Entropy and Vacuum Radiation

It is shown that entropy increase in thermodynamic systems can plausibly be accounted for by the random action of vacuum radiation. A recent calculation by Rueda using stochastic electrodynamics (SED) shows that vacuum radiation causes a particle to undergo a rapid Brownian motion about its average dynamical trajectory. It is shown that the magnitude of spatial drift calculated by Rueda can also be predicted by assuming that the average magnitudes of random shifts in position and momentum of a particle correspond to the lower limits of the uncertainty relation. The latter analysis yields a plausible expression for the shift in momentum caused by vacuum radiation. It is shown that when the latter shift in momentum is magnified in particle interactions, the fractional change in each momentum component is on the order of unity within a few collision times, for gases and (plausibly) for denser systems over a very broad range of physical conditions. So any system of particles in this broad range of conditions would move to maximum entropy, subject to its thermodynamic constraints, within a few collision times. It is shown that the spatial drift caused by vacuum radiation, as predicted by the above SED calculation, can be macroscopic in some circumstances, and an experimental test of this effect is proposed. Consistency of the above results with quantum mechanics is discussed, and it is shown that the diffusion constant associated with the above Brownian drift is the same as that used in stochastic interpretations of the Schrödinger equation.     

Features and states of microscopic particles in nonlinear quantum-mechanics systems

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schr?dinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy, momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.      

Features and states of microscopic particles in nonlinear quantum-mechanics systems

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schrödinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy,momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.     

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

We study a simple microscopic model for the one-dimensional stochastic motion of a (non-)relativistic Brownian particle, embedded into a heat bath consisting of (non-)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the non-relativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard Jüttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.     

A Stochastic Mechanics Based on Bohm‧s Theory and its Connection with Quantum Mechanics

We construct a stochastic mechanics by replacing Bohm‧s first-order ordinary differential equation of motion with a stochastic differential equation where the stochastic process is defined by the set of Bohmian momentum time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process with n-th order joint probability density in the form of products of delta functions, then the stochastic mechanics is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non-purely random process, we show that the stochastic mechanics is not equivalent to quantum mechanics. Whether the equivalence between the stochastic mechanics and quantum mechanics holds for all purely random processes but breaks down for all non-purely random processes remains an open question.     

The impact approximation in the theory of bimolecular quasi-resonant processes

The kinetic equation for a density matrix, which describes the relaxation of the internal states of encountering particles dissolved in an inert medium, has been derived under the following assumptions: a) the random motion of reacting particles in a liquid is considered to be a classical Markoffian process; b) the concentration of reacting particles is small enough. The equation obtained is shown to be the generalization of that of the familiar impact theory of pressure broadening for the case of any type of encountering particle motion. Our general formulae are concretized in accordance with the physical situations of rectilinear, diffusion, and stochastic jump motion of the encountering particles.     

The mechanism of Vavilov-Cherenkov radiation

The mechanism of generation of Vavilov-Cherenkov radiation is discussed in this article. The developers of the theory of the Vavilov-Cherenkov effect, I.E. Tamm and I.M. Frank, attributed this effect to their discovery of a new mechanism of radiation when a charged particle moves uniformly and rectilinearly in the medium. As such a mechanism presupposes the violation of the laws of conservation of energy and momentum, they proposed the abolition of these laws to account for the Vavilov-Cherenkov radiation mechanism. This idea has received a considerably wide acceptance in the creation of other theories, for example, transition radiation theory. In this paper, the radiation mechanism for the charge constant motion is demonstrated to be incorrect, because it contradicts not only the laws of conservation of energy and momentum, but also the very definitions of uniform and rectilinear motion (Newton's First Law). A consistent explanation of the Vavilov-Cherenkov radiation microscopic mechanism that does not contradict the basic laws is proposed. It is shown that the radiation arises from the interaction of the moving charge with bound charges that are spaced fairly far away from its trajectory. The Vavilov-Cherenkov radiation mechanism bears a slowing down character, but it differs fundamentally from bremsstrahlung, primarily because the Vavilov-Cherenkov radiation onset results from a two-stage process. First, the moving particle polarizes the medium; then, the already polarized atoms radiate coherently, provided that the particle velocity exceeds the phase speed of light in the medium. If the particle velocity is less than the phase speed of light in the medium, the polarized atoms return energy to the outgoing particle. In this case, radiation is not observed. Special attention is given to the relatively constant particle velocity as the condition of the coherent composition of waves. However, its motion cannot be designated as a uniform and rectilinear one in the sense of its definition by Newton's First Law, and it also contradicts the laws of conservation of energy and momentum.     

Diffusion in a Weakly Random Hamiltonian Flow

We consider the motion of a particle governed by a weakly random Hamiltonian flow. We identify temporal and spatial scales on which the particle trajectory converges to a spatial Brownian motion. The main technical issue in the proof is to obtain error estimates for the convergence of the solution of the stochastic acceleration problem to a momentum diffusion. We also apply our results to the system of random geometric acoustics equations and show that the energy density of the acoustic waves undergoes a spatial diffusion.     

Mean-square displacement of a stochastic oscillator: Linear vs quadratic noise

Using the Langevin equations, we calculated the stationary second-order moment (mean-square displacement) of a stochastic harmonic oscillator subject to an additive random force (Brownian motion in a parabolic potential) and to different types of multiplicative noise (random frequency or random damping or random mass). The latter case describes Brownian motion with adhesion, where the particles of the surrounding medium may adhere to the oscillator for some random time after the collision. Since the mass of the Brownian particle is positive, one has to use quadratic (positive) noise. For all types of multiplicative noise considered, replacing linear noise by quadratic noise leads to an increase in stability.     

New Type of Brownian Motion

We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision (Brownian motion with adhesion for a harmonically bound particle). This is another form of a stochastic oscillator, different from oscillator usually studied that is subject to a random force or having random frequency or random damping. Calculation of the first two stationary moments shows that for white multiplicative noise of week strength the second moment coincides with that of usual Brownian motion, but for symmetric dichotomous noise, the second moment may appear the same type of the “energetic” instability, which exists for white noise random frequency or damping coefficient.     

Rosette motion and string scattering of 20 MeV electrons in MgO single crystals

Analytical estimates and computer simulations were undertaken to perceive the motion of negative particles through a lattice structure, the interaction being classical binary scattering. Three distinct modes of particle motion along atomic strings were found depending on the magnitude of the transverse energy and the angular momentum L of the particle with regard to the string axis. At small and large L increased scattering on the strings as compared with random penetration dominates. At medium L and negative transverse energy (bound state particles in the attractive potential) a rosette motion along the string occurs. In this case small impact parameters to the string atoms are avoided and thus an increased penetrability of the negative particles results. The influence of thermal lattice vibrations on these motions was studied.

Experimentally, the negative particle motion modes manifested themselves in the penetration profiles of 20 MeV electrons through an 8 μm MgO single crystal.     

A Local-Realistic Model of Quantum Mechanics Based on a Discrete Spacetime

This paper presents a realistic, stochastic, and local model that reproduces nonrelativistic quantum mechanics (QM) results without using its mathematical formulation. The proposed model only uses integer-valued quantities and operations on probabilities, in particular assuming a discrete spacetime under the form of a Euclidean lattice. Individual (spinless) particle trajectories are described as random walks. Transition probabilities are simple functions of a few quantities that are either randomly associated to the particles during their preparation, or stored in the lattice nodes they visit during the walk. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. The scenarios considered to assess the model comprise of free particle, constant external force, harmonic oscillator, particle in a box, the Delta potential, particle on a ring, particle on a sphere and include quantization of energy levels and angular momentum, as well as momentum entanglement.     

Spherical particle Brownian motion in viscous medium as non-Markovian random process

The Brownian motion of a spherical particle in an infinite medium is described by the conventional methods and integral transforms considering the entrainment of surrounding particles of the medium by the Brownian particle. It is demonstrated that fluctuations of the Brownian particle velocity represent a non-Markovian random process. The features of Brownian motion in short time intervals and in small displacements are considered.     

Application of integral transforms to a description of the Brownian motion by a non-Markovian random process

The one-dimensional Brownian motion and the Brownian motion of a spherical particle in an infinite medium are described by the conventional methods and integral transforms considering the entrainment of surrounding particles of the medium by the Brownian particle. It is demonstrated that fluctuations of the Brownian particle velocity represent a non-Markovian random process. A harmonic oscillator in a viscous medium is also considered within the framework of the examined model. It is demonstrated that for rheological models, random dynamic processes are also non-Markovian in character. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 66–74, February, 2009.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.     

Angular momentum non-conserving decays in isotropic media

Various processes that are forbidden in vacuum due to angular momentum conservation can occur in a medium that is isotropic and does not carry any angular momentum. We illustrate this by considering explicitly two examples. The first one is the decay of a spin-0 particle into a photon and another spin-0 particle, using a model involving the Yukawa interactions of the scalar particles with a charged fermion field. The second one involves the decay of a neutrino into another neutrino and a graviton, in the standard model of particle interactions augmented with the linearized gravitational couplings.     

Global Behavior of Field Line and Particle Diffusion in a Stochastic Magnetic Field

Global behavior of field line diffusion in a stochastic magnetic field is obtained. Stochastic motion of particles undergoing mutural random collisions in the stochastic magnetic field is studied for the whole time range. The field line as wel as the particle diffusion coefficients are calculated to the sixth order of the relative magnitude of the fluctuating magnetic field.     

A stochastic theory for deep bed filtration accounting for dispersion and size distributions

We develop a stochastic theory for filtration of suspensions in porous media. The theory takes into account particle and pore size distributions, as well as the random character of the particle motion, which is described in the framework of the theory of continuous-time random walks (CTRW). In the limit of the infinitely many small walk steps we derive a system of governing equations for the evolution of the particle and pore size distributions. We consider the case of concentrated suspensions, where plugging the pores by particles may change porosity and other parameters of the porous medium. A procedure for averaging of the derived system of equations is developed for polydisperse suspensions with several distinctive particle sizes. A numerical method for solution of the flow equations is proposed. Sample calculations are applied to compare the roles of the particle size distribution and of the particle flight dispersion on the deposition profiles. It is demonstrated that the temporal flight dispersion is the most likely mechanism forming the experimentally observed hyperexponential character of the deposition profiles.     

Conservation laws for a vortex in a compressible nonequilibrium medium

The problem of conservation of magnitudes is considered for a vortex in a relaxing compressible medium. Heat release due to the relaxation of a nonequilibrium medium leads to the propagation of compression waves, which remove material. Traditional integrals of motion are inapplicable in this case. We pro-pose the concept of integral quantity, which is conserved with an arbitrary degree of accuracy despite the fact that waves cross the boundary of the integration domain. Based on this concept, a broad class of conservation laws is derived for axisymmetric disturbances of columnar vortices, including conservation of the circulation and total angular momentum of the vortex. For nonaxisymmetric disturbances, it is shown that the total angular momentum and properly defined energy integral are conserved. Numerical verification of the derived conservation laws is performed and the perspectives for using these conservation laws in numerical simulations are discussed.     

Generalized variational principles,global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.