A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension

The Vlasov–Fokker–Planck equation is a model for a collisional, electrostatic plasma. The approximation of this equation in one spatial dimension is studied. The equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function. The approximation method applied is the deterministic particle method described in Wollman and Ozizmir [Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. For the present linear problem an analysis of the stability and convergence of the numerical method is carried out. In addition, computations are done that verify the convergence of the numerical solution. It is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in Bouchut and Dolbeault [On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and Newtonian potentials, Differential Integral Equations 8(3) (1995) 487–514].     

Negative asymptotic topological dimension, a new condensate, and their relation to the quantized Zipf law

We introduce the notion of weight for the asymptotic topological dimension. Planck’s formula for black-body radiation is refined. We introduce the notion of negative asymptotic topological dimension (of hole dimension). The condensate in the hole dimension is applied to the quantized Zipf law for frequency dictionaries (obtained earlier by the author).     

On a Moving Boundary Solution to the Fokker-Planck Equation for Particle Transport in Turbulent Flows with Absorbing Boundaries

A Fokker—Planck equation describing the transport of particlesin turbulent flows is considered. The initial value problemwith perfectly absorbing boundary conditions on the wall issolved by introducing a characteristic line in phase space.It is found that the solution domain in phase space decomposesinto three regions which are connected by two moving boundaries,and the moving boundaries can be found explicitly. For Gaussian-typeinitial conditions, the solution of the Fokker—Planckequation can be obtained in each of the three regions by solvinga diffusion equation. An approximation procedure for the generalinitial value problem is established and the approximate solutionsequence is shown to be convergent in a certain Hilbert space.     

A new look at quarks confinement

Quarks confinement is an experimental fact. ‘tHooft and later on Gross, Wilczek and Politzer have contributed in various ways to our present considerable theoretical understanding of the problem. However, an exact water proof theoretical derivation of the problem is at best still in progress. The present note argues that to understand quarks confinement, a deeper understanding of the Planck scale physics is indispensable and shows using analytical topological arguments that absolute confinement is a result of a phase transition of quantum spacetime at the Planck scale.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.     

Exponential Decay in Time of Density of One-dimensional Quantum Navier-Stokes Equations

The long-time behavior of the particle density of the compressible quantum Navier-Stokes equations in one space dimension is studied. It is shown that the particle density converges exponentially fast to the constant thermal equilibrium state as the time tends to infinity, the decay rate is also obtained. The results hold regardless of either the bigger of the scaled Planck constant or the viscosity constant. This improves the decay results of [5] by removing the crucial assumption that the scaled Planck constant is bigger than the viscosity constant. The proof is based on the entropy dissipation method and the Bresch-Desjardins type of entropy.     

Systematic Measures of Biological Networks I: Invariant Measures and Entropy

This paper is part I of a two‐part series devoted to the study of systematic measures in a complex biological network modeled by a system of ordinary differential equations. As the mathematical complement to our previous work with collaborators, the series aims at establishing a mathematical foundation for characterizing three important systematic measures: degeneracy, complexity, and robustness, in such a biological network and studying connections among them. To do so, we consider in part I stationary measures of a Fokker‐Planck equation generated from small white noise perturbations of a dissipative system of ordinary differential equations. Some estimations of concentration of stationary measures of the Fokker‐Planck equation in the vicinity of the global attractor are presented. The relationship between the differential entropy of stationary measures and the dimension of the global attractor is also given.© 2016 Wiley Periodicals, Inc.     

Is quantum space-time infinite dimensional

The stringy uncertainty relations, and corrections thereof, were explicitly derived recently from the new relativity principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. A simple numerical argument is presented which suggests that quantum space-time may very well be infinite dimensional. A discussion of the repercussions of this new paradigm in Physics is given. A truly remarkably simple and plausible solution of the cosmological constant problem results from the new relativity principle: The cosmological constant is not a constant, in the same vein that energy in Einstein's Special Relativity is observer dependent. Finally, following El Naschie, we argue why the observed D=4 world might just be an average dimension over the infinite possible values of the quantum space-time and why the compactification mechanisms from higher to four dimensions in string theory may not be actually the right way to look at the world at Planck scales.     

Simulated Annealing Algorithms for Continuous Global Optimization: Convergence Conditions

In this paper, simulated annealing algorithms for continuous global optimization are considered. After a review of recent convergence results from the literature, a class of algorithms is presented for which strong convergence results can be proved without introducing assumptions which are too restrictive. The main idea of the paper is that of relating both the temperature value and the support dimension of the next candidate point, so that they are small at points with function value close to the current record and bounded away from zero otherwise.     

Cobordism of algebraic knots

Summary The isolated singularities of complex hypersurfaces are studied by considering the topology of the highly connected submanifolds of spheres determined by the singularity. By introducing the notion of the link of a perturbation of the singularity and using techniques of surgery theory, we are able to describe which invariants associated to a singularity can be used to determine the cobordism type of the singularity.It is shown that the cobordism type is determined by the set of weakly distinguished bases. This result is used to draw a distinction between the classical case of two variables and the higher dimensional problem. That is, we show that the result of Le which states that the cobordism and topological classifications of singularities coincide in the classical dimension does not hold for singularities of functions of more than three variables. Examples of topologically distinct but cobordant singularities are obtained using results of Ebeling.     

Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\ge 3$, including the case of a Coulomb singularity in dimension $d=3$. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects, uniform in the Planck constant ħ.     

Quiescent phases and stability

If an autonomous system of differential equations is diffusively coupled to a quiescent phase then at stationary points the stability properties change. If the coupling matrices are multiples of the identity then introducing a quiescent phase stabilizes against the onset of oscillations, in particular high frequency oscillations are damped.For arbitrary (diagonal) coupling matrices the situation gets much more complex. For dimension two it can be shown that stability at a stationary point is preserved for arbitrary rates if and only if the Jacobian is strongly stable in the sense of Turing stability.     

New results on Fokker–Planck equations of fractional order

It is shown that the fractional Fokker–Planck equations proposed recently in the literature (by merely substituting time fractional derivative for time derivative) give rise to some problems in the sense that they provide probability densities which may have negative values. In the same way, one shows that the Kramers–Moyal equation can be thought of as related to fractal processes, but it is well known that it yields also negative densities. It seems that the key of this trouble is the misuse of the Chapman Kolmogorov equation on the one hand, and of the fractional difference on the other hand. In fact, there is a complete identification between Kramers–Moyal equation and Fokker–Planck equation of fractional order. After a careful analysis, one arrives at the conclusion that the fractional derivative in Liouville–Riemann (L–R) sense should be replaced by a slightly finite fractional derivative which involves finite difference, whilst L–R fractional derivative refers to difference of infinite order. The new fractional Fokker–Planck equation so obtained is displayed, and its solution via separation of variables is outlined. It seems that there is no alternative but to work via non-standard analysis, that is to say infinitesimal discretization in time.     

On symmetries of stochastic differential equations

This note can be considered as a supplement to article [8]. Its purpose is twofold. First, to show that symmetries of Itô stochastic differential equations form a Lie algebra. Second, to provide more precise formulation of the relation between symmetries of SDEs and symmetries of the associated Fokker–Planck equation. Relation between first integrals of SDEs and symmetries of the associated Fokker–Planck equation is also considered.     

Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations

We describe conditions on non-gradient drift diffusion Fokker–Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.     

Zur Theorie endlicher geschlitzter Inzidenzgruppen

It is proved that every finite incidence group which is neither projective nor affine has at least dimension three. There is also an estimation of the dimension by means of the dimension of the affine kernel. The result can be improved under special conditions. The paper contains also two caracterizations of the commutative finite incidencegroups.     

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

A numerical method is developed for simulation of stochastic chemical reactions. The system is modeled by the Fokker–Planck equation for the probability density of the molecular state. The dimension of the domain of the equation is reduced by assuming that most of the molecular species have a normal distribution with a small variance. The numerical approximation preserves properties of the analytical solution such as non-negativity and constant total probability. The method is applied to a nine dimensional problem modelling an oscillating molecular clock. The oscillations stop at a fixed point with a macroscopic model but they continue with our two dimensional, mixed macroscopic and mesoscopic model. Dedicated to the memory of Germund Dahlquist (1925–2005). AMS subject classification (2000)  65M20, 65M60