### Attention : désormais les séminaires ont lieu tous les lundis à 10h45 en salle 523 du LPTMC - Tour 12-13

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**Gleb Oshanin (LPTMC, Paris)**

### Spectral density of an individual trajectory of a Gaussian stochastic process

In this talk I will focus on the behavior of a particular random (non-local) functional - the spectral density S(f,T) (with f being the frequency and T - the observation time) of an individual random trajectory of a stochastic Gaussian process. I will first recall the textbook definition based on the covariance function of the process, and show on several examples how diverse its functional form can be depending on the spread of a process. Then, I will specify the limitations of the standard definition and will go beyond it by considering the “noise-to-signal” ratio - the ratio of the standard deviation of S(f,T) and its mean value. Next, I will prove a simple but crucial double-sided inequality obeyed by the noise-to-signal ratio for any Gaussian process, any f and any T, and eventually will derive the full probability density function of S(f,T) in most general conditions. Lastly, for several Gaussian processes I will discuss the behavior of the frequency-frequency correlations of such random variables and will demonstrate that they may be used as a robust property permitting to distinguish between normal and anomalous diffusion.

A Squarcini, E Marinari, G Oshanin, L Peliti, and L Rondoni, Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes, Journal of Physics A: Mathematical and Theoretical 55, 405001 (2022).

A Squarcini, E Marinari, G Oshanin, L Peliti, and L Rondoni, Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes, New Journal of Physics 24, 093031 (2022).

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**Éric Bertin (LIPhy, Univ. Grenoble)**

### Large-scale descriptions of densely packed soft particles under drive

A broad class of soft matter systems, ranging from sheared emulsions or microgels to cell monolayers, may be considered as densely packed soft particles under drive, the latter being either a boundary drive or active forces at particle scale. In this talk, I'll first show how a macroscopic description in terms of stress tensor can be derived from particle dynamics for jammed soft disks under shear. Then turning to dense monolayers of self-propelled soft particles, I'll discuss how a persistent particle motility couples to the collective elastic modes to produce large-scale swirl-like motions.

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**Pavel Krapivsky (Department of Physics, Boston University)**

### Blast and splash in a cold gas

We study the response of a cold gas (all particles are initially at rest) to a sudden kick when one particle suddenly starts moving. The outcome is a spherical shock wave advancing as \(t^\frac{2}{d+2}\). The density, velocity, and temperature behind the shock are described by Euler equations. Deviations from the predictions of non-dissipative hydrodynamics arise in the central region that grows as \(t^\frac{38}{93}, ~t^\frac{2}{5}, ~t^\frac{62}{175}\) when \(d=1,2,3\). In a one-dimensional semi-infinite setting, when the left-most particle suddenly starts moving to the right, a growing number of ``splatter" particles penetrate the initially empty half-line. The total energy and momentum of the splatter particles exhibit counterintuitive behaviors.

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**Bertrand Lacroix-A-Chez-Toine (King’s College London)**

### Superposition of random plane waves in high dimension as a random landscape

Superpositions of random plane waves play important roles in the semi-classical description of quantum billiard as pointed out by Berry’s conjecture [1]. In this context, they can describe the eigenfunctions of the Laplacian operator at high energy. While their properties have been explored in depth in low spatial dimensions, we consider in this talk a large superposition of M ≫ 1 random plane waves in high dimension N ≫ 1 with M/N = α > 1. Here, we consider instead this object as a (random) energy landscape and, adding an isotropic harmonic confinement of strength μ, we characterise the ergodicity breaking in such a landscape. To characterise this property we consider two quantities: the quenched free energy and the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points of the energy landscape with the spatial dimension N. While similar high-dimensional random landscapes display topology trivialisation transition [2], whereby the complexity vanishes above some finite value of the confinement μ, the complexity vanishes only as μ → ∞ in this system [3]. One might thus expect that ergodicity is always broken at zero temperature in this model. This is confirmed and enriched by our quenched free-energy computations.

References

[1] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 2083 (1977).

[2] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[3] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, J. Math. Phys. 63 (9), 093301 (2022)

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**Jim Sethna (Cornell University)**

### Using Universal Scaling Functions

Half a century ago, Ken Wilson and Leo Kadanoff introduced the renormalization-group framework for understanding systems with emergent, fractal scale invariance. For five decades, statistical physicists have applied these techniques to equilibrium phase transitions, avalanche models, glasses and disordered systems, the onset of chaos, plastic flow in crystals, surface morphologies, fracture, … But these tools have not made a substantial impact in engineering or biology.

Even now we do not have a clear understanding of the singularity at the critical point for even the traditional equilibrium phase transitions – we know the critical exponents to high precision, but are missing complete understanding of the universal scaling functions. Also, we have not developed the tools to extend our predictions of the asymptotic behavior to a systematic approximation of the phase diagrams – analytic corrections to scaling. Sethna will begin with an introduction to how normal form theory from dynamical systems can determine the arguments of scaling functions where logarithms and exponentials intrude on the traditional power laws – the 4D Ising model, the 2D random-field Ising model, and jamming and the onset of amorphous rigidity in 2D. He will then use the 2D Ising model to show how adding analytic corrections to the critical point singularity can lead to high-precision descriptions of the entire phase diagram, and present Jaron Kent-Dobias’s derivation of a universal scaling function for the free energy in an external field, accurate to seven digits.