I. Coupled map lattices and the Standard Model

In an early publication with C.P. Dettmann Chaos Soliton. Fract. 2005 I employed well-established computational techniques (periodic-orbit expansions) to estimate a number of parameters of the standard model of particle physics predicted in Christian Beck's theory of chaotic strings, that utilizes coupled map lattices as dynamical models. The merit of that early work resides in achieving the desired estimates with an unprecedented precision of at least fourteen significant figures, essential for a high-energy theory to be testable.

II. Optimal resolution of a chaotic phase space

In my Ph.D. work with P. Cvitanović (Phys. Rev. Lett. 2010 and AIP Conf. Proc. 2012), similar techniques, based on periodic orbit theory, were developed to study the interplay of deterministic chaos with background noise. Since fractal sets, typical of a chaotic phase space, are washed out by noise, a finite resolution can be attained, the evolution operator of phase-space densities acquires a finite dimensionality, and it can be written as a tranfer matrix or a Markov graph. Initially tested on one-dimensional maps (surfaces of sections of a set of dynamical equations), more recent publications with J. M. Heninger (Phys. Rev. E 2015 and Commun. Nonlin. Sci. Numer. Simul. 2018, NSF China Grant No. 11450110057) attack the problem in higher dimensions, and address the question of the effectiveness of a perturbative approach to chaos. The primary contribution as well as the power of this method resides in the dimensionality reduction of a problem that otherwise counts infinitely many degrees of freedom. Secondly, a peculiar aspect of stochastic nonlinear dynamics is revealed, that is that noise is never effectively white, since its interplay with dynamics depends on both the past and the future noise integrated and nonlinearly convolved with deterministic evolution along every trajectory.
The project is still ongoing, and future goals include the application of our optimal-partition algorithm to Navier-Stokes' equation and other high-dimensional systems relevant to fluid-dynamics, weather science, and quantum mechanics (Entropy 2023), address symmetry reductions of the noisy state space, whose implications may go well beyond chaotic dynamics, and study the validity and generalizations of linear response theory for the noisy transport (`Fokker-Planck') operator.

III. Scar theory in quantum and classical chaos

The affinity of the stochastic dynamics formalism with quantum mechanics made me envision the possibility to extend the optimal resolution hypothesis of my PhD work to open quantum chaotic systems. For that reason, I have dedicated most of my early postodoctoral work to quantum chaos, and in particular provided a mechanistic explanation to `scars' in open systems, that is important deviations of the statistics of the eigenspectra from Random Matrix Theory (RMT) (Phys. Rev. E 2012 and 2015).
More recently, I have investigated the local density of states of an open quantum chaotic system with scarring, and derived theoretical predictions based on a mean-field approach (EPL 2019 and J. Phys. A 2022). At the same time, I have numerically discovered classical scars in noisy chaotic dynamics in collaboration with A. Shudo (Phys. Rev. E and J. Phys. A 2021, NSF China Grant No. 11750110416), as well as signatures of unstable periodic orbits in the escape rates of noisy chaotic systems (Physica D 2024). The latter classical and quantum theories and numerical results may prove impactful respectively for the fluid dynamics community and the growing quantum many-body localization or thermalization movement.
Future directions for this line of research go along studying the effects of dissipation on the spectral density and related observables (e.g. OTOCs) in systems with more degrees of freedom, of interest to atomic and molecular physics, as well as to study the effect of measurements on the spectral statistics and the signatures of localization, with possible implications to the spread/loss of information throughout the system.

IV. The fractal Weyl law at optical frequencies

In the past decade, I engaged in a fruitful collaboration with the experimental group led by Yun-Feng Xiao at Peking University. Together, we have developed techniques to accurately count statistics of chaotic resonances in optical microcavities, with the goal of testing the celebrated but still unverified fractal Weyl law on the scaling of resonant modes with the energy (Phys. Rev. E 2016 and 2017). The endeavor has proven particularly challenging, given the highly-overlapping, dissipation-plagued, nature of emission spectra. The key to success was that to leverage the phenomenon of dynamical tunneling, and count statistics of `unreadable' chaotic resonances from the clearly legible, high-quality spectrum of regular, whispering-gallery modes.
An issue that is certainly of interest to the optics/photonics community and worth future investigation is the multifractality of partially absorbing chaotic systems, such as dielectric microcavities. In simple words, a phase space is multifractal when its dimensionality cannot be characterized by only one number (there are different definitions of fractal dimensions, when they all give the same number we have a fractal, otherwise, a multifractal). The first task to accomplish would be that to measure the fractal dimension in a partially absorbing system, to then proceed with measurement and characterization of multifractality in chaotic, partially open systems, first theoretically and then experimentally.

V. Thermodynamics of chaos out of equilibrium

Chaotic systems at statistical equilibrium have been treated with the formalism of thermodynamics, and, in particular, using Gibbs' ensembles to estimate asymptotic observables. In this project, I intend to generalize some of the key definitions and relations of the Gibbs formalism to chaotic systems out of equilibrium, in the process of relaxation. Most importantly, the thermodynamic treatment will be used to relate the finite-time distributions of any integrated observable (e.g. Lyapunov exponent, diffusion coefficient, average energy) with the subleading eigenfunctions of the transfer (`Koopman') operator. I have already achieved meaningful results in these directions (Phys. Rev. E 2024 and Physica D 2024).

VI. Crackling noise in self-assembling synaptic receptors

The postsynaptic density is an essential part of the synapse. It is made of various types of proteins, and it is crucial for signal transmission. As of today, the mechanism of organization of these proteins in the so called GABA-A receptors is still obscure. The objective of this ongoing project is threefold. First, to interpret recent in-vitro experiments aimed at discovering the structure of postsynaptic receptors, as well as their mutual interaction and self-assembling. Secondly, to perform numerical simulations that reproduce the clustering process of the receptors, as observed experimentally. In particular, the self-organization of the molecules in networks is governed by power laws that would naturally lead to hypothesize a critical phenomenon. Yet, the size-controlled nature of the self-assembly mechanism observed in both experiments and preliminary numerics open the possibility of surmising a crackling behavior, that involves a broad range of (cluster) sizes. Third and last objective of this proposal is that to employ standard renormalization- group techniques to parse both the experimental and the numerical results.

updated: Feb 9 2025