Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker–Planck and Fractional Equations

The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function \(f_j(r,t)\) of a particle moving on a substrate with time delays \(\tau _j\). Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability \(P_j\) is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, \(P_j \propto f_j^{\nu - 1}\), the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, \(P_j \propto \tau _j^{-1-\alpha }\) (with \(0< \alpha < 2\)), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.     

Further application of the statistical thermodynamics of curved surfaces to scaled particle theory

The thermodynamics of curved boundary layers is combined with scaled particle theory to determine the rigid-sphere equation of state. In particular, the boundary analog of the Gibbs-Tolman-Koenig-Buff equation is solved for a rigid-sphere fluid, using the approximation that the distance between the surface of a cavity and its surface of tension is a function of the density only (the first-order approximation). This, in conjunction with several exact conditions onG, the central function of scaled particle theory, leads to an approximate rigid-sphere fluid equation of state and a qualitatively correct rigid-sphere solid equation of state. The fluid isotherm compares favorably with previous results (2.9 % error in the fourth virial coefficient), but due to the inaccuracy of the solid isotherm, no phase transition is obtained. The theory described here is to be contrasted with previous approaches in that a less arbitrary functional form forG is assumed, and the surface of tension and cavity surface are not assumed to be coincident. The cycle equation of Reiss and Tully-Smith is rederived by a simpler route and shown to be correct to all orders of cavity curvature, rather than only first order as was originally thought. A new exact condition, obtained from the compressibility equation of state, is used as a boundary condition for the cycle equation to determine the location of the equimolecular surface. This molecular calculation compares favorably (discrepancy of <2 %) with a thermodynamic calculation based on the boundary analog of the Gibbs adsorption equation and indicates the accuracy and consistency of the first-order approximation.Research supported under NSF Grant #GP-12408.     

Extended Hill equation

IfJ(z) is a periodic even function ofz with period , the equationd ²u(z)/dz² + J(z)u(z) = 0 is Hill's equation. The solution is obtained to an extended Hill equation whereJ(z) is a periodic real function ofz, using Floquet's method.     

Where do inertial particles go in fluid flows?

We derive a general reduced-order equation for the asymptotic motion of finite-size particles in unsteady fluid flows. Our inertial equation is a small perturbation of passive fluid advection on a globally attracting slow manifold. Among other things, the inertial equation implies that particle clustering locations in two-dimensional steady flows can be described rigorously by the Q parameter, i.e., by one-half of the squared difference of the vorticity and the rate of strain. Use of the inertial equation also enables us to solve the numerically ill-posed problem of source inversion, i.e., locating initial positions from a current particle distribution. We illustrate these results on inertial particle motion in the Jung-Tél-Ziemniak model of vortex shedding behind a cylinder in crossflow.     

A classical Proca particle

An elementary particle is described as a spherically symmetric solution of the Proca equations and the Einstein general relativity equations. The mass is found to be of the order of the Planck mass. If the motion of its center of mass is determined by the Dirac equations, it has a spin 1/2.This work is parallel to an earlier one involving the Klein- Gordon equation.     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

Noncommutative cumulants for stochastic differential equations and for generalized Dyson series

For the stochastic equationU=VU, Kubo's ansalze for U in the form of differential and integrodifferential equations is investigated and a newansatz as an integral equation is added. Unique solutions in terms of noncommutative W- and K-cumulants are found by elementary functional differentiation, and expressions of van Kämpen and Terwiel are recovered. For the cumulants we find simple recursion relations and prove the important cluster property. Surprisingly, it is found that the Gaussian approximation in the differential equationansatz leads to positivity problems, while this is not the case with the integral and integrodifferential equation. The cumulant expansion technique is carried over to generalized Dyson series. In a companion paper we apply our results to quantum shot noise.     

Clocks and gravity

We consider the assumption that clocks measure proper time-that is, in a gravitational field ideal clocks are governed by the equationds ²=g _ij dxⁱ dx^j-and give some theoretical and experimental constraints on clock measurements. In particular, we find that if we assume that clocks are governed by an equation of the formds ⁴=c _ijkl dxⁱ dx^j dx^k dx^l, then this equation must reduce to the quadratic equation in a weak, spherically symmetric, static gravitational field (at least to first order in the Newtonian gravitational potentialU), otherwise additional contributions to the time-delay effect of radar propagation (that are not observed) are predicted.     

A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions. Received: 11 November 1999 / Accepted: 19 May 2000     

Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advection-diffusion equations, there are very few analytical results on their stability, even for the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably, the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters. Such optimal stability is reached owing to the free (“kinetic”) relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative equilibrium functions (the distribution divided by the local mass), often labeled as “unphysical”. We prove that the non-negativity condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT sub-classes, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT sub-class more efficient than the BGK one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes.     

Macroscopic Form of the First Law of Thermodynamics for an Adibatically Evolving <Emphasis Type="Italic">Non-singular</Emphasis> Self-gravitating Fluid

We emphasize that the pressure related work appearing in a general relativistic first law of thermodynamics should involve proper volume element rather than coordinate volume element. This point is highlighted by considering both local energy momentum conservation equation as well as particle number conservation equation. It is also emphasized that we are considering here a non-singular fluid governed by purely classical general relativity. Therefore, we are not considering here any semi-classical or quantum gravity which apparently suggests thermodynamical properties even for a (singular) black hole. Having made such a clarification, we formulate a global first law of thermodynamics for an adiabatically evolving spherical perfect fluid. It may be verified that such a global first law of thermodynamics, for a non-singular fluid, has not been formulated earlier.     

Klein-Gordon thermal equation for a planck gas

In this paper the quantum hyperbolic equation formulated in our earlier paper [Found. Phys. Lett. 10, 599 (1997)] is applied to the study of the propagation of the initial thermal state of the universe. It is shown that the propagation depends on the barrier height. The Planck wall potential is introduced,V _P = ħ/8t_P = 1.125 10¹⁸ GeV, wheret _P is a Planck time. For the barrier heightV <V _P , the master thermal equation isthe modified telegrapher’sequation, and for barrier heightV >V _P the master equation is theKlein- Gordon equation. The solutions of both type equations for Cauchy boundary conditions are discussed.     

The wave properties of matter and the zeropoint radiation field

The origin of the wave properties of matter is discussed from the point of view of stochastic electrodynamics. A nonrelativistic model of a charged particle with an effective structure embedded in the random zeropoint radiation field reveals that the field induces a high-frequency vibration on the particle; internal consistency of the theory fixes the frequency of this jittering at mc²/. The particle is therefore assumed to interact intensely with stationary zeropoint waves of this frequency as seen from its proper frame of reference; such waves, identified here as de Broglie's phase waves, give rise to a modulated wave in the laboratory frame, with de Broglie's wavelength and phase velocity equal to the particle velocity. The time-independent equation that describes this modulated wave is shown to be the stationary Schrödinger equation (or the Klein-Gordon equation in the relativistic version). In a heuristic analysis appled to simple periodic cases, the quantization rules are recovered from the assumption that for a particle in a stationary state there must correspond a stationary modulation. Along an independent and complementary line of reasoning, an equation for the probability amplitude in configuration space for a particle under a general potential V(x) is constructed, and it is shown that under conditions derived from stochastic electrodynamics it reduces to Schrödinger's equation. This equation reflects therefore the dual nature of the quantum particles, by describing simultaneously the corresponding modulated waveand the ensemble of particles.     

Geometric study for the Legendre duality of generalized entropies and its application to the porous medium equation

We geometrically study the Legendre duality relation that plays an important role in statistical physics with the standard or generalized entropies. For this purpose, we introduce dualistic structure defined by information geometry, and discuss concepts arising in generalized thermostatistics, such as relative entropies, escort distributions and modified expectations. Further, a possible generalization of these concepts in a certain direction is also considered. Finally, as an application of such a geometric viewpoint, we briefly demonstrate several new results on the behavior of the solution to a nonlinear diffusion equation called the porous medium equation.     

Accumulation of particles in time-dependent thermocapillary flow in a liquid bridge. Modeling of experiments

The study addresses the phenomenon of accumulation of rigid tracer particles suspended in a time-dependent thermocapillary flow in a liquid bridge. We report the results of the three-dimensional numerical modeling of recent experiments [1,2] in a non-isothermal liquid column. Exact physical properties of both liquids and particles are used for the modeling. Two liquids are investigated: sodium nitrate (NaNO₃) and n-decane (C₁₀H₂₂). The particles are modeled as perfect spheres suspended in already well developed time-dependent thermocapillary flow. The particle dynamics is described by the Maxey-Riley equation. The results of our simulations are in excellent agreement with the experimental observations. For the first time we reproduced numerically formation of the particle accumulation structure (PAS) both under gravity and under weightlessness conditions. Our analysis confirms the experimental observations that the existence of PAS depends on the strength of the flow field, on the ratio between liquid and particle density, and on the particle size.     

Lagrangian foliations and Lax equations

If X is a bihamiltonian vector field tangent to a foliation which is Lagrangian with respect to both symplectic structures, the dynamical system x=X(x) implies a local Lax equation =[L, B], but in canonical adapted coordinates, this equation reduces to the trivial equation =0.     

Schrödinger equation as an operator acting on test functions describing observing systems and the EPR correlation

A reformulation of quantum mechanics is introduced by regarding the Schrödinger equationE(f ⁺) = 0 for the retarded particle wavef ⁺ as an operator (functional) acting on the test functionf ^– satisfying the boundary conditions of the observing system: E(f ⁺),f ^– = 0. The variational expression for the transition amplitude of a particle between the particle source and the detector naturally arises in the dual space of the particle field and the test function. In the two-slit electron interference experiment, the test function plays the role of the quantum potential which carries the information of the detector and the slit locations backwards in time, while in the Einstein-Podolsky-Rosen process the test function describes the time reversed process of a pair of spatially separated fermions with arbitrarily chosen spin orientations progressing backwards in time to form a spherically symmetric compound state. The separation of the kinematics (spin correlation and the dynamics (spacetime aspect) of the EPR process is pointed out.     

Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

A point-like particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the 1D Fokker-Planck (Kramers) equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m→0, with a series of corrections expanded in powers of m/γ, γ denotes the friction coefficient. The corrections are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.