Thesis Title: Characterisation of out-of-equilibrium phase transitions with the emergence of limit cycles.
Systems maintained out of equilibrium by a drive often exhibit a rich phenomenology, with different types of qualitative changes in behavior, corresponding to spatial or temporal structuring. Examples include systems of active particles which consume energy from their environment to move, and which may exhibit a coherent overall motion thanks to alignment interactions, like bird flocks or schools of fish. Another example of a collective phenomenon is the synchronization transition, whereby oscillators of different frequencies begin to oscillate in phase if their interactions are sufficiently strong. The description of such systems calls on the methods of non-equilibrium statistical physics.The aim of this thesis is to develop theoretical tools for studying the onset of spontaneous oscillations within the conceptual framework of non-equilibrium phase transitions. We are particularly interested in the case when microscopic degrees of freedom are not themselves oscillators, as in the study of the synchronization transition, but can nevertheless lead to the presence of collective oscillations under the influence of interactions, when the system is driven far enough from equilibrium. The study focused on models defined in terms of binary variables (called “spins”), the idea being to consider a large number of spins interacting with each other. The structure of the interaction network then plays a key role: do all spins interact with each other (a case generally referred to as “mean field” in physics), or do interactions take place between nearest neighbors on a regular network, for example? And do spins interact with each other in the same way or not? Depending on the case, one may speak of homogeneous or heterogeneous interactions, the latter representing a form of structural disorder.
With well-chosen dynamic rules, modeling a system far from equilibrium with non-reciprocal interactions, collective oscillations may appear in these models, as revealed for instance by the oscillations of the magnetization over time. This thesis has enabled us to characterize the statistical fluctuations present in these oscillating states in the “mean field” geometry. In the case of nearest-neighbor interactions, the thesis highlighted the link between the dynamics of local oscillations and their possible global synchronization. Finally, it was shown that the presence of disordered or heterogeneous interactions may give rise to “hidden” oscillations, which can nonetheless be revealed thanks to the entropy production rate, which quantifies in fine detail the non-equilibrium character of the system's dynamic state.
Key words: Statistical physics, Dynamical systems, Oscillations, Non-equilibrium phase transition, Limit cycle, Disordered systems
Doctoral School: ED PHYS - Physics
Research laboratory: Laboratoire Interdisciplinaire de Physique (LiPhy - CNRS/UGA)
Thesis supervision: Eric BERTIN
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