arXiv subjects

Marco Gherardi

Publications and source records attributed to Marco Gherardi.

36 records · Page 2Linked to original sources

arXiv2018

Zipf and Heaps laws from dependency structures in component systems

Complex natural and technological systems can be considered, on a coarse-grained level, as assemblies of elementary components: for example, genomes as sets of genes, or texts as sets of words. On one hand, the joint occurrence of components emerges from architectural and specific constraints in such systems. On the other hand, general regularities may unify different systems, such as the broadly studied Zipf and Heaps laws, respectively concerning the distribution of component frequencies and their number as a function of system size. Dependency structures (i.e., directed networks encoding the dependency relations between the components in a system) were proposed recently as a possible organizing principles underlying some of the regularities observed. However, the consequences of this assumption were explored only in binary component systems, where solely the presence or absence of components is considered, and multiple copies of the same component are not allowed. Here, we consider a simple model that generates, from a given ensemble of dependency structures, a statistical ensemble of sets of components, allowing for components to appear with any multiplicity. Our model is a minimal extension that is memoryless, and therefore accessible to analytical calculations. A mean-field analytical approach (analogous to the "Zipfian ensemble" in the linguistics literature) captures the relevant laws describing the component statistics as we show by comparison with numerical computations. In particular, we recover a power-law Zipf rank plot, with a set of core components, and a Heaps law displaying three consecutive regimes (linear, sub-linear and saturating) that we characterize quantitatively.

physics.soc-ph↗

arXiv2017

Jack on a Devil's staircase

We review a simple mechanism for the formation of plateaux in the fractional quantum Hall effect. It arises from a map of the microscopic Hamiltonian in the thin torus limit to a lattice gas model, solved by Hubbard. The map suggests a Devil's staircase pattern, and explains the observed asymmetries in the widths. Each plateau is a new ground state of the system: a periodic Slater state in the thin torus limit. We provide the unitary operator that maps such limit states to the full, effective ground states with same filling fraction. These Jack polynomials generalise Laughlin's ansatz, and are exact eigenstates of the Laplace-Beltrami operator. Why are Jacks sitting on the Devil's staircase? This is yet an intriguing problem.

cond-mat.str-el↗

arXiv2017

Intermittent transport of bacterial chromosomal loci

The short-time dynamics of bacterial chromosomal loci is a mixture of subdiffusive and active motion, in the form of rapid relocations with near-ballistic dynamics. While previous work has shown that such rapid motions are ubiquitous, we still have little grasp on their physical nature, and no positive model is available that describes them. Here, we propose a minimal theoretical model for loci movements as a fractional Brownian motion subject to a constant but intermittent driving force, and compare simulations and analytical calculations to data from high-resolution dynamic tracking in E. coli. This analysis yields the characteristic time scales for intermittency. Finally, we discuss the possible shortcomings of this model, and show that an increase in the effective local noise felt by the chromosome associates to the active relocations.

cond-mat.soft↗

arXiv2017

Current quantization and fractal hierarchy in a driven repulsive lattice gas

Driven lattice gases are widely regarded as the paradigm of collective phenomena out of equilibrium. While such models are usually studied with nearest-neighbor interactions, many empirical driven systems are dominated by slowly decaying interactions such as dipole-dipole and Van der Waals forces. Motivated by this gap, we study the non-equilibrium stationary state of a driven lattice gas with slow-decayed repulsive interactions at zero temperature. By numerical and analytical calculations of the particle current as a function of the density and of the driving field, we identify (i) an abrupt breakdown transition between insulating and conducting states, (ii) current quantization into discrete phases where a finite current flows with infinite differential resistivity, and (iii) a fractal hierarchy of excitations, related to the Farey sequences of number theory. We argue that the origin of these effects is the competition between scales, which also causes the counterintuitive phenomenon that crystalline states can melt by increasing the density.

cond-mat.stat-mech↗

arXiv2017

Cell-to-cell variability and robustness in S-phase duration from genome replication kinetics

Genome replication, a key process for a cell, relies on stochastic initiation by replication origins, causing a variability of replication timing from cell to cell. While stochastic models of eukaryotic replication are widely available, the link between the key parameters and overall replication timing has not been addressed systematically.We use a combined analytical and computational approach to calculate how positions and strength of many origins lead to a given cell-to-cell variability of total duration of the replication of a large region, a chromosome or the entire genome.Specifically, the total replication timing can be framed as an extreme-value problem, since it is due to the last region that replicates in each cell. Our calculations identify two regimes based on the spread between characteristic completion times of all inter-origin regions of a genome. For widely different completion times, timing is set by the single specific region that is typically the last to replicate in all cells. Conversely, when the completion time of all regions are comparable,an extreme-value estimate shows that the cell-to-cell variability of genome replication timing has universal properties. Comparison with available data shows that the replication program of three yeast species falls in this extreme-value regime.

q-bio.GN↗

arXiv2016

Balancing building and maintenance costs in growing transport networks

The costs associated to the length of links impose unavoidable constraints to the growth of natural and artificial transport networks. When future network developments can not be predicted, building and maintenance costs require competing minimization mechanisms, and can not be optimized simultaneously. Hereby, we study the interplay of building and maintenance costs and its impact on the growth of transportation networks through a non-equilibrium model of network growth. We show cost balance is a sufficient ingredient for the emergence of tradeoffs between the network's total length and transport effciency, of optimal strategies of construction, and of power-law temporal correlations in the growth history of the network. Analysis of empirical ant transport networks in the framework of this model suggests different ant species may adopt similar optimization strategies.

physics.soc-ph↗

arXiv2016

Devil's staircase phase diagram of the fractional quantum Hall effect in the thin-torus limit

After more than three decades the fractional quantum Hall effect still poses challenges to contemporary physics. Recent experiments point toward a fractal scenario for the Hall resistivity as a function of the magnetic field. Here, we consider the so-called thin-torus limit of the Hamiltonian describing interacting electrons in a strong magnetic field, restricted to the lowest Landau level, and we show that it can be mapped onto a one-dimensional lattice gas with repulsive interactions, with the magnetic field playing the role of a chemical potential. The statistical mechanics of such models leads to interpret the sequence of Hall plateaux as a fractal phase diagram, whose landscape shows a qualitative agreement with experiments.

cond-mat.str-el↗

arXiv2016

Measuring logic complexity can guide pattern discovery in empirical systems

We explore a definition of complexity based on logic functions, which are widely used as compact descriptions of rules in diverse fields of contemporary science. Detailed numerical analysis shows that (i) logic complexity is effective in discriminating between classes of functions commonly employed in modelling contexts; (ii) it extends the notion of canalisation, used in the study of genetic regulation, to a more general and detailed measure; (iii) it is tightly linked to the resilience of a function's output to noise affecting its inputs. We demonstrate its utility by measuring it in empirical data on gene regulation, digital circuitry, and propositional calculus. Logic complexity is exceptionally low in these systems. The asymmetry between "on" and "off" states in the data correlates with the complexity in a non-null way; a model of random Boolean networks clarifies this trend and indicates a common hierarchical architecture in the three systems.

physics.data-an↗

arXiv2014

Soft bounds on diffusion produce skewed distributions and Gompertz growth

Constraints can affect dramatically the behavior of diffusion processes. Recently, we analyzed a natural and a technological system and reported that they perform diffusion-like discrete steps displaying a peculiar constraint, whereby the increments of the diffusing variable are subject to configuration-dependent bounds. This work explores theoretically some of the revealing landmarks of such phenomenology, termed "soft bound". At long times, the system reaches a steady state irreversibly (i.e., violating detailed balance), characterized by a skewed "shoulder" in the density distribution, and by a net local probability flux, which has entropic origin. The largest point in the support of the distribution follows a saturating dynamics, expressed by the Gompertz law, in line with empirical observations. Finally, we propose a generic allometric scaling for the origin of soft bounds. These findings shed light on the impact on a system of such "scaling" constraint and on its possible generating mechanisms.

cond-mat.stat-mech↗

arXiv2014

A Parafermionic Generalization of the Jaynes Cummings Model

We introduce a parafermionic version of the Jaynes Cummings Hamiltonian, by coupling $k$ Fock parafermions (nilpotent of order $F$) to a 1D harmonic oscillator, representing the interaction with a single mode of the electromagnetic field. We argue that for $k=1$ and $F\leq 3$ there is no difference between Fock parafermions and quantum spins $s=\frac{F-1}{2}$. We also derive a semiclassical approximation of the canonical partition function of the model by assuming $\hbar$ to be small in the regime of large enough total number of excitations $n$, where the dimension of the Hilbert space of the problem becomes constant as a function of $n$. We observe in this case an interesting behaviour of the average of the bosonic number operator showing a single crossover between regimes with different integer values of this observable. These features persist when we generalize the parafermionic Hamiltonian by deforming the bosonic oscillator with a generic function $Φ(x)$; the $q-$deformed bosonic oscillator corresponds to a specific choice of the deformation function $Φ$. In this particular case, we observe at most $k(F-1)$ crossovers in the behavior of the mean bosonic number operator, suggesting a phenomenology of superradiance similar to the $k-$atoms Jaynes Cummings model.

math-ph↗

arXiv2013

Theta-point polymers in the plane and Schramm-Loewner evolution

We study the connection between polymers at the theta temperature on the lattice and Schramm-Loewner chains with constant step length in the continuum. The latter realize a useful algorithm for the exact sampling of tricritical polymers, where finite-chain effects are excluded. The driving function computed from the lattice model via a radial implementation of the zipper method is shown to converge to Brownian motion of diffusivity kappa=6 for large times. The distribution function of an internal portion of walk is well approximated by that obtained from Schramm-Loewner chains. The exponent of the correlation length nu and the leading correction-to scaling exponent Delta_1 measured in the continuum are compatible with nu=4/7 (predicted for the theta point) and Delta_1=72/91 (predicted for percolation). Finally, we compute the shape factor and the asphericity of the chains, finding surprising accord with the theta-point end-to-end values.

cond-mat.stat-mech↗

arXiv2013

q-deformed Loewner evolution

The Loewner equation, in its stochastic incarnation introduced by Schramm, is an insightful method for the description of critical random curves and interfaces in two-dimensional statistical mechanics. Two features are crucial, namely conformal invariance and a conformal version of the Markov property. Extensions of the equation have been explored in various directions, in order to expand the reach of such a powerful method. We propose a new generalization based on q-calculus, a concept rooted in quantum geometry and non-extensive thermodynamics; the main motivation is the explicit breaking of the Markov property, while retaining scale invariance in the stochastic version. We focus on the deterministic equation and give some exact solutions; the formalism naturally gives rise to multiple mutually-intersecting curves. A general method of simulation is constructed - which can be easily extended to other q-deformed equations - and is applied to both the deterministic and the stochastic realms. The way the $q\neq 1$ picture converges to the classical one is explored as well.

cond-mat.stat-mech↗

arXiv2013

Hybrid deterministic/stochastic algorithm for large sets of rate equations

We propose a hybrid algorithm for the time integration of large sets of rate equations coupled by a relatively small number of degrees of freedom. A subset containing fast degrees of freedom evolves deterministically, while the rest of the variables evolves stochastically. The emphasis is put on the coupling between the two subsets, in order to achieve both accuracy and efficiency. The algorithm is tested on the problem of nucleation, growth and coarsening of clusters of defects in iron, treated by the formalism of cluster dynamics. We show that it is possible to obtain results indistinguishable from fully deterministic and fully stochastic calculations, while speeding up significantly the computations with respect to these two cases.

cond-mat.mtrl-sci↗

arXiv2013

Hard and soft bounds in the evolution of Ubuntu packages. A lesson for species body masses?

Open-source software is a complex system; its development depends on the self-coordinated action of a large number of agents. This study follows the size of the building blocks, called "packages", of the Ubuntu Linux operating system over its entire history. The analysis reveals a multiplicative diffusion process, constrained by size-dependent bounds, driving the dynamics of the package-size distribution. A formalization of this into a quantitative model is able to match the data without relying on any adjustable parameters, and generates definite predictions. Finally, we formulate the hypothesis that a similar non-stationary mechanism could be shaping the distribution of mammal body sizes.

physics.soc-ph↗

arXiv2012

Influence of homology and node-age on the growth of protein-protein interaction networks

Proteins participating in a protein-protein interaction network can be grouped into homology classes following their common ancestry. Proteins added to the network correspond to genes added to the classes, so that the dynamics of the two objects are intrinsically linked. Here, we first introduce a statistical model describing the joint growth of the network and the partitioning of nodes into classes, which is studied through a combined mean-field and simulation approach. We then employ this unified framework to address the specific issue of the age dependence of protein interactions, through the definition of three different node wiring/divergence schemes. Comparison with empirical data indicates that an age-dependent divergence move is necessary in order to reproduce the basic topological observables together with the age correlation between interacting nodes visible in empirical data. We also discuss the possibility of nontrivial joint partition/topology observables.

q-bio.MN↗

arXiv2010

Geometrical Properties of Two-Dimensional Interacting Self-Avoiding Walks at the Theta-Point

We perform a Monte Carlo simulation of two-dimensional N-step interacting self-avoiding walks at the theta point, with lengths up to N=3200. We compute the critical exponents, verifying the Coulomb-gas predictions, the theta-point temperature T_theta = 1.4986(11), and several invariant size ratios. Then, we focus on the geometrical features of the walks, computing the instantaneous shape ratios, the average asphericity, and the end-to-end distribution function. For the latter quantity, we verify in detail the theoretical predictions for its small- and large-distance behavior.

cond-mat.soft↗

arXiv2010

Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.

cond-mat.stat-mech↗

arXiv2009

Whole-plane self-avoiding walks and radial Schramm-Loewner evolution: a numerical study

We numerically test the correspondence between the scaling limit of self-avoiding walks (SAW) in the plane and Schramm-Loewner evolution (SLE) with k=8/3. We introduce a discrete-time process approximating SLE in the exterior of the unit disc and compare the distribution functions for an internal point in the SAW and a point at a fixed fractal variation on the SLE, finding good agreement. This provides numerical evidence in favor of a conjecture by Lawler, Schramm and Werner. The algorithm turns out to be an efficient way of computing the position of an internal point in the SAW.

cond-mat.stat-mech↗