From Geometry of Hamiltonian Dynamics to Topology of Phase Transitions: A Review.

In this review work, we outline a conceptual path that, starting from the numerical investigation of the transition between weak chaos and strong chaos in Hamiltonian systems with many degrees of freedom, comes to highlight how, at the basis of equilibrium phase transitions, there must be major changes in the topology of submanifolds of the phase space of Hamiltonian systems that describe systems that exhibit phase transitions. In fact, the numerical investigation of Hamiltonian flows of a large number of degrees of freedom that undergo a thermodynamic phase transition has revealed peculiar dynamical signatures detected through the energy dependence of the largest Lyapunov exponent, that is, of the degree of chaoticity of the dynamics at the phase transition point. The geometrization of Hamiltonian flows in terms of geodesic flows on suitably defined Riemannian manifolds, used to explain the origin of deterministic chaos, combined with the investigation of the dynamical counterpart of phase transitions unveils peculiar geometrical changes of the mechanical manifolds in correspondence to the peculiar dynamical changes at the phase transition point.

Topological origin of the protein folding transition.

In this paper, a geometrical and thermodynamical analysis of the global properties of the potential energy landscape of a minimalistic model of a polypeptide is presented. The global geometry of the potential energy landscape is supposed to contain relevant information about the properties of a given sequence of amino acids, that is, to discriminate between a random heteropolymer and a protein. By considering the SH3 and PYP protein-sequences and their randomized versions it turns out that, in addition to the standard signatures of the folding transition-discriminating between protein sequences of amino acids and random heteropolymer sequences-also peculiar geometric signatures of the equipotential hypersurfaces in configuration space can discriminate between proteins and random heteropolymers.

Experimental evidence for long-distance electrodynamic intermolecular forces.

Both classical and quantum electrodynamics predict the existence of dipole-dipole long-range electrodynamic intermolecular forces; however, these have never been hitherto experimentally observed. The discovery of completely new and unanticipated forces acting between biomolecules could have considerable impact on our understanding of the dynamics and functioning of the molecular machines at work in living organisms. Here, using two independent experiments, on the basis of different physical effects detected by fluorescence correlation spectroscopy and terahertz spectroscopy, respectively, we demonstrate experimentally the activation of resonant electrodynamic intermolecular forces.

Agent-based models for detecting the driving forces of biomolecular interactions.

Agent-based modelling and simulation have been effectively applied to the study of complex biological systems, especially when composed of many interacting entities. Representing biomolecules as autonomous agents allows this approach to bring out the global behaviour of biochemical processes as resulting from local molecular interactions. In this paper, we leverage the capabilities of the agent paradigm to construct an in silico replica of the glycolytic pathway; the aim is to detect the role that long-range electrodynamic forces might have on the rate of glucose oxidation.

Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions.

Different arguments led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide open question.

Transition between Random and Periodic Electron Currents on a DNA Chain.

By resorting to a model inspired to the standard Davydov and Holstein-Fröhlich models, in the present paper we study the motion of an electron along a chain of heavy particles modeling a sequence of nucleotides proper to a DNA fragment. Starting with a model Hamiltonian written in second quantization, we use the Time Dependent Variational Principle to work out the dynamical equations of the system. It can be found that, under the action of an external source of energy transferred to the electron, and according to the excitation site, the electron current can display either a broad frequency spectrum or a sharply peaked frequency spectrum.

Energy transfer to the phonons of a macromolecule through light pumping.

In the present paper we address the problem of the energy downconversion of the light absorbed by a protein into its internal vibrational modes. We consider the case in which the light receptors are fluorophores either naturally co-expressed with the protein or artificially covalently bound to some of its amino acids. In a recent work [Phys.

Publisher Correction: Collective behavior of oscillating electric dipoles.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric.

By identifying Hamiltonian flows with geodesic flows of suitably chosen Riemannian manifolds, it is possible to explain the origin of chaos in classical Newtonian dynamics and to quantify its strength. There are several possibilities to geometrize Newtonian dynamics under the action of conservative potentials and the hitherto investigated ones provide consistent results. However, it has been recently argued that endowing configuration space with the Jacobi metric is inappropriate to consistently describe the stability/instability properties of Newtonian dynamics because of the nonaffine parametrization of the arc-length with physical time.

Collective behavior of oscillating electric dipoles.

We investigate the dynamics of a population of identical biomolecules mimicked as electric dipoles with random orientations and positions in space and oscillating with their intrinsic frequencies. The biomolecules, beyond being coupled among themselves via the dipolar interaction, are also driven by a common external energy supply. A collective mode emerges by decreasing the average distance among the molecules as testified by the emergence of a clear peak in the power spectrum of the total dipole moment.

Riemannian-geometric entropy for measuring network complexity.

A central issue in the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate with a-in principle, any-network a differentiable object (a Riemannian manifold) whose volume is used to define the entropy.

Persistent homology analysis of phase transitions.

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called mean-field XY model and by the ϕ^{4} lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.

Random walk of passive tracers among randomly moving obstacles.

Background: This study is mainly motivated by the need of understanding how the diffusion behavior of a biomolecule (or even of a larger object) is affected by other moving macromolecules, organelles, and so on, inside a living cell, whence the possibility of understanding whether or not a randomly walking biomolecule is also subject to a long-range force field driving it to its target.

Method: By means of the Continuous Time Random Walk (CTRW) technique the topic of random walk in random environment is here considered in the case of a passively diffusing particle among randomly moving and interacting obstacles.

Results: The relevant physical quantity which is worked out is the diffusion coefficient of the passive tracer which is computed as a function of the average inter-obstacles distance.

Topology driven modeling: the IS metaphor.

In order to define a new method for analyzing the immune system within the realm of Big Data, we bear on the metaphor provided by an extension of Parisi's model, based on a mean field approach. The novelty is the multilinearity of the couplings in the configurational variables. This peculiarity allows us to compare the partition function [Formula: see text] with a particular functor of topological field theory-the generating function of the Betti numbers of the state manifold of the system-which contains the same global information of the system configurations and of the data set representing them.

Possible role of electrodynamic interactions in long-distance biomolecular recognition.

The issue of retarded long-range resonant interactions between two molecules with oscillating dipole moments is revisited within the framework of classical electrodynamics. By taking advantage of a theorem in complex analysis, we present a simple method to calculate the frequencies of the normal modes, which are then used to estimate the interaction potential. The possibility that such interactions play a non-negligible role in ensuring the effective functioning of the biomolecular functions is investigated.

Experimental detection of long-distance interactions between biomolecules through their diffusion behavior: numerical study.

The dynamical properties and diffusive behavior of a collection of mutually interacting particles are numerically investigated for two types of long-range interparticle interactions: Coulomb-electrostatic and dipole-electrodynamic. It is shown that when the particles are uniformly distributed throughout the accessible space, the self-diffusion coefficient is always lowered by the considered interparticle interactions, irrespective of their attractive or repulsive character. This fact is also confirmed by a simple model to compute the correction to the Brownian diffusion coefficient due to the interactions among the particles.

Experimental assessment of the contribution of electrodynamic interactions to long-distance recruitment of biomolecular partners: Theoretical basis.

Highly specific spatiotemporal interactions between cognate molecular partners essentially sustain all biochemical transactions in living matter. That such an exquisite level of accuracy may result from encountering forces solely driven by thermal diffusive processes is unlikely. Here we propose a yet unexplored strategy to experimentally tackle the long-standing question of a possibly active recruitment at a distance of cognate partners of biomolecular reactions via the action of resonant electrodynamic interactions.

Riemannian geometry of Hamiltonian chaos: hints for a general theory.

We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam beta model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down.

Hamiltonian dynamics of homopolymer chain models.

The Hamiltonian dynamics of chains of nonlinearly coupled particles is numerically investigated in two and three dimensions. Simple, off-lattice homopolymer models are used to represent the interparticle potentials. Time averages of observables numerically computed along dynamical trajectories are found to reproduce results given by the statistical mechanics of homopolymer models.

Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics.

The elsewhere surmized topological origin of phase transitions is given here important evidence through the analytic study of an exactly solvable model for which both topology of submanifolds of configuration space and thermodynamics are worked out. The model is a mean-field one with a k-body interaction. It undergoes a second-order phase transition for k=2 and a first-order one for k >2 .

Weak and strong chaos in Fermi-Pasta-Ulam models and beyond.

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased.

Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas.

It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a turbulent electric field across the confining magnetic field in controlled thermonuclear fusion devices.

Theorem on the origin of phase transitions.

For physical systems described by smooth, finite-range, and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that, unless the equipotential hypersurfaces of configuration space Sum(v)=[(q(1),...

Exact result on topology and phase transitions at any finite N.

We study analytically the topology of a family of submanifolds of the configuration space of the mean-field XY model, computing also a topological invariant (the Euler characteristic). We prove that a particular topological change of these submanifolds is connected to the phase transition of this system, and exists also at finite N. The present result is the first analytic proof that a phase transition has a topological origin and provides a key to a possible better understanding of the origin of phase transitions at their deepest level, as well as to a possible definition of phase transitions at finite N.