On a stochastic nonlocal conservation law in a bounded domain

In this paper, we are concerned with the Dirichlet boundary value problem for a multi-dimensional nonlocal conservation law involving a multiplicative stochastic perturbation in a bounded domain. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we establish the existence and uniqueness of entropy solution to the boundary value problem

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Interpreting Quantum Mechanics in Terms of Random Discontinuous Motion of Particles

This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and dynamical collapse theories, and briefly analyze the problem of the incompatibility between quantum mechanics and special relativity

Origin of Quantum Mechanical Results and Life: A Clue from Quantum Biology

Although quantum mechanics can accurately predict the probability distribution of outcomes in an ensemble of identical systems, it cannot predict the result of an individual system. All the local and global hidden variable theories attempting to explain individual behavior have been proved invalid by experiments (violation of Bell’s inequality) and theory. As an alternative, Schrodinger and others have hypothesized existence of free will in every particle which causes randomness in individual results. However, these free will theories have failed to quantitatively explain the quantum mechanical results. In this paper, we take the clue from quantum biology to get the explanation of quantum mechanical distribution. Recently it was reported that mutations (which are quantum processes) in DNA of E. coli bacteria instead of being random were biased in a direction such that the chance of survival of the bacteria is increased. Extrapolating it, we assume that all the particles including inanimate fundamental particles have a will and that is biased to satisfy the collective goals of the ensemble. Using this postulate, we mathematically derive the correct spin probability distribution without using quantum mechanical formalism (operators and Born’s rule) and exactly reproduce the quantum mechanical spin correlation in entangled pairs. Using our concept, we also mathematically derive the form of quantum mechanical wave function of free particle which is conventionally a postulate of quantum mechanics. Thus, we prove that the origin of quantum mechanical results lies in the will (or consciousness) of the objects biased by the collective goal of ensemble or universe. This biasing by the group on individuals can be called as “coherence” which directly represents the extent of life present in the ensemble. So, we can say that life originates out of establishment of coherence in a group of inanimate particles

Quantum Cybernetics: A New Perspective for Nelson's Stochastic Theory, Nonlocality, and the Klein-Gordon Equation

The Klein-Gordon equation is shown to be equivalent to coupled partial differential equations for a sub-quantum Brownian movement of a ''particle'', which is both passively affected by, and actively affecting, a diffusion process of its generally nonlocal environment. This indicates circularly causal, or ''cybernetic'', relationships between ''particles'' and their surroundings. Moreover, in the relativistic domain, the original stochastic theory of Nelson is shown to hold as a limiting case only, i.e., for a vanishing quantum potential.Comment: 21 pages; published in Phys. Lett. A 296 (2002) 1 -