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Physics, Volume 3, Issue 3 (September 2021) – 19 articles

Cover Story (view full-size image): The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically, the problem is reduced to calculating the “energy” of the ground state in the Schrödinger equation with a complicated potential. A general method is proposed and applied to obtain the bottom-part spectrum of such equations based on the approximation of the potential by square wells. A possible generalization of the approach to other types of nonlinear diffusion equations is discussed. View this paper
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Article
Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle
Physics 2021, 3(3), 790-798; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030049 - 19 Sep 2021
Viewed by 665
Abstract
The classical uncertainty principle inequalities are imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle is reformulated in terms of proper space–time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression [...] Read more.
The classical uncertainty principle inequalities are imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle is reformulated in terms of proper space–time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirms the need for a minimum length of space–time line element in the geodesic, which depends on a Lorentz-covariant geodesic-derived scalar. In agreement with quantum gravity theories, GeUP imposes a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone is found around the singularity where uncertainty in space-time diverged to infinity. Full article
(This article belongs to the Section Astronomy, Astrophysics and Planetology)
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Article
Local Regions with Expanding Extra Dimensions
Physics 2021, 3(3), 781-789; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030048 - 13 Sep 2021
Viewed by 420
Abstract
In this paper possible spatial domains, containing expanding extra dimensions, are studied. It is demonstrated that these domains are predicted in the framework of f(R) gravity (where R is the scalar curviture) and could appear due to quantum fluctuations during [...] Read more.
In this paper possible spatial domains, containing expanding extra dimensions, are studied. It is demonstrated that these domains are predicted in the framework of f(R) gravity (where R is the scalar curviture) and could appear due to quantum fluctuations during inflation. The interior of the domains is shown to be characterized by the multidimensional curvature ultimately tending to zero and a slowly growing size of the extra dimensions. Full article
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Article
Statistical Scrutiny of Particle Spectra in ep Collisions
Physics 2021, 3(3), 757-780; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030047 - 08 Sep 2021
Viewed by 403
Abstract
Charged particle multiplicity distributions in positron–proton deep inelastic scattering at a centre-of-mass energy s = 300 GeV, measured in the hadronic centre-of-mass frames and in different pseudorapidity windows are studied in the framework of two statistical distributions, the shifted Gompertz distribution and the [...] Read more.
Charged particle multiplicity distributions in positron–proton deep inelastic scattering at a centre-of-mass energy s = 300 GeV, measured in the hadronic centre-of-mass frames and in different pseudorapidity windows are studied in the framework of two statistical distributions, the shifted Gompertz distribution and the Weibull distribution. Normalised moments, normalised factorial moments and the H-moments of the multiplicity distributions are determined. The phenomenon of oscillatory behaviour of the counting statistics and the Koba-Nielsen-Olesen (KNO) scaling behaviour are investigated. This is the first such analysis using these data. In addition, projections of the two distributions for the expected average charged multiplicities obtainable at the proposed future ep colliders. Full article
(This article belongs to the Special Issue Statistical Approaches in High Energy Physics)
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Communication
Enhanced Chiral Mie Scattering by a Dielectric Sphere within a Superchiral Light Field
Physics 2021, 3(3), 747-756; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030046 - 02 Sep 2021
Viewed by 399
Abstract
A superchiral field, which can generate a larger chiral signal than circularly polarized light, is a promising mechanism to improve the capability to characterize chiral objects. In this paper, Mie scattering by a chiral sphere is analyzed based on the T-matrix method. The [...] Read more.
A superchiral field, which can generate a larger chiral signal than circularly polarized light, is a promising mechanism to improve the capability to characterize chiral objects. In this paper, Mie scattering by a chiral sphere is analyzed based on the T-matrix method. The chiral signal by circularly polarized light can be obviously enhanced due to the Mie resonances. By employing superchiral light illumination, the chiral signal is further enhanced by 46.8% at the resonance frequency. The distribution of the light field inside the sphere is calculated to explain the enhancement mechanism. The study shows that a dielectric sphere can be used as an excellent platform to study the chiroptical effects at the nanoscale. Full article
(This article belongs to the Special Issue Dedication to Professor Michael Tribelsky: 50 Years in Physics)
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Article
Classical Chaos Described by a Density Matrix
Physics 2021, 3(3), 739-746; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030045 - 01 Sep 2021
Viewed by 528
Abstract
In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is [...] Read more.
In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states. Full article
(This article belongs to the Section Statistical Physics and Nonlinear Phenomena)
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Article
Vadasz Number Effects on Convection in a Vertical Rotating Porous Layer, Placed Far from Axis of Rotation, and Subjected to Internal Heat Generation and Centrifugal Jitter
Physics 2021, 3(3), 728-738; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030044 - 01 Sep 2021
Viewed by 552
Abstract
The flow and heat transfer in a rotating vertical porous layer, placed far from the axis of rotation, and subjected to internal heat generation and centrifugal jitter, is considered. The linear stability theory is used to determine the convection threshold, in terms of [...] Read more.
The flow and heat transfer in a rotating vertical porous layer, placed far from the axis of rotation, and subjected to internal heat generation and centrifugal jitter, is considered. The linear stability theory is used to determine the convection threshold, in terms of the critical Rayleigh number. Typical liquids used in engineering applications and heavy liquid metals are used to demonstrate conditions at which the Vadasz number is sufficiently small to warrant the retention of the time derivative in the momentum equation. When considering low amplitude and high frequency approximation, the results show that vibration has a stabilizing effect on the onset of convection. The impact of increasing the Vadasz number is to stabilize the convection, in addition to reducing the transition point from synchronous to subharmonic solutions. In summary, when the Vadasz number is large, centrifugal jitter has no impact on the convection stability criteria. In contrast, when the Vadasz number is small, centrifugal jitter impacts the convection stability criteria. Full article
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Article
Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential
Physics 2021, 3(3), 715-727; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030043 - 30 Aug 2021
Viewed by 494
Abstract
The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically, the problem is reduced to the calculation of the “energy” of the ground state in the Schrödinger equation with a complicated potential. A [...] Read more.
The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically, the problem is reduced to the calculation of the “energy” of the ground state in the Schrödinger equation with a complicated potential. A general method to obtain the bottom-part spectrum of such equations based on the approximation of the potential by square wells is proposed and applied. Possible generalization of the approach to other types of nonlinear diffusion equations is discussed. Full article
(This article belongs to the Special Issue Dedication to Professor Michael Tribelsky: 50 Years in Physics)
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Article
Effects of Quantum Metric Fluctuations on the Cosmological Evolution in Friedmann-Lemaitre-Robertson-Walker Geometries
Physics 2021, 3(3), 689-714; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030042 - 24 Aug 2021
Cited by 2 | Viewed by 514
Abstract
In this paper, the effects of the quantum metric fluctuations on the background cosmological dynamics of the universe are considered. To describe the quantum effects, the metric is assumed to be given by the sum of a classical component and a fluctuating component [...] Read more.
In this paper, the effects of the quantum metric fluctuations on the background cosmological dynamics of the universe are considered. To describe the quantum effects, the metric is assumed to be given by the sum of a classical component and a fluctuating component of quantum origin . At the classical level, the Einstein gravitational field equations are equivalent to a modified gravity theory, containing a non-minimal coupling between matter and geometry. The gravitational dynamics is determined by the expectation value of the fluctuating quantum correction term, which can be expressed in terms of an arbitrary tensor Kμν. To fix the functional form of the fluctuation tensor, the Newtonian limit of the theory is considered, from which the generalized Poisson equation is derived. The compatibility of the Newtonian limit with the Solar System tests allows us to fix the form of Kμν. Using these observationally consistent forms of Kμν, the generalized Friedmann equations are obtained in the presence of quantum fluctuations of the metric for the case of a flat homogeneous and isotropic geometry. The corresponding cosmological models are analyzed using both analytical and numerical method. One finds that a large variety of cosmological models can be formulated. Depending on the numerical values of the model parameters, both accelerating and decelerating behaviors can be obtained. The obtained results are compared with the standard ΛCDM (Λ Cold Dark Matter) model. Full article
(This article belongs to the Special Issue New Advances in Quantum Geometry)
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Article
Splitting of the Magnetic Loss Peak of Composites under External Magnetic Field
Physics 2021, 3(3), 678-688; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030041 - 23 Aug 2021
Viewed by 420
Abstract
Composite materials filled with ferromagnetic inclusions are useful in the development of various microwave devices. The performance of such devices is determined both by material properties (such as the saturation magnetization and the permeability) and by the demagnetization effects. The paper is devoted [...] Read more.
Composite materials filled with ferromagnetic inclusions are useful in the development of various microwave devices. The performance of such devices is determined both by material properties (such as the saturation magnetization and the permeability) and by the demagnetization effects. The paper is devoted to the study of the demagnetization effect on the permeability measurements of composites under external magnetic bias. The microwave permeability of composites filled with flake sendust (Fe-Si-Al alloy) particles is measured as a function of frequency and the external magnetic field. The measurements are carried out by the Nicolson–Ross–Weir technique in a 7/3 coaxial line in the frequency range of 0.1 to 20 GHz by a vector network analyzer. It is found that the magnetic loss peak is split under external fields of more than 1.5 kOe. The main aim of this paper is to study the causes of this splitting and to interpret the observed magnetic loss peaks. To study this effect, the samples of various thicknesses and the samples with isotropic and anisotropic orientations of particles are measured. The particles in the anisotropic samples are oriented by a strong uniform magnetic field. At a small fraction of inclusions, the permanent magnetic field is demagnetized on the individual particles rather than the whole sample. The splitting of the magnetic loss peak of the isotropic sample is caused by different orientations of particles in the sample. At a high fraction of inclusions, the permanent magnetic field is demagnetized on the whole sample and the magnetic loss peak of the isotropic sample is not split. The saturation magnetization of the material is found by measurements under the external magnetic field of the anisotropic sample. Full article
(This article belongs to the Section Applied Physics)
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Article
Application of Regge Theory to Astronomical Objects
Physics 2021, 3(3), 669-677; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030040 - 15 Aug 2021
Viewed by 435
Abstract
Using the model based on the Regge-like laws, new analytical formulas are obtained for the moment of inertia, the rotation frequency, and the radius of astronomical non-exotic objects (planets, stars, galaxies, and clusters of galaxies). The rotation frequency and moment of inertia of [...] Read more.
Using the model based on the Regge-like laws, new analytical formulas are obtained for the moment of inertia, the rotation frequency, and the radius of astronomical non-exotic objects (planets, stars, galaxies, and clusters of galaxies). The rotation frequency and moment of inertia of a neutron star and the observable Universe are estimated. The estimates of the average numbers of stars and galaxies in the observable Universe are given. The Darwin instability effect in the binary systems (di-planets, di-stars, and di-galaxies) is also analyzed. Full article
(This article belongs to the Section Astronomy, Astrophysics and Planetology)
Article
Scaling Conjecture Regarding the Number of Unknots among Polygons of N≫1 Edges
Physics 2021, 3(3), 664-668; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030039 - 12 Aug 2021
Viewed by 431
Abstract
The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of N1 edges has only logarithmic, and not power law corrections to the known leading exponential law: [...] Read more.
The conjecture is made based on a plausible, but not rigorous argument, suggesting that the unknot probability for a randomly generated self-avoiding polygon of N1 edges has only logarithmic, and not power law corrections to the known leading exponential law: Punknot(N)expN/N0+o(lnN) with N0 being referred to as the random knotting length. This conjecture is consistent with the numerical result of 2010 by Baiesi, Orlandini, and Stella. Full article
(This article belongs to the Special Issue Dedication to Professor Michael Tribelsky: 50 Years in Physics)
Article
Design of Switchable On/Off Subpixels for Primary Color Generation Based on Molybdenum Oxide Gratings
Physics 2021, 3(3), 655-663; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030038 - 12 Aug 2021
Viewed by 814
Abstract
Structural color emerges from the interaction of light with structured matter when its dimension is comparable to the incident wavelength. The reflected color can be switched by controlling such interaction with materials whose properties can be changed through external stimuli such as electrical, [...] Read more.
Structural color emerges from the interaction of light with structured matter when its dimension is comparable to the incident wavelength. The reflected color can be switched by controlling such interaction with materials whose properties can be changed through external stimuli such as electrical, optical, or thermal excitation. In this research, a molybdenum oxide (MoOx) reflective grating to get a switchable on/off subpixel is designed and analyzed. The design is based on subpixel on and off states that could be controlled through the oxidation degree of MoOx. A suitable combination of three of these subpixels, optimized to get a control of primary colors, red, green, and blue, can lead to a pixel which can cover a wide range of colors in the color space for reflective display applications. Full article
(This article belongs to the Special Issue Dedication to Professor Michael Tribelsky: 50 Years in Physics)
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Review
Luminosity Measurements at the LHC at CERN Using Medipix, Timepix and Timepix3 Devices
Physics 2021, 3(3), 579-654; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030037 - 11 Aug 2021
Viewed by 383
Abstract
The precise determination of the luminosity is essential for many analyses in physics based on the data from the particle accelerator Large Hadron Collider (LHC) at CERN. There are different types of detectors used for the luminosity measurements. The focus of this review [...] Read more.
The precise determination of the luminosity is essential for many analyses in physics based on the data from the particle accelerator Large Hadron Collider (LHC) at CERN. There are different types of detectors used for the luminosity measurements. The focus of this review is on luminosity measurements with hybrid-pixel detectors and the progress made over the past decade. The first generations of detectors of the Medipix and Timepix families had frame-based readout, while Timepix3 has a quasi-continuous readout. The applications of the detectors are manifold, and in particular, the detectors have been operated in the harsh environment of the LHC. The excellent performance in detecting high fluxes of elementary particles made these detectors ideal tools to measure the delivered luminosity resulting from proton–proton collisions. Important aspects of this review are the performance improvements in relative luminosity measurements from one detector generation to another, the long-term stability of the measurements, absolute luminosity measurements, material activation (radiation-induced) corrections, and the measurement of luminosity from neutron counting. Rather than bunch-average luminosity provided by previous detector generations, owing to the excellent time-resolution, Timepix3 measured the luminosity of individual proton bunches that are 25 ns apart. This review demonstrates the large progress in the precision of luminosity measurements during LHC Run-1 and Run-2 operations using hybrid-pixel detectors, and thus their importance for luminosity measurements in the future of LHC operations. Full article
(This article belongs to the Section High Energy Physics)
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Article
Critical Angle Refractometry for Lossy Media with a Priori Known Extinction Coefficient
Physics 2021, 3(3), 569-578; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030036 - 03 Aug 2021
Viewed by 809
Abstract
Critical angle refractometry is an established technique for determining the refractive index of liquids and solids. For transparent samples, the critical angle refractometry precision is limited by incidence angle resolution. For lossy samples, the precision is also affected by reflectance measurement error. In [...] Read more.
Critical angle refractometry is an established technique for determining the refractive index of liquids and solids. For transparent samples, the critical angle refractometry precision is limited by incidence angle resolution. For lossy samples, the precision is also affected by reflectance measurement error. In the present study, it is demonstarted that reflectance error can be practically eliminated, provided that the sample’s extinction coefficient is a priori known with sufficient accuracy (typically, better than 5%) through an independent measurement. Then, critical angle refractometry can be as precise with lossy media as with transparent ones. Full article
(This article belongs to the Section Applied Physics)
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Communication
Cosmological Formation of (2 + 1)-Dimensional Soliton Structures in Models Possessing Potentials with Local Peaks
Physics 2021, 3(3), 563-568; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030035 - 02 Aug 2021
Viewed by 447
Abstract
Production of domain walls and string-like solitons in the model with two real scalar fields and potential with at least one saddle point and a local maximum is considered. The model is regarded as 2-dimensional spatial slices of 3-dimensional entire structures. It is [...] Read more.
Production of domain walls and string-like solitons in the model with two real scalar fields and potential with at least one saddle point and a local maximum is considered. The model is regarded as 2-dimensional spatial slices of 3-dimensional entire structures. It is shown that, in the early Universe, both types of solitons may appear. In addition, the qualitative estimate of the domain walls and strings formation probabilities is presented. It is found that the probability of the formation of string-like solitons is suppressed compared to that of domain walls. Full article
(This article belongs to the Special Issue Beyond the Standard Models of Physics and Cosmology)
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Article
Spatially Developing Modes: The Darcy–Bénard Problem Revisited
Physics 2021, 3(3), 549-562; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030034 - 30 Jul 2021
Viewed by 375
Abstract
In this paper, the instability resulting from small perturbations of the Darcy–Bénard system is explored. An analysis based on time–periodic and spatially developing Fourier modes is adopted. The system under examination is a horizontal porous layer saturated by a fluid. The two impermeable [...] Read more.
In this paper, the instability resulting from small perturbations of the Darcy–Bénard system is explored. An analysis based on time–periodic and spatially developing Fourier modes is adopted. The system under examination is a horizontal porous layer saturated by a fluid. The two impermeable and isothermal plane boundaries are considered to have different temperatures, so that the porous layer is heated from below. The spatial instability for the system is defined by taking into account both the spatial growth rate of the perturbation modes and their propagation direction. A comparison with the neutral stability condition determined by using the classical spatially periodic and time–evolving Fourier modes is performed. Finally, the physical meaning of the concept of spatial instability is discussed. In contrast to the classical analysis, based on spatially periodic modes, the spatial instability analysis, involving time–periodic Fourier modes, is found to lead to the conclusion that instability occurs whenever the Rayleigh number is positive. Full article
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Article
A Local and Time Resolution of the COVID-19 Propagation—A Two-Dimensional Approach for Germany Including Diffusion Phenomena to Describe the Spatial Spread of the COVID-19 Pandemic
Physics 2021, 3(3), 536-548; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030033 - 07 Jul 2021
Cited by 1 | Viewed by 505
Abstract
The understanding of factors that affect the dissemination of a viral infection is fundamental to help combat it. For instance, during the COVID-19 pandemic that changed the lives of people all over the world, one observes regions with different incidences of cases. One [...] Read more.
The understanding of factors that affect the dissemination of a viral infection is fundamental to help combat it. For instance, during the COVID-19 pandemic that changed the lives of people all over the world, one observes regions with different incidences of cases. One can speculate that population density might be one of the variables that affect the incidence of cases. In populous areas, such as big cities or congested urban areas, higher COVID-19 incidences could be observed than in rural regions. It is natural to think that if population density is such an important factor, then a gradient or difference in population density might lead to a diffusion process that will proceed until equilibrium is reached. The aim of this paper consists of the inclusion of a diffusion concept into the COVID-19 modeling. With this concept, one covers a gradient-driven transfer of the infection next to epidemic growth models (SIR-type models). This is discussed for a certain period of the German situation based on the quite different incidence data for the different federal states of Germany. With this ansatz, some phenomena of the actual development of the pandemic are found to be confirmed. The model provides a possibility to investigate certain scenarios, such as border-crossings or local spreading events, and their influence on the COVID-19 propagation. The resulting information can be a basis for the decisions of politicians and medical persons in charge of managing a pandemic. Full article
(This article belongs to the Special Issue Physics Methods in Coronavirus Pandemic Analysis)
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Review
Scintillation in Low-Temperature Particle Detectors
Physics 2021, 3(3), 473-535; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030032 - 01 Jul 2021
Viewed by 820
Abstract
Inorganic crystal scintillators play a crucial role in particle detection for various applications in fundamental physics and applied science. The use of such materials as scintillating bolometers, which operate at temperatures as low as 10 mK and detect both heat (phonon) and scintillation [...] Read more.
Inorganic crystal scintillators play a crucial role in particle detection for various applications in fundamental physics and applied science. The use of such materials as scintillating bolometers, which operate at temperatures as low as 10 mK and detect both heat (phonon) and scintillation signals, significantly extends detectors performance compared to the conventional scintillation counters. In particular, such low-temperature devices offer a high energy resolution in a wide energy interval thanks to a phonon signal detection, while a simultaneous registration of scintillation emitted provides an efficient particle identification tool. This feature is of great importance for a background identification and rejection. Combined with a large variety of elements of interest, which can be embedded in crystal scintillators, scintillating bolometers represent powerful particle detectors for rare-event searches (e.g., rare alpha and beta decays, double-beta decay, dark matter particles, neutrino detection). Here, we review the features and results of low-temperature scintillation detection achieved over a 30-year history of developments of scintillating bolometers and their use in rare-event search experiments. Full article
(This article belongs to the Special Issue Radiation Spectroscopy with Solid Scintillators for Rare Events)
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Article
SIR-PID: A Proportional–Integral–Derivative Controller for COVID-19 Outbreak Containment
Physics 2021, 3(3), 459-472; https://0-doi-org.brum.beds.ac.uk/10.3390/physics3030031 - 27 Jun 2021
Viewed by 716
Abstract
Ongoing social restrictions, including social distancing and lockdown, adopted by many countries to inhibit spread of the the COVID-19 epidemic, must attempt to find a trade-off between induced economic damage, healthcare system collapse, and the costs in terms of human lives. Applying and [...] Read more.
Ongoing social restrictions, including social distancing and lockdown, adopted by many countries to inhibit spread of the the COVID-19 epidemic, must attempt to find a trade-off between induced economic damage, healthcare system collapse, and the costs in terms of human lives. Applying and removing restrictions on a system with a given latency as represented by an epidemic outbreak (and formally comparable with mechanical inertia), may create critical instabilities, overshoots, and strong oscillations in the number of infected people around the desirable set-point, defined in a practical way as the maximum number of hospitalizations acceptable by a given healthcare system. A good understanding of the system reaction to any change of the input control variable can be reasonably achieved using a proportional–integral–derivative controller (PID), which is a widely used technique in various physics and technological applications. In this paper, this control theory to is proposed to be applied epidemiology, to understand the reaction of COVID-19 propagation to social restrictions and to reduce epidemic damages through the correct tuning of the containment policy. Regarding the synthesis of this interdisciplinary approach, the extended to the susceptible–infectious–recovered (SIR) model name “SIR-PID” is suggested. Full article
(This article belongs to the Special Issue Physics Methods in Coronavirus Pandemic Analysis)
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