Space-time Diffusion Research Articles

Current experimental upper bounds on spacetime diffusion

A theory describing the dynamics of quantum systems interacting on a classical spacetime was recently put forward by Oppenheim Quantum states may retain their coherence, at the cost of some amount of stochasticity of the spacetime metric, characterized by a spacetime diffusion parameter. Here, we report existing experimental upper bounds on such spacetime diffusion, based on a review of several types of experiments with very low force noise over a broad range of test masses from single atoms to several kilograms. We find an upper bound at least 15 orders of magnitude lower as compared to the initial bounds for explicit models presented by Oppenheim . The results presented here provide a path forward for future experiments that can help evaluate classical-quantum theories. Published by the American Physical Society 2024

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity

We consider two interacting systems when one is treated classically while the other system remains quantum. Consistent dynamics of this coupling has been shown to exist, and explored in the context of treating space-time classically. Here, we prove that any such hybrid dynamics necessarily results in decoherence of the quantum system, and a breakdown in predictability in the classical phase space. We further prove that a trade-off between the rate of this decoherence and the degree of diffusion induced in the classical system is a general feature of all classical quantum dynamics; long coherence times require strong diffusion in phase-space relative to the strength of the coupling. Applying the trade-off relation to gravity, we find a relationship between the strength of gravitationally-induced decoherence versus diffusion of the metric and its conjugate momenta. This provides an experimental signature of theories in which gravity is fundamentally classical. Bounds on decoherence rates arising from current interferometry experiments, combined with precision measurements of mass, place significant restrictions on theories where Einstein’s classical theory of gravity interacts with quantum matter. We find that part of the parameter space of such theories are already squeezed out, and provide figures of merit which can be used in future mass measurements and interference experiments.

Fully Jacobi–Galerkin algorithm for two-dimensional time-dependent PDEs arising in physics

Herein, a pure shifted Jacobi–Galerkin (SJG) method is offered for handling linear two-dimensional space-time diffusion and telegraph equations but by considering their integrated forms. The semi-analytic approximate solution is expanded in spatial and temporal variables as polynomials bases built as a linear combination of the shifted Jacobi Polynomials (JPs). Expanding the exact solution regarding these polynomials satisfies the homogeneous initial and boundary conditions. The proposed Jacobi–Galerkin (JG) procedure yields exponential convergence rates when the solution is smooth enough. A careful investigation of the convergence analysis of the suggested approximate triple series expansion examines the offered numerical algorithm. Two test illustrative examples are given to prove the high accuracy of the proposed scheme.

Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives

Abstract In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

Late-time dark energy and Hubble tension

AbstractWe extend Einstein’s theory of general relativity by introducing stochastic elements in addition to the usual fields and apply it to explore late-time redshift. The stochastic perturbation of spacetime enforces an effective minimum length (ML) to give us a cosmological constant naturally derived from the diffusive nature of spacetime and a redshift driven by both the geometry of spacetime as well as its diffusive nature. In this new theory, “dark energy” is the manifestation of fundamental uncertainty caused by ML of spacetime. The new theory converges to the minimalΛ\LambdaCDM model in the era after the Big Bang, when the geometry dominates over the diffusive character of spacetime. However, as the Hubble parameter decreases in value over time, there is a period during which the diffusive effects play an increasingly important role. For later times, as the universe approaches its minimum total energy density, the resulting redshift obtains significant contributions from both the geometry, captured by the Hubble parameter “HH,” and spacetime diffusion, captured by a new parameter “DD,” the diffusive equivalent toHH. Hence, the new theory presented here is particularly important during the later times in whichHHdiminishes and becomes comparable toDD. The theory suggests that the Hubble tension might be relieved by the diffusive character of spacetime. In order to compare the early time Hubble parameter estimates to the late-time estimates, we must recognize the contribution diffusion makes to the redshift observations and further reformulate luminosity distance and its kinematic expression to account for the effects of diffusion in addition to geometry. We perform a simple analysis of Type Ia supernovae observations with distances calibrated using Cepheids to obtain estimates for the new diffusion parameter. Based on these results, the new theory places the universe well inside a vacuum-dominated regime with a small and diminishing diffusion parameter.

Environmental-Economic Unit Commitment With Robust Diffusion Control of Gas Pollutants

Air pollution is one of the main environmental issues. This paper proposes an environmental-economic unit commitment model (EEUC) incorporating a novel robust diffusion control of gas pollutants. The control mode evaluates the space-time diffusion of gas emissions, and then reduces the contribution of the thermal power generation to the gas pollutant concentration. To handle the meteorological condition uncertainty in the gas diffusion, the proposed EEUC is derived as a constraint-based robust optimization problem. The strategy optimizes the expected objective of the empirical distribution, together with a limitation on the worst objective induced by the uncertainty set, such that the overall effectiveness is prioritized with an adjustable robustness. To tackle the nonconvex gas diffusion, an improved scenario-based uncertainty set composed of both the historical observation and sampled scenarios is developed, using a Nataf conversion-based data-driven method. Finally, a tailored column-and-constraint generation solution is developed to solve the uncertain problem. Case studies demonstrate that, the robust diffusion control can reduce the air pollution in the meteorological condition uncertainty, and the overall effectiveness and robustness are balanced by the constraint-based robust optimization.

A dynamic GIS space-time diffusion model to tackle COVID-19 emergency

BackgroundSurveillance and containment of the spread of COVID-19 requires the use of advanced geographic information science and technology (GIS&T) to map the spread and eventually to guide interventions. A dynamic space-time diffusion model in a GIS environment was developed and succesfully tested in Rome, Italy.MethodsInformation on cases of SARS-CoV-2 infection confirmed by molecular diagnostics from Feb 25 to Sep 26 2020 (collected by a large Local Health Unit of Rome, Italy) was used to test a GIS simulator model able to monitor the spatial diffusion and temporal evolution of the spread of the disease. Data included information on: sex, date and place of birth, healthcare facility of hospitalization, date of notification, start date and end date of isolation, date of recovery (both clinical and laboratory confirmed), residence address.ResultsGlobally, 3,056 cases were geocoded and analysed. The spatio-temporal analysis of the first 45 days since 25 Feb 2020 shows that the spread of COVID-19 was very fast (1,230 cases recorded on 11 Apr) and spatially widespread. Number of cases was highest in the city centre with clusters, thickets and axes in different sub-municipal areas. A slowdown occurred the following month, confirming the positive effect of the lockdown. This effect continued until 11 Jun with a small increase in the number of cases (+10.9%). The period up to 26 Sep is paradigmatic of the second wave, with a continuous increase in cases that spread from the city centre to the suburbs.ConclusionsUsing geocoding process and a detailed GIS mapping it is possible to identify streets, buildings and census sections where the number of cases is high and tends to increase rapidly and, at the same time, it is possible to distinguish clusters and axes that should be kept immediately under special observation as potential pools of super-diffusion. Development of its use in near-real time could bring significant advantages in controlling the spread of COVID-19.Key messages The use of GIS technology is fundamental for mapping the spread of COVID-19.A greater effort should be made by institutions to increase the digitisation of health data and the possibility of using them for both research and surveillance purposes.

DYNAMIC SPACE-TIME DIFFUSION SIMULATOR IN A GIS ENVIRONMENT TO TACKLE THE COVID-19 EMERGENCY. TESTING A GEOTECHNOLOGICAL APPLICATION IN ROME

In this paper, we start from a contextualization about the measures used to contain the COVID-19 diffusion and the need to promote geotechnological proposals, data sharing and homogenous centralised systems for data collection and analysis. Successively, we present the “Dynamic Space-Time Diffusion Simulator in a GIS Environment to Tackle the COVID-19 Emergency” that we have elaborated on the basis of the data provided by the UOC Hygiene and Public Health Service – Local Health Unit Rome 1. Particularly, after describing the main technical process able to predispose the dynamic simulator, we underline the possible added value that it can provide in terms of infection surveillance and monitoring, precision preparedness, support to decision making and territorial screening. For this demonstrative application, we have extracted from the simulator some groups of four digital screenshots which are able to show synoptic photographs in temporal perspective concerning the total number of cases of COVID-19 in Rome (Italy) for the period February 25th-September 26th. Specifically we have selected:-four screenshots for the period February 25th-June 11th, to provide significant evidence about the first three months and a half;-four screenshots for the period March 1st – March 29th, to add an insight into the geographically and statistically meaningful month of March;-four screenshots for the period June 12th-September 26th, to supply an efficacious geovisualisation of the last three months and a half available;-four screenshots for the period February 25th-September 26th, to show a cumulative elaboration aimed at geolocating all the cases recorded in the seven months examined;-four screenshots for the period March 26th-September 26th, with a distinction about the first and second data sets, for a detailed (cumulative) zoom. This simulator, elaborated for the COVID-19 emergency, can be replicated in any circumstance for which specific data and information are available for the scientific community, shared and progressively updated in order to provide a productive contribution to the identification of serious infectious disease clusters, patterns and trends, and quickly respond to specific needs. © 2021, Asociatia Geographia Technica. All rights reserved.

Variational Bayesian inversion for the reaction coefficient in space-time nonlocal diffusion equations

In the paper, a variational Bayesian method is used to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data. To show the posterior measure to be well-defined, we rigorously prove that the forward operator is continuous with respect to the unknown reaction field. Then, gradient-based prior information is proposed to explore oscillation features in the reaction coefficient. Moreover, the Bayesian inverse problem is shown to be well-posed in Hellinger distance. To accurately characterize the posterior density using uncorrelated samples, an efficient variational Bayesian method is used to estimate the reaction coefficient in the nonlocal models. A few numerical results are presented to illustrate the efficacy of the proposed approach and confirm some theoretic discoveries.

A new generalized Hilfer-type fractional derivative with applications to space-time diffusion equation

This paper is concerned to present and apply a new generalized fractional derivative, that is the Generalized Hilfer-type (GH) fractional derivative. This derivative unifies various previously defined fractional derivatives of the types Hilfer-Katugampola, Hilfer-Hadamard, Caputo-Hadamard, Hadamard, Hilfer, Riemann-Liouville, Caputo etc into a single form. The mellin transform of this new GH fractional derivative is obtained. As an application, a generalized space-time fractional diffusion equation is constructed using the newly obtained GH fractional derivative in the time-variable and the already existing Riesz fractional derivative in the space-variable. For the solution of this new generalized fractional diffusion equation, the similarity transformations and the mellin transform methods are used where an explicit solution in terms of the Fox’s H-function is derived. The diffusive behavior and the role of various parameters involved in the newly obtained GH fractional derivative are described with the help of 3D plots.

Fatal self-injury in the United States, 1999-2018: Unmasking a national mental health crisis.

BackgroundSuicides by any method, plus ‘nonsuicide’ fatalities from drug self-intoxication (estimated from selected forensically undetermined and ‘accidental’ deaths), together represent self-injury mortality (SIM)—fatalities due to mental disorders or distress. SIM is especially important to examine given frequent undercounting of suicides amongst drug overdose deaths. We report suicide and SIM trends in the United States of America (US) during 1999–2018, portray interstate rate trends, and examine spatiotemporal (spacetime) diffusion or spread of the drug self-intoxication component of SIM, with attention to potential for differential suicide misclassification.MethodsFor this state-based, cross-sectional, panel time series, we used de-identified manner and underlying cause-of-death data for the 50 states and District of Columbia (DC) from CDC's Wide-ranging Online Data for Epidemiologic Research. Procedures comprised joinpoint regression to describe national trends; Spearman's rank-order correlation coefficient to assess interstate SIM and suicide rate congruence; and spacetime hierarchical modelling of the ‘nonsuicide’ SIM component.FindingsThe national annual average percentage change over the observation period in the SIM rate was 4.3% (95% CI: 3.3%, 5.4%; p<0.001) versus 1.8% (95% CI: 1.6%, 2.0%; p<0.001) for the suicide rate. By 2017/2018, all states except Nebraska (19.9) posted a SIM rate of at least 21.0 deaths per 100,000 population—the floor of the rate range for the top 5 ranking states in 1999/2000. The rank-order correlation coefficient for SIM and suicide rates was 0.82 (p<0.001) in 1999/2000 versus 0.34 (p = 0.02) by 2017/2018. Seven states in the West posted a ≥ 5.0% reduction in their standardised mortality ratios of ‘nonsuicide’ drug fatalities, relative to the national ratio, and 6 states from the other 3 major regions a >6.0% increase (p<0.05).InterpretationDepiction of rising SIM trends across states and major regions unmasks a burgeoning national mental health crisis. Geographic variation is plausibly a partial product of local heterogeneity in toxic drug availability and the quality of medicolegal death investigations. Like COVID-19, the nation will only be able to prevent SIM by responding with collective, comprehensive, systemic approaches. Injury surveillance and prevention, mental health, and societal well-being are poorly served by the continuing segregation of substance use disorders from other mental disorders in clinical medicine and public health practice.FundingThis study was partially funded by the National Centre for Injury Prevention and Control, US Centers for Disease Control and Prevention (R49CE002093) and the US National Institute on Drug Abuse (1UM1DA049412–01; 1R21DA046521-01A1).

Effects of Surface Viscosity on Breakup of Viscous Threads.

In addition to surface tension lowering and Marangoni stresses, surfactants also induce surface rheological effects when they deform against themselves at fluid interfaces. Because surface viscosities are functions of surfactant concentration, surface rheological stresses can compete with capillary, Marangoni, and bulk stresses in surfactant-laden free surface flows with breakup. To elucidate the effects of surface rheology, we examine the breakup of a Stokes thread covered with a monolayer of insoluble surfactant when either surfactants are convected away from the space-time singularity or diffusion is dominant. Surprisingly, in both limits, surface rheological effects always enter the dominant balance of forces and alter the thread's thinning rate. Moreover, if surfactants are convected away from the singularity, we provide an analytical expression for thinning rate that explicitly depends on surface rheological parameters, providing a simple route for measuring surface viscosity.

A comprehensive framework for studying diffusion patterns of imported dengue with individual-based movement data

ABSTRACTInternational communication and global cooperation have greatly accelerated the worldwide spread of dengue fever, increasing the impact of imported cases on dengue outbreaks in non-naturally endemic areas. Existing studies mostly focus on describing the quantitative relationship between imported cases and local transmission but ignore the space-time diffusion mode of imported cases under the influence of individual mobility. In this paper, we propose a comprehensive framework at a fine scale to establish the disease transmission network and a mathematical model, which constructs ‘source-sink’ links between the imported and indigenous cases on a regular grid with a spatial resolution of 1 km to explore the diffusion pattern and spatiotemporal heterogeneity of imported cases. An application to Guangzhou, China, reveals the main flow and transmission path of imported cases under the influence of human movement and identifies the spatiotemporal distribution of transmission speed according to the time lag of each source-sink link. In addition, we demonstrate that using individual-based movement data and socio-economic factors to study human mobility and imported cases can help to understand the driving forces of dengue spread. Our research provides a comprehensive framework for the analysis of early dengue transmission patterns with benefits to similar urban applications.

Synergetic Power-Gas Flow With Space-Time Diffusion Control of Air Pollutants Using a Convex Multi-Objective Optimization

Accurate and effective control mode of pollutant emissions is the key to alleviate the environmental impacts of energy supply systems. For this purpose, this paper establishes a precise space-time diffusion (STD) model of air pollutants associated with emission sources and atmosphere environment factors, and proposes a novel STD control of air pollutant emissions. Compared to the traditional emission quantity control, the proposed STD control takes the air pollutant diffusion and space-varying environmental protection demands into account, which can truly reduce the harm of air pollutant emissions. In view of an increasing interdependence between power system and natural gas system, a novel multi-objective optimal power-gas flow (MOPGF) with STD control of air pollutants is constructed. To solve the MOPGF with conflicting objectives and nonconvex gas transmission constraints, a generalized membership optimization (GMO) incorporating penalty convex-concave procedure (PCCP) based convexification is developed. GMO aims to coordinate conflicting objectives and PCCP convexifies the nonconvex gas dynamics, such that a high-quality solution can be obtained. Finally, simulations are carried out to verify the effectiveness of the STD control to reduce the influence of generation emissions, and the superiority of the convex multi-objective optimization for the MOPGF.

Modeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments

In the Fourier heat conduction equation, when the flux definition is expressed as the product of a constant diffusivity and the temperature gradient, the characteristic length scale evolves as the square root of time. However, if we replace the 1st order transient and gradient terms in the Fourier equation with fractional derivatives and/or define a non-linear spatially dependent diffusivity, it is possible to generate an anomalous space-time scaling, i.e., a scaling where the time exponent differs from the expected value of 1/2. To compare and contrast the possible consequences of using fractional calculus along with a non-linear flux, we investigate a space-time fractional heat diffusion equation that involves a non-linear diffusivity. Following presentation of the governing non-linear fractional equation, we arrive at a space-time scaling that accounts for the combined anomalous contributions of memory (fractional derivative in time), non-locality (fractional derivative in space), and a non-linear diffusivity. We demonstrate how this scaling can manifest in a physical setting by considering the analytical solution of a non-linear fractional space-time diffusion equation, a limit case Stefan problem related to moisture infiltration into a porous media. A direct physically realizable simulation of this process shows how the anomalous space-time scaling is explicitly related to measures of both the memory and non-linearity in the system. Overall, the findings from this work clearly show how the definition of a non-linear diffusivity might contribute to anomalous diffusion behavior and suggests that, in modeling a particular observation, the roles of fractional derivatives and a suitably defined non-linear diffusivity are interchangeable.

Numerical simulation of the nuclear reactors kinetics behavior using method of lines based on the GDQ method

In this paper, for the first time, numerical technique using Method of Lines (MOL) based on the Generalized Differential Quadrature method (GDQ) is developed and applied for two energy groups of reactor kinetics and one group of precursor delayed neutrons. Also, the presented method (MOL-GDQ) has been developed for 3D transient benchmark nuclear reactor with two neutron energy groups and six delayed precursor groups. The basic idea of the MOL is to replace the spatial (boundary value) derivatives in the Partial Differential Equations (PDE) with algebraic approximations. In other words, with only one remaining independent variable, we have a system of Ordinary Differential Equations (ODEs) that approximates the original PDE. The advantages of the GDQ method lie in its easy use and flexibility with regard to arbitrary grid spacing. Compared to the conventional low-order numerical techniques such as the finite element and finite difference methods, the GDQ method can yield accurate solutions with relatively much fewer grid points. The numerical technique is applied to three-dimensional space-time neutron diffusion equations with average one group of delayed neutrons in the different nuclear reactors. Also, the presented method (MOL-GDQ) is developed for 3D transient benchmark nuclear reactor with two neutron energy groups and six delayed precursor groups. Simulation results are presented to demonstrate the effectiveness of the proposed method in terms of performance and stability. The results of numerical technique are discussed and compared with the results of traditional methods and strong citations to demonstrate the accuracy of it.

Relativistic analysis of stochastic kinematics.

The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient measured in a frame Σ moving with velocity w with respect to the rest frame of the stochastic process is inversely proportional to the third power of the Lorentz factor γ(w)=(1-w^{2}/c^{2})^{-1/2}. Subsequently, higher-dimensional processes are analyzed and it is shown that the diffusivity tensor in a moving frame becomes nonisotropic: The diffusivities parallel and orthogonal to the velocity of the moving frame scale differently with respect to γ(w). The analysis of discrete space-time diffusion processes permits one to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed.

Export diversification and economic development: A dynamic spatial data analysis

AbstractThis paper contributes to the empirical literature on the relationship between “export variety” (export diversification) and economic development by relaxing the assumption of cross‐country independence and allowing for spatial diffusion of shocks in observed and unobserved factors. Export variety is measured for a balanced panel of 114 countries (1992–2012) using very detailed information on their exports (HS 6‐digit product level). The estimation results of a dynamic spatial panel data model confirm the relevance of spatial network effects in export diversification: indirect effects (spatial spillovers) strongly reinforce direct effects, while spatial proximity to large countries accelerates the diversification process. In about 10 years the whole space–time diffusion of the diversification shock is widely completed. We reveal that the long‐run spillover impact from European countries is much higher than from other countries such as the United States, Japan, or the BRICS (Brazil, Russia, India, China, and South Africa).

Asymptotically compatible schemes for space-time nonlocal diffusion equations

In this paper, we study a space-time nonlocal diffusion model that reduces to the classical diffusion equation in the local limit. Firstly, we show the uniqueness and existence of the weak solution of the nonlocal model, and study the local limit of the nonlocal model as horizon parameters approach zero. Particularly, it is shown that the convergence is uniform at a rate of O(δ+σ2), under certain regularity assumptions on initial and source data. Next we propose a fully discrete scheme, by exploiting the quadrature-based finite difference method in time and the Fourier spectral method in space, and show its stability. The numerical scheme is proved to be asymptotically compatible so that it preserves the local limiting behavior at the discrete level. Numerical experiments are provided to illustrate the theoretical results.