arXiv subjects

Mauro Bologna

Publications and source records attributed to Mauro Bologna.

At least 19 recordsLinked to original sources

arXiv2026

Exact and limit results for the CTRW in presence of drift and position dependent noise intensity

Continuous-time random walks (CTRWs) with drift and position-dependent jumps provide a general framework for describing a wide range of natural and engineered systems. We analyze the stochastic differential equation associated with this class of models, in which the driving noise consists of spike (shot) events, and we derive two exact analytical results. First, we obtain a closed-form expression for the $n$-time correlation functions of The noise, expressed as a sum over all $2^{n-1}$ ordered partitions of the observation times (Proposition 2). Second, using the $G$-cumulant formalism, we derive an \emph{exact} non-local master equation (ME) for the probability density function of the CTRW variable, valid without invoking diffusive limits, fractional scaling assumptions, or closure hypotheses (Proposition 3). In interaction representation, this ME retains the same structural form as that of the standard CTRW without drift or position-dependent jumps. Our main result is the emergence of a \emph{universal local master equation}: at long times, the exact non-local ME is universally and accurately approximated by a time-local ME whose only coefficient is the instantaneous renewal rate $R(t)$. From this equation, exact in the well known Poissonian case, both local and global properties of the PDF can be readily inferred. For example, the temporal behavior of the PDF is directly controlled by that of the rate function $R(t)$: if the waiting-time distribution decays as a power law with exponent $μ>2$, then $R(t)\to const$ and the system converges to the Poissonian equilibrium. By contrast, for $μ<2$, the rate decays in time and the effective diffusion induced by the noise slowly weakens, without leading to a stationary state. Numerical experiments confirm its remarkable accuracy even far beyond regimes where a naive time-scale separation would justify it.

cond-mat.stat-mech↗

arXiv2026

Noise-induced resonant acceleration of a charge in an intermittent magnetic field: an exact solution for ergodic and non-ergodic fluctuations

We study the diffusion of a charged particle in a magnetic field subject to stochastic dichotomous fluctuations. The associated induced electric field gives rise to non-trivial dynamical regimes. In particular, when the mean magnetic field vanishes, the particle remains confined within a finite radius, regardless of the fluctuation statistics. For a non-zero mean field, we show, using a density approach for Poissonian fluctuations, that the particle undergoes an exponential regime of accelerated diffusion. Crucially and more generally, adopting a trajectory-based formalism, we derive an exact analytical solution valid for arbitrary waiting-time distributions, including non-Poissonian and non-ergodic cases. Even rare, abrupt field reversal are shown to trigger exponential acceleration of the particle's diffusion. We demonstrate that this behaviour stems from noise exciting resonance bands present for periodic fluctuations, and we propose noise-induced resonant acceleration as a robust and efficient charge acceleration mechanism, potentially more effective than Fermi's classic model for cosmic acceleration.

cond-mat.dis-nn↗

arXiv2025

Universal behaviors of the multi-time correlation functions of random processes with renewal: the step noise case (the random velocity of a Lévy walk)

Stochastic processes with renewal properties are powerful tools for modeling systems where memory effects and long-time correlations play a significant role. In this work, we study a broad class of renewal processes where a variable's value changes according to a prescribed Probability Density Function (PDF), $p(ξ)$, after random waiting times $θ$. This model is relevant across many fields, including classical chaos, nonlinear hydrodynamics, quantum dots, cold atom dynamics, biological motion, foraging, and finance. We derive a general analytical expression for the $n$-time correlation function by averaging over process realizations. Our analysis identifies the conditions for stationarity, aging, and long-range correlations based on the waiting time and jump distributions. Among the many consequences of our analysis, two new key results emerge. First, for Poissonian waiting times, the correlation function quickly approaches that of telegraphic noise. Second, for power-law waiting times with $μ>2$, , \emph{any $n$-time correlation function asymptotically reduces to the two-time correlation evaluated at the earliest and latest time points}. This second result reveals a universal long-time behavior where the system's full statistical structure becomes effectively two-time reducible. Furthermore, if the jump PDF $p(ξ)$ has fat tails, this convergence becomes independent of the waiting time PDF and is significantly accelerated, requiring only modest increases in either the number of realizations or the trajectory lengths. Building upon earlier work that established the universality of the two-point correlation function (i.e., a unique formal expression depending solely on the variance of $ξ$ and on the waiting-time PDF), the present study extends that universality to the full statistical description of a broad class of renewal-type stochastic processes.

cond-mat.stat-mech↗

arXiv2024

Colored Stochastic Multiplicative Processes with Additive Noise Unveil a Third-Order PDE, Defying Conventional FPE and Fick-Law Paradigms

Research on stochastic differential equations (SDE) involving both additive and multiplicative noise has been extensive. In situations where the primary process is driven by a multiplicative stochastic process, additive white noise typically represents an intrinsic and unavoidable fast factor, including phenomena like thermal fluctuations, inherent uncertainties in measurement processes, or rapid wind forcing in ocean dynamics. This work focuses on a significant class of such systems, particularly those characterized by linear drift and multiplicative noise, extensively explored in the literature. Conventionally, multiplicative stochastic processes are also treated as white noise in existing studies. However, when considering colored multiplicative noise, the emphasis has been on characterizing the far tails of the probability density function (PDF), regardless of the spectral properties of the noise. In the absence of additive noise and with a general colored multiplicative SDE, standard perturbation approaches lead to a second-order PDE known as the Fokker-Planck Equation (FPE), consistent with Fick's law. This investigation unveils a notable departure from this standard behavior when introducing additive white noise. At the leading order of the stochastic process strength, perturbation approaches yield a \textit{third-order PDE}, irrespective of the white noise intensity. The breakdown of the FPE further signifies the breakdown of Fick's law. Additionally, we derive the explicit solution for the equilibrium PDF corresponding to this third-order PDE Master Equation. Through numerical simulations, we demonstrate significant deviations from outcomes derived using the FPE obtained through the application of Fick's law.

math.ST↗

arXiv2023

Influence of an environment changing in time on Crucial Events: the earthquake prototype

This paper is devoted to the study of the interaction between two distinct forms of non-stationary processes, which we will refer to as non-stationarity of first and second kind. The non-stationarity of first kind is caused by criticality-generated events that we call crucial events. Crucial events signal ergodicity breaking emerging from the interaction between the units of the complex system under study, indicating that the non stationarity of first kind has internal origin. The non-stationarity of second kind is due to the influence on the system of interest of an environment changing in time, thereby implying an external origin. In this paper we show that the non-stationarity of first kind, measured by an inverse power law index μ is characterized by singularities at μ = 2 and μ = 3. We realize the interaction between the non-stationarity of first kind and the non-stationarity of second kind with a model frequently adopted to study earthquakes, namely, a system of mainshocks, assumed to be crucial events, generating a cascade of after-shocks simulating the changing in time environment. We prove that the after-shocks significantly affects the detection of anomalous scaling, with this effect weakening as the value μ approaches μ = 2.5. We argue that this result is a consequence of the fact that the states μ = 2 and μ = 3 are the borders between different statistical regimes, where a sort of phase transition occurs, with μ = 2.5 being a state sufficiently far from both transition regimes. We conclude this paper with the observation that the earthquakes should be interpreted as resulting from the interaction between many geophysical units generating criticality, with the non-stationary events of second kind affecting conveniently short time regions between two consecutive crucial events.

physics.geo-ph↗

arXiv2023

Unveiling Pseudo-Crucial Events in Noise-Induced Phase Transitions

Noise-induced phase transitions are common in various complex systems, from physics to biology. In this article, we investigate the emergence of crucial events in noise-induced phase transition processes and their potential significance for understanding complexity in such systems. We utilize the first-passage time technique and coordinate transformations to study the dynamics of the system and identify crucial events. Furthermore, we employ Diffusion Entropy Analysis, a powerful statistical tool, to characterize the complexity of the system and quantify the information content of the identified events. Our results show that the emergence of crucial events is closely related to the complexity of the system and can provide insight into its behavior. This approach may have applications in diverse fields, such as climate modeling, financial markets, and biological systems, where understanding the emergence of crucial events is of great importance.

physics.data-an↗

arXiv2021

Effect of decreasing population growth-rate on deforestation and population sustainability

We consider the effect of non-constant parameters on the human-forest interaction logistic model coupled with human technological growth introduced in "Deforestation and world population sustainability: a quantitative analysis"[1]. In recent years in fact, a decrease in human population growth rate has emerged which can be measured to about 1.7% drop per year since 1960 value which coincides with latest UN projections for next decades up to year 2100 [2]. We therefore consider here the effect of decreasing human population growth-rate on the aforementioned model and we evaluate its effect on the probability of survival of human civilisation without going through a catastrophic collapse in population. We find that for realistic values of the human population carrying capacity of the earth (measured by parameter beta) this decrease would not affect previous results leading to a low probability of avoiding a catastrophic collapse. For larger more optimistic values of beta instead, a decrease in growth-rate would tilt the probability in favour of a positive outcome, i.e. from 10-20% up to even 95% likelihood of avoiding collapse.

q-bio.PE↗

arXiv2021

Effect of ergodic and non-ergodic fluctuations on a charge diffusing in a stochastic Magnetic Field

In this paper, we study the basic problem of a charged particle in a stochastic magnetic field. We consider dichotomous fluctuations of the magnetic field {where the sojourn time in one of the two states are distributed according to a given waiting time distribution either with Poisson or non-Poisson statistics, including as well the case of distributions with diverging mean time between changes of the field}, corresponding to an ergodicity breaking condition. We provide analytical and numerical results for all cases evaluating the average and the second moment of the position and velocity of the particle. We show that the field fluctuations induce diffusion of the charge with either normal or anomalous properties, depending on the statistics of the fluctuations, with distinct regimes from those observed, e.g., in standard Continuous Time Random Walk models.

cond-mat.dis-nn↗

arXiv2021

Revisiting the Memristor Concept within Basic Circuit Theory

In this paper we revisit the memristor concept within circuit theory. We start from the definition of the basic circuit elements, then we introduce the original formulation of the memristor concept and summarize some of the controversies on its nature. We also point out the ambiguities resulting from a non rigorous usage of the flux linkage concept. After concluding that the memristor is not a fourth basic circuit element, prompted by recent claims in the memristor literature, we look into the application of the memristor concept to electrophysiology, realizing that an approach suitable to explain the observed inductive behavior of the giant squid axon had already been developed in the 1960s, with the introduction of "time-variant resistors." We also discuss a recent memristor implementation in which the magnetic flux plays a direct role, concluding that it cannot strictly qualify as a memristor, because its $v-i$ curve cannot exactly pinch at the origin. Finally, we present numerical simulations of a few memristors and memristive systems, focusing on the behavior in the $φ-q$ plane. We show that, contrary to what happens for the most basic memristor concept, for general memristive systems the $φ-q$ curve is not single-valued or not even closed.

eess.SY↗

arXiv2020

Deforestation and world population sustainability: a quantitative analysis

In this paper we afford a quantitative analysis of the sustainability of current world population growth in relation to the parallel deforestation process adopting a statistical point of view. We consider a simplified model based on a stochastic growth process driven by a continuous time random walk, which depicts the technological evolution of human kind, in conjunction with a deterministic generalised logistic model for humans-forest interaction and we evaluate the probability of avoiding the self-destruction of our civilisation. Based on the current resource consumption rates and best estimate of technological rate growth our study shows that we have very low probability, less than 10% in most optimistic estimate, to survive without facing a catastrophic collapse.

q-bio.PE↗

arXiv2019

About the foundation of the Kubo Generalized Cumulants theory. A revisited and corrected approach

More than fifty years ago, in a couple of seminal works Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept, implicitly introduced by Laplace in 1810. Kubo's idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo's idea, today not yet fully exploited. It results that for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo-Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it {\em does not} depend on the specific nature of the moments, but just on the definition of the generalized cumulants.

math-ph↗

arXiv2019

Intelligence of small groups

Dunbar hypothesized that $150$ is the maximal number of people with whom one can maintain stable social relationships. We explain this effect as being a consequence of a process of self-organization between $N$ units leading their social system to the edge of phase transition, usually termed criticality. Criticality generates events, with an inter-event time interval distribution characterized by an inverse power law (IPL) index $μ_{S}<2$. These events break ergodicity and we refer to them as crucial events. The group makes decisions and the time persistence of each decision is given by another IPL distribution with IPL index $μ_{R}$, which is different from $μ_{S}$ if $N\neq 150$. We prove that when the number of interacting individuals is equal to $150$, these two different IPL indexes become identical, with the effect of generating the Kardar Parisi Zhang (KPZ) scaling $δ=1/3$. We argue this to be an enhanced form of intelligence, which generates efficient information transmission within the group. We prove the inflrmation transmission efficiency is maximal when $N=150$, the Dunbar number.

nlin.AO↗

arXiv2018

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate a closed formula that has several applications in solving problems ranging from linear differential equations, quantum mechanics to non-linear differential equations.

cond-mat.stat-mech↗

arXiv2017

Non-Poisson Renewal Events and Memory

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out of equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same non-exponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin re-crossings and establishing the corresponding waiting time probability density function (PDF), $ψ(t)$. In the former case, with no crucial events, $ψ(t)$ is exponential and in the latter case, with crucial events, $ψ(t)$ is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.

physics.data-an↗

arXiv2016

Ergodicity Breaking and Localization

We study the joint action of the non-Poisson renewal events (NPR) yielding Continuous Time Random Walk (CTRW) with index alpha < 1 and two different generators of Hurst coefficient H not equal to 0.5, one generating fractional Brownian motion (FBM) and another scaled Brownian motion (SBM). We discuss the ergodicity breaking emerging from these joint actions and we find that in both cases the adoption of time averages leads to localization. In the case of the joint action of NPR and SBM, localization occurs when SBM would produce sub-diffusion. The joint action of NPR and FBM, on the contrary, may lead to localization when FBM would be a source of super-diffusion. We argue that the second effect might require a refinement of the theoretical perspective about determinism and randomness.

cond-mat.stat-mech↗

arXiv2016

Non-Ergodic Complexity Management

Linear response theory, the backbone of non-equilibrium statistical physics, has recently been extended to explain how and why non-ergodic renewal processes are insensitive to simple perturbations, such as in habituation. It was established that a permanent correlation resulted between an external stimulus and the response of a complex system generating non-ergodic renewal processes, when the stimulus is a similar non-ergodic process. This is the principle of complexity management, whose proof relies on ensemble distribution functions. Herein we extend the proof to the non-ergodic case using time averages and a single time series, hence making it usable in real life situations where ensemble averages cannot be performed because of the very nature of the complex systems being studied.

nlin.AO↗

arXiv2015

Analytical estimation of the Earth's magnetic field scale

In this paper we analytically estimate the magnetic field scale of planets with physical core conditions similar to that of Earth from a statistical point of view. We evaluate the magnetic field on the basis of the physical parameters of the center of the planet, such as density, temperature, and core size. We look at the contribution of the Peltier-Seebeck effect on the magnetic field, showing that an electrical thermal current can exist in a rotating fluid sphere. Finally, we apply our calculations to Earth and Jupiter. In each case we show that the thermal generation of currents leads to a magnetic field scale comparable to the observed fields of the two planets.

physics.geo-ph↗

arXiv2015

Exact analytical approach to differential equations with variable coefficients

This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the analytical expression presented in the paper. The formalism can be easily extended to the infinite dimensional case such as the quantum time-dependent Hamiltonian problem.

math.CA↗