Rare events in first-passage and exit-time statistics of jump processes can play a decisive role in triggering anomalous reactions and extreme responses in a wide range of systems. This is particularly relevant when jump lengths or waiting times follow broad, heavy-tailed distributions, for which rare events are not exponentially suppressed and can significantly affect macroscopic observables....
In recent times, several rigorous results on simple random exchange models were proved. In this talk, after a short review of the literature, I focus on a subset of results in a recent paper in which we discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under...
Deviations from the standard laws of Brownian motion, the linear time dependence
of the mean squared displacement and the Gaussian probability density function,
are quite commonly observed in an adundance of systems. The physical mechanisms
for these anomalies are non-universal, prompting the need for different stochastic
models along with their identification from measured time series of...
By analyzing physically meaningful solutions of a generalized one-dimensional Fokker-Planck equation, I show that a broad class of random motions can be described within a unified framework. The approach recovers a variety of stochastic processes, ranging from Brownian diffusion to active motion, and including anomalous transport, stochastic resetting, and trapping mechanisms.
Noise-driven dynamics in single-well potentials can produce bimodal, or more generally multimodal, stationary states. The emergence of multimodality depends sensitively on the properties of the driving noise. Such behavior is typically associated with systems subject to strong fluctuations, for example those driven by Lévy noise, or with systems driven by temporally correlated noise, such as...
Continuous-time random walks (CTRWs) with stochastic resetting provide a natural framework for modelling anomalous transport under intermittent restart mechanisms. In this seminar, we present recent asymptotic results for CTRWs with resetting under suitable space–time scaling limits. For heavy-tailed waiting times and jumps in the domain of attraction of stable laws, we show convergence to an...
Many real-world infrastructures, from sensor and road networks to power grids, are spatially embedded and anisotropic, with constraints on the maximum number of links each node can establish. Such systems can be represented as anisotropic limited-degree networks, in which each node forms at most $q$ outgoing links preferentially oriented along a fixed direction. By increasing the node density...
Low-rank matrix inference is a central problem in high-dimensional statistics, machine learning, and statistical physics. In the classical spiked random matrix setting, a rank-one signal is corrupted by dense random noise, and the celebrated BBP transition marks the point at which the signal becomes detectable by principal component analysis. In many applications, however, the corrupting noise...
First, we consider the so-called preferential attachment random graphs, which appear extensively in the mathematics, physics, and computer science literature. We then present some variants in which the attachment mechanism is not of pure preferential type, or in which the initial degrees with which the nodes appears in the graph are random. In particular, regarding the latter case, we address...
I will consider the thermodynamic properties of an information engine that uses
feedback control to extract work from a manipulated stochastic system.
I will discuss the fluctuation theorems that involve the information associated with the feedback-controlled stochastic trajectories. Such an information turns out to be based on the first-passage-time distribution.
I will then discuss the...
The butterfly effect, introduced about half century ago by E. Lorenz, is now part of the
folklore of chaos and of the pop culture.
We show how for the understanding of a realistic scenario in fully developed turbulence,
one must to go beyond the pure mathematical study of infinitesimal perturbations of the Lyapunov exponent.
In particular it is necessary to take into account:
a)...
The study of thermalisation in isolated systems was pioneered by Fermi-Pasta-Ulam-Tsingou (FPUT). In this talk I will present a new avenue towards the understanding of thermalisation, the presence of a power-law in the Fourier energy spectrum. A universal scaling exponent is obtained by mapping the FPUT model onto a pair of Burgers equations. Energy is transferred to higher Fourier modes like...
We formulate coarse-grained dynamics and irreversibility from an observer-centered perspective within nonequilibrium statistical mechanics. Here, an observer is specified by a resolved subspace of observables, which determines the associated relevant ensemble, unresolved degrees of freedom, noise, and memory.
We analyze how different choices of observer modify the reduced dynamics and compare...
Fluctuation-responce relations (FRRs), or fluctuation-dissipation theorems (FDTs), are an important tool for understanding and predicting the response of physical systems to external perturbations. They come in different kinds and flavors. Thus, in equilibrium, the linear FDT of the first kind connects the linear response of a system to an external perturbation with properties of spontaneous...
This presentation generalizes and expands the theory for a particular branching process (see Kimmel and Axelrod (2015) Branching processes in biology) used to model cancerous tumors. The process was first introduced by us in Ernst et al. (2018) Adv Appl Prob 50A: 99-114. The motivation were data from Perez-Garcia et al. Nat Phys (2020) 16: 1232 who found super-exponential growth examples among...
It is well established that the brain spontaneously fluctuates through a very large number of states. Nevertheless, despite its relevance to understanding brain function, still is often ignored that its origin formally correspond to critical phenomena. We discuss the most recent results at large and small scale that are consistent with the view that such ubiquitous fluctuations are critical.
The brain is a complex system whose multiscale organization supports cognitive functioning. In neurodegeneration in particular, alterations in brain structure are accompanied by a broad cognitive decline, motivating the question of whether changes to that in multiscale brain organization track these impairments. Multifractal measures provide a promising description of brain organization that...
Sticky diffusion processes on bounded domains can spend finite time (and finite mean time) on the lower-dimensional space given by the boundary. Once the process hits the boundary, then it starts again after a random amount of time. While on the boundary it can stay or move according to dynamics that are different from those in the interior. Such processes may be characterized by a...
Superdiffusion is a process in which diffusing molecules can make anomalously long jumps with relatively high probability. This process occurs, among others, in turbulent media and various biological processes related to cell migration. Superdiffusion is often described by the equation with the fractional Riesz derivative with respect to a spatial variable. This derivative is a...
In this talk we give an explicit solution to Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\nu_k$, for $k$ non-negative integer and $\nu > 0$. The solution is obtained by connecting the differential equation with the roots of the characteristic polynomial and it is expressed in terms of Mittag-Leffler-type functions. Under some additional...
Biological neural networks solve cognitive tasks with varying levels of complexity. However, it remains unclear how specific structural and functional features of these networks are related to increasing difficulty of the problems to be solved. How do network motifs change when the cognitive cost is increased? We address this question by evolving Artificial Neural Networks (ANNs) under the...
We study two variants of the Blume-Emery-Grifith model with long range mean-field-like interaction and random disorder, described by different Hamiltonians. In one variant the system is uniformly populated by $N$ spins and the disorder is represented by $\textit{i.i.d}$ random variables chosen with probability $p$. In the other one, we consider $N_s\sim Bin(N,p)$ $\textit{strong}$ spins, i.e.,...
Vertex models have been widely used to study mechanical phase transitions in confluent tissues. In the classical formulation, the normalized cell perimeter, or shape index, serves as a key geometric control parameter: increasing the shape index drives a transition from solid-like to fluid-like tissue behavior as cells become more elongated. Here, we extend the passive energy of the vertex...
We investigate the escape properties of a Feller diffusion process confined to a finite interval within an exactly solvable framework. The dynamics correspond to a particle in a shifted harmonic potential with state-dependent diffusivity, which lifts the symmetry of the escape events. By assigning the boundaries as extinction (near-zero, fluctuation-suppressed) and outbreak (enhanced-noise)...
Turing patterns are an example of thermodynamical systems out of equilibrium, exhibiting symmetry breaking and self-organization. Turing in 1952 showed theoretically that in reaction-diffusion under certain conditions a spatially homogeneous stable system can be destabilized by diffusion. Then, a large-scale spatially periodic static pattern emerges. In 1990, Cassettes et al., made the first...
We investigate a Brownian information engine driven by space-dependent diffusivity. The working protocol is described by overdamped Langevin dynamics within a harmonic confinement and is controlled via resetting. The geometry of the space-dependent diffusivity has a significant impact on the information processing. If the fluctuation increases, while moving away from the potential's center...
Thermodynamic uncertainty relations (TURs) bound the precision of thermodynamic currents in autonomous nonequilibrium steady states and constrain the trade-off between power, efficiency, and constancy in heat engines. We study a minimal autonomous heat engine composed of a discrete ratchet that performs work against a constant bias and an underdamped harmonic oscillator acting as an internal...
Odd systems, characterised by broken time-reversal or parity symmetry, exhibit striking transport phenomena due to transverse responses. In this talk, I will introduce the concept of odd diffusion, a generalisation of diffusion in two-dimensional systems that incorporates antisymmetric tensor components. Focusing on systems of interacting particles, I present analytical results on effective...
Linear regression is one of the simplest and most widely used tools to learn patterns from data: it fits a set of coefficients so that a linear combination of predictors best matches observed responses. The quality of the fit is measured by the residual sum of squares, the total squared mismatch between predictions and data, whose minimum defines the training loss. For Gaussian data, the...
The noisy voter model is a widely used framework for stochastic opinion dynamics in finite populations, where changes in individual opinions are driven by two competing mechanisms: spontaneous opinion changes and social imitation. Although the model is originally formulated in terms of discrete states and transition rates, its continuum approximation can be related to Fokker-Planck-type...
We consider a discrete-time random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process (we consider both light- and fat-tailed inter-reset distributions). Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes....
Liquid crystals (LCs) are a mesophase that combines the properties of liquids and solids. They are commonly found in nature and are a basis of liquid-crystal display (LCD) technology. The two most recent generations LC observed in the $21^{st}$ century, including phases such as twist-bend ($N_{TB}$), splay ($N_S$), and ferroelectric ($N_F$) nematics, are the subject of ongoing scientific...
Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing insights into causality, structure, and hidden patterns within complex systems. In physical systems with many degrees of freedom, such as gases, liquids, and...
DNA methylation is an epigenetic modification that adds a methyl group to cytosine within CpG dinucleotides. CpG-rich regions often occur in gene promoters and regulatory elements. Methylation at these sites is a relatively stable, heritable mark that modulates transcription, chromatin state and genome stability [1]. Dysregulation is linked to several diseases including cancer [2].
Each CpG...
