Universal singularities of anomalous diffusion in the Richardson class.

Inhomogeneous environments are rather ubiquitous in nature, often implying anomalies resulting in deviation from Gaussianity of diffusion processes. While sub- and superdiffusion are usually due to contrasting environmental features (hindering or favoring the motion, respectively), they are both observed in systems ranging from the micro- to the cosmological scale. Here we show how a model encompassing sub- and superdiffusion in an inhomogeneous environment exhibits a critical singularity in the normalized generator of the cumulants. The singularity originates directly and exclusively from the asymptotics of the non-Gaussian scaling function of displacement, and the independence from other details confers it a universal character. Our analysis, based on the method first applied by Stella et al. [Phys. Rev. Lett. 130, 207104 (2023)10.1103/PhysRevLett.130.207104], shows that the relation connecting the scaling function asymptotics to the diffusion exponent characteristic of processes in the Richardson class implies a nonstandard extensivity in time of the cumulant generator. Numerical tests fully confirm the results.

Anomalous Dynamical Scaling Determines Universal Critical Singularities.

Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at a large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.

Entropic measure unveils country competitiveness and product specialization in the World trade web.

We show how the Shannon entropy function can be used as a basis to set up complexity measures weighting the economic efficiency of countries and the specialization of products beyond bare diversification. This entropy function guarantees the existence of a fixed point which is rapidly reached by an iterative scheme converging to our self-consistent measures. Our approach naturally allows to decompose into inter-sectorial and intra-sectorial contributions the country competitivity measure if products are partitioned into larger categories. Besides outlining the technical features and advantages of the method, we describe a wide range of results arising from the analysis of the obtained rankings and we benchmark these observations against those established with other economical parameters. These comparisons allow to partition countries and products into various main typologies, with well-revealed characterizing features. Our methods have wide applicability to general problems of ranking in bipartite networks.

Topological Disentanglement of Linear Polymers under Tension.

We develop a theoretical description of the topological disentanglement occurring when torus knots reach the ends of a semiflexible polymer under tension. These include decays into simpler knots and total unknotting. The minimal number of crossings and the minimal knot contour length are the topological invariants playing a key role in the model. The crossings behave as particles diffusing along the chain and the application of appropriate boundary conditions at the ends of the chain accounts for the knot disentanglement. Starting from the number of particles and their positions, suitable rules allow reconstructing the type and location of the knot moving on the chain Our theory is extensively benchmarked with corresponding molecular dynamics simulations and the results show a remarkable agreement between the simulations and the theoretical predictions of the model.

Exact Coarse Graining Preserves Entropy Production out of Equilibrium.

The entropy production rate associated with broken time-reversal symmetry provides an essential characterization of nanosystems out of equilibrium, from driven colloidal particles to molecular motors. Limited access to the dynamical states is generally expected to hinder the correct estimation of this observable. Here we show how memoryless jump processes can be coarse grained, exactly preserving its average and fluctuations at stationarity. This supports univocal applicability of fluctuation theorems for entropy and allows inference of the genuine thermodynamics together with inaccessible process details.

Topological Disentanglement Dynamics of Torus Knots on Open Linear Polymers.

We simulate and study the topological disentanglement occurring when torus knots reach the ends of a semiflexible open polymer (decay into simpler knots or unknotting). Through a rescaling procedure and the application of appropriate boundary conditions, we show that the full unknotting process can be understood in terms of point-like particles representing essential crossings, diffusing on the support [0, 1]. We address the bending and configurational free energy drives on the diffusion process, together with the scaling properties of the effective diffusion and friction coefficients. Agreement with simulations suggests universal features for these two model parameters.

Growth dynamics and complexity of national economies in the global trade network.

We explore the quantitative nexus among economic growth of a country, diversity and specialization of its productions, and evolution in time of its basket of exports. To this purpose we set up a dynamic model and construct economic complexity measures based on panel data concerning up to 1238 exports of 223 countries for 21 years. Key statistical features pertaining to the distribution of resources in the different exports of each country reveal essential in both cases. The parameters entering the evolution model, combined with counterfactual analyses of synthetic simulations, give novel insight into cooperative effects among different productions and prospects of growth of each economy. The complexity features emerging from the analysis of dynamics are usefully compared with gross domestic product per capita (GDPpc) and with an original measure of the efficiency of the economic systems. This measure, whose construction starts from an estimate of bare diversity in terms of Shannon's entropy function, is made fully consistent with the degree of specialization of the products. Comparisons of this measure with the model parameters allow clear distinctions, from multiple perspectives, among developed, emerging, underdeveloped and risky economies.

Data Driven Approach to the Dynamics of Import and Export of G7 Countries.

The dynamics of imports plus exports of 226 product classes by the G7 countries between 1962 and 2000 is described in terms of stochastic differential equations. The model allows interesting comparisons among the different economies related to the compositions of the national baskets. Synthetic solutions can also be used to estimate hidden and unexploited growth potentials. These prerogatives are strictly connected with the fact that a network structure is at the basis of the model. Such a network expresses the mutual influences of different products through resource transfers, and is a key ingredient producing cooperative growth effects which can be quantified and distinguished from those generated by deterministic drifts and representing direct resource inputs. An analysis of this network, which differs substantially from those previously considered within the economic complexity approach, allows to estimate the centrality of different products in each national basket, highlighting the most essential commodities for each economy. Solutions of the model give the possibility of performing counterfactual analyses aimed at estimating how much the growth of each country could have profited from a general strengthening, or weakening, of the links in the same products network.

Export dynamics as an optimal growth problem in the network of global economy.

We analyze export data aggregated at world global level of 219 classes of products over a period of 39 years. Our main goal is to set up a dynamical model to identify and quantify plausible mechanisms by which the evolutions of the various exports affect each other. This is pursued through a stochastic differential description, partly inspired by approaches used in population dynamics or directed polymers in random media. We outline a complex network of transfer rates which describes how resources are shifted between different product classes, and determines how casual favorable conditions for one export can spread to the other ones. A calibration procedure allows to fit four free model-parameters such that the dynamical evolution becomes consistent with the average growth, the fluctuations, and the ranking of the export values observed in real data. Growth crucially depends on the balance between maintaining and shifting resources to different exports, like in an explore-exploit problem. Remarkably, the calibrated parameters warrant a close-to-maximum growth rate under the transient conditions realized in the period covered by data, implying an optimal self organization of the global export. According to the model, major structural changes in the global economy take tens of years.

Role of trapping and crowding as sources of negative differential mobility.

Increasing the crowding in an environment does not necessarily trigger negative differential mobility of strongly pushed particles. Moreover, the choice of the model, in particular the kind of microscopic jump rates, may be very relevant in determining the mobility. We support these points via simple examples and we therefore address recent claims saying that crowding in an environment is likely to promote negative differential mobility. Trapping of tagged particles enhanced by increasing the force remains the mechanism determining a drift velocity not monotonous in the driving force.

Scaling symmetry, renormalization, and time series modeling: the case of financial assets dynamics.

We present and discuss a stochastic model of financial assets dynamics based on the idea of an inverse renormalization group strategy. With this strategy we construct the multivariate distributions of elementary returns based on the scaling with time of the probability density of their aggregates. In its simplest version the model is the product of an endogenous autoregressive component and a random rescaling factor designed to embody also exogenous influences. Mathematical properties like increments' stationarity and ergodicity can be proven. Thanks to the relatively low number of parameters, model calibration can be conveniently based on a method of moments, as exemplified in the case of historical data of the S&P500 index. The calibrated model accounts very well for many stylized facts, like volatility clustering, power-law decay of the volatility autocorrelation function, and multiscaling with time of the aggregated return distribution. In agreement with empirical evidence in finance, the dynamics is not invariant under time reversal, and, with suitable generalizations, skewness of the return distribution and leverage effects can be included. The analytical tractability of the model opens interesting perspectives for applications, for instance, in terms of obtaining closed formulas for derivative pricing. Further important features are the possibility of making contact, in certain limits, with autoregressive models widely used in finance and the possibility of partially resolving the long- and short-memory components of the volatility, with consistent results when applied to historical series.

Geometry and topology of knotted ring polymers in an array of obstacles.

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, RG, and that of the region containing the knot, R(G,k), are small compared to the distance b between the obstacles, the knot is weakly localized and RG scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where R(G)>b>R(G,k), the geometry of the polymer becomes branched. When R(G,k) exceeds b, the knot delocalizes and becomes also branched. In this regime, RG is independent of knot type. We discuss the implications of this behavior for gel electrophoresis experiments on knotted DNA in weak fields.

The size of knots in polymers.

Circular DNA in viruses and bacteria is often knotted. While mathematically problematic, the determination of the knot size is crucial for the study of the physical and biological behaviour of long macromolecules. Here, we review work on the size distribution of these knots under equilibrium conditions. We discuss knot localization in good and poor solvents, or in polymers that are adsorbed on a surface. We also discuss recent evidence that knot size is a crucial quantity in relaxation processes of knotted polymers.

Central limit theorem for anomalous scaling due to correlations.

We derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables. The theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the Lévy-Gnedenko uncorrelated case. In particular, correlated anomalous diffusion is mapped onto Lévy diffusion. Under suitable assumptions, the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling.

Scaling of a collapsed polymer globule in two dimensions.

Extensive Monte Carlo data analysis gives clear evidence that collapsed linear polymers in two dimensions fall in the universality class of athermal, dense self-avoiding walks, as conjectured by Duplantier [Phys. Rev. Lett. 71, 4274 (1993)].10.1103/PhysRevLett.71.4274 However, the boundary of the globule has self-affine roughness and does not determine the anticipated nonzero topological boundary contribution to entropic exponents. Scaling corrections are due to subleading contributions to the partition function corresponding to polymer configurations with one end located on the globule-solvent interface.

Intensity thresholds and the statistics of the temporal occurrence of solar flares.

Introducing thresholds to analyze time series of emission from the Sun enables a new and simple definition of solar flare events and their interoccurrence times. Rescaling time by the rate of events, the waiting and quiet time distributions both conform to scaling functions that are independent of the intensity threshold over a wide range. The scaling functions are well-described by a two-parameter function, with parameters that depend on the phase of the solar cycle. For flares identified according to the current, standard definition, similar behavior is found.

Scaling in DNA unzipping models: denaturated loops and end segments as branches of a block copolymer network.

For a model of DNA denaturation, exponents describing the distributions of denaturated loops and unzipped end segments are determined by exact enumeration and by Monte Carlo simulations in two and three dimensions. The loop distributions are consistent with first-order thermal denaturation in both cases. Results for end segments show a coexistence of two distinct power laws in the relative distributions, which is not foreseen by a recent approach in which DNA is treated as a homogeneous network of linear polymer segments. This unexpected feature, and the discrepancies with such an approach, are explained in terms of a refined scaling picture in which a precise distinction is made between network branches representing single-stranded and effective double-stranded segments.

Roles of stiffness and excluded volume in DNA denaturation.

The nature and the universal properties of DNA thermal denaturation are investigated by Monte Carlo simulations. For suitable lattice models we determine the exponent c describing the decay of the probability distribution of denaturated loops of length l, P approximately l(-c). If excluded volume effects are fully taken into account, c = 2.10(4) is consistent with a first order transition. The stiffness of the double stranded chain has the effect of sharpening the transition, if it is continuous, but not of changing its order and the value of the exponent c, which is also robust with respect to inclusion of specific base-pair sequence heterogeneities.