Session C18: Kadanoff Prize symposium

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I describe the use of minimal renormalization-group inspired/justified models in non-equilibrium statistical physics and biology. Using examples chosen from my own work in pattern formation, turbulence and biology, I point out that minimal models are able to make specific parameter-free predictions of appropriately-chosen experimental quantities, an outcome of universality in somewhat surprising settings. I argue that these considerations account for the unreasonable effectiveness of theoretical physics in describing seemingly complex phenomena.   

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Topological phases were discovered in condensed matter physics and recently extended to classical physics such as topological mechanical metamaterials. Their study and realization in soft-matter and biological systems has only started to develop. In this talk we discuss how topological phases may determine the behavior of nonlinear dynamical systems that arise, for example, in population dynamics. We have shown that topological phases can be realized with the anti-symmetric Lotka-Volterra equation (ALVE). The ALVE is a paradigmatic model system in population dynamics and governs, for example, the evolutionary dynamics of zero-sum games, such as the rock-paper-scissors game [1], but also describes the condensation of non-interacting bosons in driven-dissipative set-ups [2]. We have shown that for the ALVE, defined on a one-dimensional chain of rock-paper-scissors cycles, robust polarization emerges at the chain’s edge [3]. The system undergoes a transition from left to right polarization as the control parameter passes through a critical value. At the critical point, solitary waves are observed. We found that the polarization states are topological phases and that this transition is indeed a topological phase transition. Remarkably, this phase transition falls into symmetry class D within the “ten-fold way” classification scheme of gapped free-fermion systems, which also applies, for example, to one-dimensional topological superconductors. Beyond the observation of topological phases in the ALVE, it might be possible to generalize the approach of our work to other dynamical systems in biological physics whose attractors are nonlinear oscillators or limit cycles.

[1] J. Knebel, T. Krüger, M. F. Weber, and E. Frey, Phys. Rev. Lett. 110, 168106 (2013).
[2] J. Knebel, M. F. Weber, T. Krüger, and E. Frey, Nature Communications 6, 6977 (2015).
[3] J. Knebel, P. M. Geiger, and E. Frey, Phys. Rev. Lett. (in press) [arXiv:2009.01780].   

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When does a diffusing particle first reach a specified location? The probability that such a particle first reaches a given location is known as the first-passage probability. First-passage ideas underlie a wide range of phenomena, such as the triggering of integrate-and-fire neurons in the brain, when a fluctuating voltage level in a neuron reaches a triggering threshold, to the execution of stock options, when the price of a stock first reaches a strike price. In this talk, I discuss some basic first-passage ideas and present a few idealized models with possible applications to mechanical systems and to finance.   

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Extreme value statistics (EVS) concerns the study of the statistics of the
maximum or the minimum of a set of random variables. This is an important
problem for any time-series and has applications in climate, finance,
sports, all the way to physics of disordered systems where one is
typically interested in the statistics of the ground state energy. While the EVS
of `uncorrelated' variables is well understood, not much is known for strongly
correlated random variables. Only recently this subject has gained much
importance both in statistical physics and in probability theory. In this
talk, I will give an overview of the developments in physics and mathematics.
In particular, I will discuss some recent developements on the EVS of
strongly correlated systems and discuss several examples.