bacteria and cell clusters, to insect swarms and up to vertebrate groups. It entails concepts as diverse as coordination, emergence, interaction, information, cooperation, decision-making, and synchronization. Amid this jumble, however, we cannot help noting many similarities between collective behavior in biological systems and collective behavior in statistical physics, even though none of these organisms remotely looks like an Ising spin. Such similarities, though somewhat qualitative, are startling, and regard mostly the emergence of global dynamical patterns qualitatively different from individual behavior, and the development of system-level order from local interactions. It is therefore tempting to describe collective behavior in biology within the conceptual framework of statistical physics, in the hope to extend to this new fascinating field at least part of the great predictive power of theoretical physics.
Since the turn of the 20-th century Brownian noise has continuously disclosed a rich variety of phenomena in and around physics. The understanding of this jittering motion of suspended microscopic particles has undoubtedly helped to reinforce and substantiate those pillars on which the basic modern physical theories are resting: Its formal description provided the key to great achievements in statistical mechanics, the foundations of quantum mechanics and also astrophysical phenomena, to name but a few. Recent progress of Brownian motion theory involves (i) the description of relativistic Brownian motion and its impact for relativistic thermodynamics, or (ii) its role for fluctuation theorems and symmetry relations that constitute the pivot of those recent developments for nonequilibrium thermodynamics beyond the linear response.
Although noise commonly is hold as the enemy of order, it in fact also can be of constructive influence. The phenomena of Stochastic Resonance and Brownian motors present two such archetypes wherein random Brownian dynamics together with unbiased nonequilibrium forces beneficially cooperate in enhancing detection and/or in facilitating directed transmission of information. The applications range from information processing devices in physics, chemistry, and physical biology to new hardware for medical rehabilitation. Particularly, additional non-equilibrium disturbances enable the rectification of haphazard Brownian noise so that quantum and classical objects can be directed along on a priori designed routes (i.e. Brownian motors). We conclude with an outlook of future prospects, trends and unsolved issues.