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arXiv2008

RNA Secondary Structures: Complex Statics and Glassy Dynamics

Models for RNA secondary structures (the topology of folded RNA) without pseudo knots are disordered systems with a complex state-space below a critical temperature. Hence, a complex dynamical (glassy) behavior can be expected, when performing Monte Carlo simulation. Interestingly, in contrast to most other complex systems, the ground states and the density of states can be computed in polynomial time exactly using transfer matrix methods. Hence, RNA secondary structure is an ideal model to study the relation between static/thermodynamic properties and dynamics of algorithms. Also they constitute an ideal benchmark system for new Monte Carlo methods. Here we considered three different recent Monte Carlo approaches: entropic sampling using flat histograms, optimized-weights ensembles, and ParQ, which estimates the density of states from transition matrices. These methods were examined by comparing the obtained density of states with the exact results. We relate the complexity seen in the dynamics of the Monte Carlo algorithms to static properties of the phase space by studying the correlations between tunneling times, sampling errors, amount of meta-stable states and degree of ultrametricity at finite temperature.

cond-mat.dis-nn↗
arXiv2007

Anderson localization in quantum chaos: scaling and universality

The one parameter scaling theory is a powerful tool to investigate Anderson localization effects in disordered systems. In this paper we show this theory can be adapted to the context of quantum chaos provided that the classical phase space is homogeneous, not mixed. The localization problem in this case is defined in momentum, not in real space. We then employ the one parameter scaling theory to: a) propose a precise characterization of the type of classical dynamics related to the Wigner-Dyson and Poisson statistics which also predicts in what situations Anderson localization corrections invalidate the relation between classical chaos and random matrix theory encoded in the Bohigas-Schmit-Giannoni conjecture, b) to identify the universality class associated with the metal-insulator transition in quantum chaos. In low dimensions it is characterized by classical superdiffusion, in higher dimensions it has in general a quantum origin as in the case of disordered systems. We illustrate these two cases by studying 1$d$ kicked rotors with non analytical potentials and a 3$d$ kicked rotor with a smooth potential.

cond-mat.dis-nn↗
arXiv2007

Finite size effects and symmetry breaking in the evolution of networks of competing Boolean nodes

The effects of the finite size of the network on the evolutionary dynamics of a Boolean network are analyzed. In the model considered, Boolean networks evolve via a competition between nodes that punishes those in the majority. It is found that finite size networks evolve in a fundamentally different way than infinitely large networks do. The symmetry of the evolutionary dynamics of infinitely large networks that selects for canalizing Boolean functions is broken in the evolutionary dynamics of finite size networks. In finite size networks there is an additional selection for input inverting Boolean functions that output a value opposite to the majority of input values. These results are revealed through an empirical study of the model that calculates the frequency of occurrence of the different possible Boolean functions. Classes of functions are found to occur with the same frequency. Those classes depend on the symmetry of the evolutionary dynamics and correspond to orbits of the relevant symmetry group. The empirical results match analytic results, determined by utilizing Polya's theorem, for the number of orbits expected in both finite size and infinitely large networks. The reason for the symmetry breaking in the evolutionary dynamics is found to be due to the need for nodes in finite size networks to behave differently in order to cooperate so that the system collectively performs as well as possible. The results suggest that both finite size effects and symmetry are important for understanding the evolution of real-world complex networks, including genetic regulatory networks.

cond-mat.stat-mech↗
arXiv2008

Correlation Inequalities for Spin Glass in one Dimension

We prove two inequalities for the direct and truncated correlation for the nearest-neighboor one-dimensional Edwards-Anderson model with symmetric quenched disorder. The second inequality has the opposite sign of the GKS inequality of type II. In the non symmetric case with positive average we show that while the direct correlation keeps its sign the truncated one changes sign when crossing a suitable line in the parameter space. That line separates the regions satisfying the GKS second inequality and the one proved here.

math-ph↗
arXiv2007

How to measure the effective action for disordered systems

In contrast to standard critical phenomena, disordered systems need to be treated via the Functional Renormalization Group. The latter leads to a coarse grained disorder landscape, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We review recent progress on how the non-analytic effective action can be measured both in simulations and experiments, and confront theory with numerical work.

cond-mat.dis-nn↗
arXiv2010

Funnelling related phase transitions in networks

Funnelling effect, in the context of searching on networks, precisely indicates that the search takes place through a few specific nodes. We define the funnelling capacity $f$ of a node as the fraction of successful dynamic paths through it with a fixed target. We claim that there is a funnelling effect if the distribution $D(f)$ of the fraction of nodes with funnelling capacity $f$ shows a power law behaviour. We find that a transition from a search with funnelling effect to that without any such effect can exist and is driven by the parameters defining the network.

cond-mat.dis-nn↗
arXiv2008

Transient and Equilibrium Synchronization in Complex Neuronal Networks

Transient and equilibrium synchronizations in complex neuronal networks as a consequence of dynamics induced by having sources placed at specific neurons are investigated. The basic integrate-and-fire neuron is adopted, and the dynamics is estimated computationally so as to obtain the activation at each node along each instant of time. In the transient case, the dynamics is implemented so as to conserve the total activation entering the system. In our equilibrium investigations, the internally stored activation is limited to the value of the respective threshold. The synchronization of the activation of the network is then quantified in terms of its normalized entropy. The equilibrium investigations involve the application of a number of complementary characterization methods, including spectra and Principal Component Analysis, as well as of an equivalent model capable of reproducing both the transient and equilibrium dynamics. The potential of such concepts and measurements is explored with respect to several theoretical models, as well as for the neuronal network of \emph{C. elegans}. A series of interesting results are obtained and discussed, including the fact that all models led to a transient period of synchronization, whose specific features depend on the topological structures of the networks. The investigations of the equilibrium dynamics revealed a series of remarkable insights, including the relationship between spiking oscillations and the hierarchical structure of the networks and the identification of twin correlation patterns between node degree and total activation, implying that hubs of connectivity are also hubs of integrate-and-fire activation.

q-bio.NC↗
arXiv2008

Exact results and new insights for models defined over small-world networks. First and second order phase transitions. I: General result

We present, as a very general method, an effective field theory to analyze models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it gives the exact critical behavior and the exact critical surfaces and percolation thresholds, and provide a clear and immediate (also in terms of calculation) insight of the physics. The underlying structure of the non random part of the model, i.e., the set of spins staying in a given lattice L_0 of dimension d_0 and interacting through a fixed coupling J_0, is exactly taken into account. When J_0\geq 0, the small-world effect gives rise to the known fact that a second order phase transition takes place, independently of the dimension d_0 and of the added random connectivity c. However, when J_0<0, a completely different scenario emerges where, besides a spin glass transition, multiple first- and second-order phase transitions may take place.

cond-mat.dis-nn↗
arXiv2008

Exact results and new insights for models defined over small world networks. First and second order phase transitions. II: Applications

We apply a novel method (presented in part I) to solve several small-world models for which the method can be applied analytically: the Viana-Bray model (which can be seen as a 0 or infinite dimensional small-world model), the one-dimensional chain small-world model, and the small-world spherical model in generic dimension. In particular, we analyze in detail the one-dimensional chain small-world model with negative short-range coupling showing that in this case, besides a second-order spin glass phase transition, there are two critical temperatures corresponding to first- or second-order phase transitions.

cond-mat.dis-nn↗
arXiv2008

Influence of spin and interactions on quantum dots and nano-wires

In this dissertation we use sophisticated numerical methods in order to examine ground-state (GS) properties of two types of quantum systems with electron electron interactions: A quantum dot (QD) and a nano-wire. In the first half of the work we study a system of a single level coupled to a one-dimensional wire with interacting spinless electrons, when the wire is either clean or disordered. We utilize the density-matrix renormalization-group (DMRG) method to investigate the influence of the level on several thermodynamic properties of the clean interacting wire, which can be in one of two phases: Tomonaga-Luttinger liquid and charge density wave phases. When the wire is disordered, we investigate the Friedel oscillations, exploring the difference between the two phases and comparing them to the clean non-interacting case, for which we develop an exact formula for the oscillations. In the second half of the dissertation we study two cases of an isolated two-dimensional QD. We begin by an investigation of a new numerical method, the particle-hole DMRG (PH-DMRG), which is used to calculate the GS energy of a disordered QD consisting of interacting spinless electrons. We show that this method is much more accurate than the Hartree-Fock method, and we suggest an improvement of the algorithm, which reduces the error rate by almost 30 percents. Finally we study the magnetization of a QD with spin 1/2 electrons, in the presence of spin-orbit coupling and interactions. We calculate the g-factor and the expectation values of the spin operators in the GS, and find that when the QD is occupied by an even number of electrons, the GS can have a finite magnetization.

cond-mat.str-el↗
arXiv2008

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.

cs.CC↗
arXiv2008

What Is There Between Any Two Nodes in a Complex Network?

This article focuses on the identification of the number of paths with different lengths between pairs of nodes in complex networks and how, by providing comprehensive information about the network topology, such an information can be effectively used for characterization of theoretical and real-world complex networks, as well as for identification of communities.

physics.soc-ph↗
arXiv2010

Shear transformation zone analysis of shear reversal during granular flow

Experimental measurements of the onset of granular flow are directly compared to predictions of the "shear transformation zone" (STZ) theory of amorphous plasticity. The STZ equations make it possible, on a coarse grained level, to incorporate the anisotropy of the particle contact network and its dynamic evolution including changes in the contact network direction under shear reversal. The STZ theory and experiment show qualitative agreement in the transient and steady state shear forces and velocity profiles during the start of shear and shear reversal.

cond-mat.stat-mech↗
arXiv2008

Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise disordered porous media

Stress and strain fields in a two-dimensional pixelwise disordered system are computed by a Fast Fourier Transform method. The system, a model for a ductile damaged medium, consists of an elastic-perfectly matrix containing void pixels. Its behavior is investigated under equibiaxial or shear loading. We monitor the evolution with loading of plastically deformed zones, and we exhibit a nucleation / growth / coalescence scenario of the latter. Identification of plastic ``clusters'' is eased by using a discrete Green function implementing equilibrium and continuity at the level of one pixel. Observed morphological regimes are put into correspondence with some features of the macroscopic stress / strain curves.

cond-mat.mtrl-sci↗
arXiv2008

Variable range hopping in thin film with large dielectric constant

In a film with large dielectric constant $κ$ the electric field of an electron spreads inside the film before exiting the film at large distances of order of $κd$ ($d$ is the film width). This leads to the logarithmic Coulomb repulsion between electrons and modifies the shape of the Coulomb gap in the density of localized states in a doped film. As a result the variable range hopping conductivity in such a film has a peculiar temperature dependence, where the domain of the $\ln σ(T) \propto (T_0/T)^{p}$ dependence, with the index $p \simeq 0.7$, is sandwiched between the two domains with $p = 1/2$.

cond-mat.dis-nn↗
arXiv2008

Spin Glass Identities and the Nishimori Line

For a general spin glass model with asymmetric couplings we prove a family of identities involving expectations of generalized overlaps and magnetizations in the quenched state. Those identities holds pointwise in the Nishimori line and are reached at the rate of the inverse volume while, in the general case, they can be proved in integral average.

cond-mat.dis-nn↗