Antonio Politi

Thermal conduction in classical low-dimensional lattices

Abstract Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to simple mathematical models consisting of coupled oscillators on a lattice. The role of lattice dimensionality on the breakdown of the Fouriers law is discussed and some universal quantitative aspects are emphasized: the divergence of the finite-size thermal conductivity is characterized by universal laws in one and two dimensions. Equilibrium and non-equilibrium molecular dynamics methods are presented along with a critical survey of previous numerical results. Analytical results for the non-equilibrium dynamics can be obtained in the harmonic chain where the role of disorder and localization can be also understood. The traditional kinetic approach, based on the Boltzmann–Peierls equation is also briefly sketched with reference to one-dimensional chains. Simple toy models can be defined in which the conductivity is finite. Anomalous transport in integrable non-linear systems is briefly discussed. Finally, possible future research themes are outlined.

Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.

Statistical description of chaotic attractors: The dimension function

A method for the investigation of fractal attractors is developed, based on statistical properties of the distributionP(δ, n) of nearest-neighbor distancesδ between points on the attractor. A continuous infinity of dimensions, called dimension function, is defined through the moments ofP(δ, n). In particular, for the case of self-similar sets, we prove that the dimension function DF yields, in suitable points, capacity, information dimension, and all other Renyi dimensions. An algorithm to compute DF is derived and applied to several attractors. As a consequence, an estimate of nonuniformity in dynamical systems can be performed, either by direct calculation of the uniformity factor, or by comparison among various dimensions. Finally, an analytical study of the distributionP(δ, n) is carried out in some simple, meaningful examples.

Characterizing dynamics with covariant lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.

Ki-67 acts as a biological surfactant to disperse mitotic chromosomes.

Eukaryotic genomes are partitioned into chromosomes that form compact and spatially well-separated mechanical bodies during mitosis. This enables chromosomes to move independently of each other for segregation of precisely one copy of the genome to each of the nascent daughter cells. Despite insights into the spatial organization of mitotic chromosomes and the discovery of proteins at the chromosome surface, the molecular and biophysical bases of mitotic chromosome structural individuality have remained unclear. Here we report that the proliferation marker protein Ki-67 (encoded by the MKI67 gene), a component of the mitotic chromosome periphery, prevents chromosomes from collapsing into a single chromatin mass after nuclear envelope disassembly, thus enabling independent chromosome motility and efficient interactions with the mitotic spindle. The chromosome separation function of human Ki-67 is not confined within a specific protein domain, but correlates with size and net charge of truncation mutants that apparently lack secondary structure. This suggests that Ki-67 forms a steric and electrostatic charge barrier, similar to surface-active agents (surfactants) that disperse particles or phase-separated liquid droplets in solvents. Fluorescence correlation spectroscopy showed a high surface density of Ki-67 and dual-colour labelling of both protein termini revealed an extended molecular conformation, indicating brush-like arrangements that are characteristic of polymeric surfactants. Our study thus elucidates a biomechanical role of the mitotic chromosome periphery in mammalian cells and suggests that natural proteins can function as surfactants in intracellular compartmentalization.

On the anomalous thermal conductivity of one-dimensional lattices

The divergence of the thermal conductivity in the thermodynamic limit is thoroughly investigated. The divergence law is consistently determined with two different numerical approaches based on equilibrium and nonequilibrium simulations. A possible explanation in the framework of linear-response theory is also presented, which traces back the physical origin of this anomaly to the slow diffusion of the energy of long-wavelength Fourier modes. Finally, the results of dynamical simulations are compared with the predictions of mode-coupling theory.

High-dimensional chaos in delayed dynamical systems

Abstract We introduce a general class of iterative delay maps to model high-dimensional chaos in dynamical systems with delayed feedback. The class includes as particular cases systems with a linear local dynamics. We report analytic and numerical results on the scaling laws of Lyapunov spectra with a number of degrees of freedom. Invariant measure is computed through a self-consistent Frobenius-Perron formalism, which allows also a recalculation of the maximum Lyapunov exponent in good agreement with the one measured directly.

Collective chaos in pulse-coupled neural networks

We study the dynamics of two symmetrically coupled populations of identical leaky integrate-and-fire neurons characterized by an excitatory coupling. Upon varying the coupling strength, we find symmetry-breaking transitions that lead to the onset of various chimera states as well as to a new regime, where the two populations are characterized by a different degree of synchronization. Symmetric collective states of increasing dynamical complexity are also observed. The computation of the the finite-amplitude Lyapunov exponent allows us to establish the chaoticity of the (collective) dynamics in a finite region of the phase plane. The further numerical study of the standard Lyapunov spectrum reveals the presence of several positive exponents, indicating that the microscopic dynamics is high-dimensional.

From anomalous energy diffusion to levy walks and heat conductivity in one-dimensional systems

The evolution of infinitesimal, localized perturbations is investigated in a one-dimensional diatomic gas of hard-point particles (HPG) and thereby connected to energy diffusion. As a result, a Levy walk description, which was so far invoked to explain anomalous heat conductivity in the context of noninteracting particles is here shown to extend to the general case of truly many-body systems. Our approach does not only provide firm evidence that energy diffusion is anomalous in the HPG, but proves definitely superior to direct methods for estimating the divergence rate of heat conductivity which turns out to be

Unpredictable Behaviour in Stable Systems

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