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The mixed p-spin model : selecting, following and losing states

Abstract : The main driving notion behind my thesis research is to explore the connection between the dynamics and the static in a prototypical model of glass transition, i.e. the mean-field p-spin spherical model. This model was introduced more than 30 years ago with the purpose of offering a simplified model that had the same equilibrium dynamical slowing down, theoretically described a few years earlier by mode-coupling theory. Over the years, the p-spin spherical model has shown to be a very meaningful and promising model, capable of describing many equilibrium and out-of-equilibrium aspects of glasses. Eventually it came to be considered as a prototypical model of glassiness. Having such a simple but rich reference model allows a coherent examination of a subject, in our case the glass behavior, which presents a very intricate phenomenology. Thus, the main purpose is not to have a quantitative prediction of the phenomena, but rather a broader view with a strong analytical basis. In this sense the p-spin model has assumed a role for disordered systems which is comparable to that of the Ising model for understanding ferromagnetism. My research is a natural path to reinforce our knowledge and comprehension of this model. In the first chapter, we provide a general introduction to supercooled liquids and their phenomenology. The introduction is brief, and the main goal is to give a general overview, mainly from the point of view of the Random First Order Transition, while considering other perspectives on the subject and attempting to provide a ‘fair' starting bibliography to whomever wants to study supercooled liquids. The last section focuses on the Potential Energy Landscape paradigm (PEL), which in my view, gives a very solid modelization of glassy phenomenology, and shares many aspects with mean-field analysis. In the second chapter, the p-spin spherical model is presented in details. The equilibrium analysis is performed with the replica formalism, with a focus on the ultrametric structure. Then, different tools to study its free energy landscape are introduced: the TAP approach, the Franz-Parisi potential and the Monasson method. These three different ways of selecting states are carefully contrasted and their analogies and differences are underlined, in particular highlighting the different behavior played by pure and mixed p-spin models. Then the equilibrium dynamics is discussed, and a selection of classical results on the dynamical slowing down are analyzed by numerical integration. To conclude, the out-of-equilibrium dynamics in the two temperature protocol is analyzed. This shows two different regimes, the state following and the aging. For both, an asymptotic analysis and a numerical integration are performed and compared. A strong emphasis is given to the possibility of describing the asymptotic dynamics with a static potential. The third chapter presents all the new results that emerged during my research. The study focuses on the two temperature protocol, starting in equilibrium and setting the second temperature to zero, which corresponds to a gradient descent dynamics. This protocol is especially interesting because it corresponds to the search of inherent structure of the energy landscape. The integrated dynamics, depending on the starting temperature, shows three different regimes, one that corresponds to a new phase, which shows aging together with memory of the initial condition. This new phase is not present in pure p-spin models, only in mixed ones. In order to theoretically describe this new phase, a constrained analysis of the stationary points of the energy landscape is performed. A numerical simulation of the system is also presented to confirm this new scenario.
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Giampaolo Folena. The mixed p-spin model : selecting, following and losing states. Disordered Systems and Neural Networks [cond-mat.dis-nn]. Université Paris-Saclay; Università degli studi La Sapienza (Rome), 2020. English. ⟨NNT : 2020UPASS060⟩. ⟨tel-02883385⟩

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