Department of Physics • University of Cologne

Understanding the response of a disordered solid to an externally imposed forcing or deformation is important in order to characterize the transitions between rigid and flowing states in a wide variety of soft matter systems, such as the yielding transition of an amorphous solid under shear. Such complex phenomena are already present in the "AQS" regime where thermal effects are negligible, the response to forcing is quasi-static, and the dynamics proceeds via triggering of mechanical instabilities. A key insight underlying my recent research has been the observation that the AQS conditions permit a rigorous representation of the driven dynamics in term of a directed state-transition graph. This AQS graph represents the response of the system to any deformation protocol, providing thereby a bird's-eye (or map-like) view of all the possible dynamics, which in turn is encoded in the topology of the AQS graph. At the same time, the AQS graph is a discrete random structure that lends itself to a mathematical treatment. In this talk I will describe the energy landscape of a sheared amorphous solid via AQS transition graphs, highlighting key features, such as hysteresis, and showing how various topological properties of this graph can be captured in terms of stochastic models.

The theory of large deviations provides probabilistic and analytical tools for describing the exponential decay rate of very small probabilities in terms of variational formulas. In the development of this theory, driving forces came from various applications in physics, like random path models with interactions having long or short memories, and from static interacting particle systems like the interacting Bose gas. We will explain how to apply the theory to such examples and how much it can contribute to a rigorous understanding of the underlying physics. The audience should be familiar with the most fundamental concepts of measure theory based probability and with the concept of weak convergence of probability measures.

The theory of large deviations provides probabilistic and analytical tools for describing the exponential decay rate of very small probabilities in terms of variational formulas. In the development of this theory, driving forces came from various applications in physics, like random path models with interactions having long or short memories, and from static interacting particle systems like the interacting Bose gas. We will explain how to apply the theory to such examples and how much it can contribute to a rigorous understanding of the underlying physics. The audience should be familiar with the most fundamental concepts of measure theory based probability and with the concept of weak convergence of probability measures.

The time evolution of the probability distribution of a Markov process is given by a generator in a fashion that is formally very similar to the time evolution of the (complex) probability amplitude of a quantum system in imaginary time. This correspondence allows for the import of technical tools and notions from quantum mechanics to the study of stochastic processes (and vice versa) which has proved to be useful particularly in the context of stochastic interacting particles and quantum spin systems. This mini-course begins with an introduction into the formal aspects of this relationship. In the second part we discuss as applications: (a) Symmetry and duality, (b) Large deviations and Non-hermitian quantum mechanics, and, briefly, (c) Large-scale behaviour.

The time evolution of the probability distribution of a Markov process is given by a generator in a fashion that is formally very similar to the time evolution of the (complex) probability amplitude of a quantum system in imaginary time. This correspondence allows for the import of technical tools and notions from quantum mechanics to the study of stochastic processes (and vice versa) which has proved to be useful particularly in the context of stochastic interacting particles and quantum spin systems. This mini-course begins with an introduction into the formal aspects of this relationship. In the second part we discuss as applications: (a) Symmetry and duality, (b) Large deviations and Non-hermitian quantum mechanics, and, briefly, (c) Large-scale behaviour.