Free Particle Research Articles

Thermodynamics of a Brownian particle in a nonconfining potential.

We consider the overdamped Brownian dynamics of a particle starting inside a square potential well which, upon exiting the well, experiences a flat potential where it is free to diffuse. We calculate the particle's probability distribution function (PDF) at coordinate x and time t,P(x,t), by solving the corresponding Smoluchowski equation. The solution is expressed by a multipole expansion, with each term decaying t^{1/2} faster than the previous one. At asymptotically large times, the PDF outside the well converges to the Gaussian PDF of a free Brownian particle. The average energy, which is proportional to the probability of finding the particle inside the well, diminishes as E∼1/t^{1/2}. Interestingly, we find that the free energy of the particle, F, approaches the free energy of a freely diffusing particle, F_{0}, as δF=F-F_{0}∼1/t, i.e., at a rate faster than E. We provide analytical and computational evidence that this scaling behavior of δF is a general feature of Brownian dynamics in nonconfining potential fields. Furthermore, we argue that δF represents a diminishing entropic component which is localized in the region of the potential, and which diffuses away with the spreading particle without being transferred to the heat bath.

Susceptible-Infected-Recovered model study using free particle dynamics

A study on the epidemiologic Susceptible-Infected-Recovered (SIR) model is presented using free particle dynamics. The study is performed using a computational model consisting of randomly allocated particles in a closed domain which are free to move inrandom directions with the ability to collide into each other. The transmission rules for the particle–particle interactions are based on the main viral infection mechanisms, resulting in real–time results of the number of susceptible, infected, and recovered particles within a population of N= 200 particles. The results are qualitatively compared with a differential equation SIR model in terms of the transmission rate β, recovery rate γ, and the basic reproductive number R0, yielding overall good results. The effect of the particle density ρ on R0 is also studied to analyze how an infectious disease spreads over different types of populations. The versatility of the proposed free–particle–dynamics SIR model allows to simulate different scenarios, such as social distancing, commonly referredto as quarantine, no social distancing measures, and a mixture of the former and the latter. It is found that by implementing early relaxation of social distancing measures before the number of infected particles reaches zero, could lead to subsequent outbreaks such as the particular events observed in different countries due to the ongoing COVID–19 health crisis

Surface Charge Accumulation on a Real Size Epoxy Insulator With Bouncing Metal Particle Under DC Voltage

Free metal particle can affect surface charge accumulation on epoxy insulator installed in gas-insulated transmission line (GIL) or gas-insulated switchgear (GIS), which results in reduction of surface flashover voltage thus threatening the safe operation of the power equipment. This article reports on the charge accumulation behavior on a real size epoxy insulator with bouncing metal particle under DC voltage. A coaxial electrode system was designed for the test under -30 kV, a metal particle-launching system was employed to inject the bouncing linear particle to the insulator surface. The charge distribution was measured by a Kelvin-type probe, and the influence of the particle on the charge accumulation was analyzed with the help of electric field simulation. The results showed that there were mainly two moving patterns of the bouncing particle: the charges were accumulated with speckle shape and the charged particle that hit the edge of electrode had more remarkable influence on the charge accumulation than other particles. It is suggested that the charge distribution feature is dependent upon the collision point and the trajectory of the bouncing particle.

Wave-particle duality and the zitterbewegung

In previous work, the Hamilton-Jacobi equation has been associated with the metrics of general relativity and shown to be a generalized Dirac equation for quantum mechanics. This lends itself to a natural definition of wave-particle duality in quantum mechanics. This theory is now further developed to show that a free spinless quantum particle moving with velocity v obeys the standard wave equation of electro-magnetism. We also discuss the implications for the zitterbewegung problem and its relationship to isotropy. Moreover, it is also shown that for the theory to be consistent, the momentum defined by the Hamilton-.!acobi function presupposes the existence of a universal parameter internal to the system. In the case of particles with mass this invariant can be defined by dX = dt/m(t) where t has the units of time and m = m(t) has the units of mass.

Effect of Annealing at High Temperatures on Microstructural Formation of 15Cr-25Ni Austenitic Stainless Steel Cast

Microstructure of 15Cr-25Ni Austenite stainless steel (ASS) cast formed at various high temperatures has been investigated in this study. The steel was synthesized from scrap materials containing Cr and Ni elements, respectively, about 15 and 25 wt.%. This steel designed to have high corrosion resistance or high-temperature oxidation is a steel candidate for evaporator materials in a waste incinerator, and their operating temperature is used for annealing temperatures in this study. The variation of annealing temperatures was carried out at temperatures of 400, 600, 800, 950, and 1078°C for a time of 30 minutes. The ASS as-cast and annealed have a similar microstructure dominated by the dendritic structure of γ-FeNi austenite with the crystal structure of the face-centered cubic interdendritic region of a eutectic structure containing free particle zone and (Fe, Cr)7C3 island compounds. The short dendritic arm spacings for ASS after annealing at temperatures of 400, 600, and 800°C were measured to have about 61.71, 62.11, and 57.85 mm, respectively. By contrast, the longer dendritic arm spacings were obtained after annealing at temperatures of 950 and 1078°C about 70.29 and 197.04 μm, respectively. In addition to higher micro-strains, the short dendritic arm spacings formed after annealing at 400-800° C lead in a relatively higher hardness profile, whereas the longer dendritic arm spacings obtained after annealing at 950 and 1078°C result in a low hardness profile in the ASS.

Application of Smooth Particle Hydrodynamics to Particular Flow Cases Solved by Saint-Venant Equations

Smoothed particle hydrodynamics (SPH) is a Lagrangian mesh free particle method which has been developed and widely applied to different areas in engineering. Recently, the SPH method has also been used to solve the shallow water equations, resulting in (SPH-SWEs) formulations. With the significant developments made, SPH-SWEs provide an accurate computational tool for solving problems of wave propagation, flood inundation, and wet-dry interfaces. Capabilities of the SPH method to solve Saint-Venant equations have been tested using a SPH-SWE code to simulate different hydraulic test cases. Results were compared to other established and commercial hydraulic modelling packages that use Eulerian approaches. The test cases cover non-uniform steady state profiles, wave propagation, and flood inundation cases. The SPH-SWEs simulations provided results that compared well with other established and commercial hydraulic modeling packages. Nevertheless, SPH-SWEs simulations experienced some drawbacks such as loss of inflow water volume of up to 2%, for 2D flood propagation. Simulations were carried out using an open source solver, named SWE-SPHysics.

A new look at quantal time evolution

Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free particle (with or without spin-orbit interaction). For the simple harmonic oscillator, however, the fact that the wavepacket remains a Gaussian as time progresses has to be inferred on the basis of the skewness and kurtosis as opposed to a study of all the relevant moments.

Coherent states of systems with pure continuous energy spectra

While dealing with a Hamiltonian with continuous spectrum we use a tridiagonal method involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonality measure to show that the obtained coherent states can be recast in the Gazeau-Klauder approach. The Hamiltonian of the $\ell$-wave free particle is treated as an example to illustrate the method.

Parameter Id of Metal Hi-pressure State Equation

Abstract In this study, parameters of the Grüneisen equation of state for GH4169 alloy were calculated based on multi-scale impact technology and first-principles calculation methods. The calculated parameters are consistent with the results of Liu et al., which primarily verifies the accuracy of the method. The AUTODYN software was used for numerical simulation of dynamic plate impact experiments. The parameters of the Grüneisen equation of GH4169 alloy were used as input to verify its accuracy. Comparing and analysing the speed of the free surface particle and the actual experimental measurement point at the same position, it is concluded that the simulated value is consistent with the experimental value. The morphology of the flying piece and the target have the same characteristics, which proves that Grüneisen equation of state parameters obtained by proposed parameter identification method are practical and reliable.

Path integral of the relativistic point particle in Minkowski space

In this article, we analyze the fundamental global and local symmetries involved in the action for the free relativistic point particle in Minkowski space. Moreover, we identify a hidden local symmetry, whose explicit consideration and factorization utilizing of a Fujikawa prescription leads to the construction of relativistic two-point correlation functions that satisfy the Chapman-Kolmogorov identity. By means of a detailed topological analysis, we find three different relativistic correlation functions (orthochronous, spacelike, and Feynman) which are obtained from the exclusive integration of paths within different sectors in Minkowski space. Finally, the connection of this approach to the Feynman checkerboard construction is explored.

Statistical system based on p-adic numbers

We propose statistical systems based on p-adic numbers. In the systems, the Hamiltonian is a standard real number which is given by a map from the p-adic numbers. Therefore we can introduce the temperature as a real number and calculate the thermodynamical quantities like free energy, thermodynamical energy, entropy, specific heat, etc. Although we consider a very simple system, which corresponds to a free particle moving in one dimensional space, we find that there appear the behaviors like phase transition in the system. Usually in order that a phase transition occurs, we need a system with an infinite number of degrees of freedom but in the system where the dynamical variable is given by p-adic number, even if degree of freedom is unity, there might occur the phase transition.

Fully Symmetric Relativistic Quantum Mechanics and Its Physical Implications

A new formulation of relativistic quantum mechanics is presented and applied to a free, massive, and spin-zero elementary particle in the Minkowski spacetime. The reformulation requires that time and space, as well as the timelike and spacelike intervals, are treated equally, which makes the new theory fully symmetric and consistent with the special theory of relativity. The theory correctly reproduces the classical action of a relativistic particle in the path integral formalism, and allows for the introduction of a new quantity called vector-mass, whose physical implications for nonlocality, the uncertainty principle, and quantum vacuum are described and discussed.

A Comparative Investigation of the Intrinsic Mobility of the Natural Body in the Mulla Sadra’s Philosophy with the Continuous Evolution of Quantum “Particles”

In quantum theory, objects (particles) are intrinsically evolving and changing based on the Uncertainty Principle, and the particle state is described by a time-dependent wave packet such that the wave packet corresponding to even a free particle is successively evolvable; a quantum particle has an indefinite and transformative state. In the relativistic Quantum Fields Theory, instead of the concepts of particle and wave packet, a continuous object named “field” with permanent fluctuation in its ground state (the quantum vacuum) is considered as the original concept. In Mulla Sadra’s philosophy, based on the Substantial Motion Theory, the subject of continuous evolution and change in the natural body appears as the intrinsic mobility of the natural body. In this paper, we are going to present a comparative investigation of this topic in the two areas of study mentioned. Initially, we present some necessary topics in physics including the motion of fundamental particles, the motion of quantum objects (quantum non-localizability), Quantum Fields Theory, quantum vacuum, and the creation and annihilation of particles in the Quantum Fields Theory; then, we consider some necessary topics in some fields of study in philosophy about the natural body and the nature and the identity of physical bodies including the ideas and the special opinion of Mulla Sadra. Particularly and with more detail, we pay attention to the Sadrian philosophy about the evolution and intrinsic mobility of the natural body based on his famous Substantial Motion Theory. With a comparative study, and by considering Sadrian philosophy as the fundamental philosophical basis for natural sciences, particularly physics, we try to compare the intrinsic motion of natural bodies in Mulla Sadra’s philosophy with the intrinsic evolution of quantum objects in modern physics. Our research method is based on and uses standard resources including books and research papers about Mulla Sadra’s thoughts and theories and the internationally well-known books and papers on standard quantum physics and quantum fields theory. The conclusive remarks are as follows: The transformation of quantum particles or the creation and annihilation of particles in the theory of quantum fields agrees with this point that the intrinsic mobility of the natural body in Mulla Sadra’s philosophy doesn’t reason on the philosophical existence and/or loss of existence. That a quantum particle doesn’t have a constant status in two consecutive instants is in agreement with the Sadrian Substantial Motion Theory that explains ‘motion’ in the ‘bed’ of time. Finally, we should mention that although the intrinsic motion of the natural body in Mulla Sadra’s philosophy is based on its completion while the creation and annihilation of particles in Quantum Fields Theory may be better compared with the emergence and extinction well known in peripatetic philosophy, by considering the background quantum vacuum field as the main basis of the so-called “particles” which is continuously in evolution and permanent fluctuation and is the basis for the creation and annihilation of particles, the comparison between the modern Quantum Fields Theory and Mulla Sadra’s philosophy is done well.

A standard energy eigenvalue problem for directly solving the stationary states of quantum billiards via boundary integral analysis

By using the series expansions of the Bessel functions for the real and imaginary parts of the free particle Green’s function for the two-dimensional stationary states of quantum billiards, it can be shown that some components of the Green’s function are redundant which can be eliminated to make the boundary integral equation for the wave function of free particles inside quantum billiards independent of the wave numbers. This development leads to a much faster search for the energy eigenvalues of quantum billiards by scanning their wave numbers or the formulation of the standard eigenvalue problem which can directly be solved for the energy eigenvalues. Some numerical examples were used to demonstrate that the proposed technique is accurate, computationally effective and straightforward to be applied in practice.

Quantum Implications of Non-Extensive Statistics

Exploring the analogy between quantum mechanics and statistical mechanics, we formulate an integrated version of the Quantropy functional. With this prescription, we compute the propagator associated to Boltzmann–Gibbs statistics in the semiclassical approximation as K=F(T)exp(iScl/ℏ). We determine also propagators associated to different nonadditive statistics; those are the entropies depending only on the probability S± and Tsallis entropy Sq. For S±, we obtain a power series solution for the probability vs. the energy, which can be analytically continued to the complex plane and employed to obtain the propagators. Our work is motivated by the work of Nobre et al. where a modified q-Schrödinger equation is obtained that provides the wave function for the free particle as a q-exponential. The modified q-propagator obtained with our method leads to the same q-wave function for that case. The procedure presented in this work allows to calculate q-wave functions in problems with interactions determining nonlinear quantum implications of nonadditive statistics. In a similar manner, the corresponding generalized wave functions associated to S± can also be constructed. The corrections to the original propagator are explicitly determined in the case of a free particle and the harmonic oscillator for which the semiclassical approximation is exact, and also the case of a particle with an infinite potential barrier is discussed.

Conformal generation of an exotic rotationally invariant harmonic oscillator

An exotic rotationally invariant harmonic oscillator (ERIHO) is constructed by applying a non-unitary isotropic conformal bridge transformation (CBT) to a free planar particle. It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant $g$. The model reveals the Euclidean ($|g|<1$) and Minkowskian ($|g|>1$) phases separated by the phases $g=+1$ and $g=-1$ of the Landau problem in the symmetric gauge with opposite orientation of the magnetic field. A hidden symmetry emerges in the system at rational values of $g$. Its generators, together with the Hamiltonian and angular momentum produce non-linearly deformed $\mathfrak{u}(2)$ and $\mathfrak{gl}(2,{\mathbb R})$ algebras in the cases of $0<|g|<1$ and $\infty>|g|>1$, which transmute one into another under the inversion $g\rightarrow -1/g$. Similarly, the true, $\mathfrak{u}(2)$, and extended conformal, $\mathfrak{gl}(2,{\mathbb R})$, symmetries of the isotropic Euclidean oscillator ($g=0$) interchange their roles in the isotropic Minkowskian oscillator ($|g|=\infty$), while two copies of the $\mathfrak{gl}(2,{\mathbb R})$ algebra of analogous symmetries mutually transmute in Landau phases. We show that the ERIHO system is transformed by a peculiar unitary transformation into the anisotropic harmonic oscillator generated, in turn, by anisotropic CBT. The relationship between the ERIHO and the subcritical phases of the harmonically extended Landau problem, as well as with a plane isotropic harmonic oscillator in a uniformly rotating reference frame, is established.

5-Dimensional SO(1,4)-invariant action as an origin to the Magueijo-Smolin Doubly Special Relativity proposal

In this paper we discuss how the Magueijo-Smolin Doubly Special Relativity proposal may be obtained from a singular Lagrangian action. The deformed energy-momentum dispersion relation rises as a particular gauge, whose covariance imposes the non-linear Lorentz group action. Moreover, the additional invariant scale is present from the beginning as a coupling constant to a gauge auxiliary variable. The geometrical meaning of the gauge fixing procedure and its connection to the free relativistic particle are also described.

Conformal bridge in a cosmic string background

Hidden symmetries of non-relativistic mathfrak{so}left(2,1right)cong mathfrak{sl}left(2,mathrm{mathbb{R}}right) invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the “geometrical parameter” α, determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at α = q/k with q, k = 1, 2, . . ., which for q > 1 are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of α; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when α is an integer, that reveals a quantum anomaly.

Solutions to linear dissipative quantum systems

We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.

Schrödinger's equation as a diffusion equation

Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β2=ℏ/2m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability density. Considering these two equations as if they were the basic equations of quantum mechanics, we apply them to several one-dimensional quantum systems. We show the dispersive properties in the probability densities of stationary states of a particle in a rigid box and in harmonic potential; quasi-classical Gaussian probability densities of a free particle; and coherent states and squeezed states of the harmonic oscillator. We also present the soliton as a quantum mechanical representation of a free particle. We discuss the meaning of the diffusion coefficient β2 for each quantum system using a density plot.