Plank Equation Research Articles

Statistical dynamics of regional populations and economies

Quantitative analysis of human behavior and social development is becoming a hot spot of some interdisciplinary studies. A statistical analysis on the population and GDP of 150 cities in China from 1990 to 2013 is conducted. The result indicates the cumulative probability distribution of the populations and that of the GDPs obeying the shifted power law, respectively. In order to understand these characteristics, a generalized Langevin equation describing variation of population is proposed, which is based on the correlations between population and GDP as well as the random fluctuations of the related factors. The equation is transformed into the Fokker–Plank equation to express the evolution of population distribution. The general solution demonstrates a transition of the distribution from the normal Gaussian distribution to a shifted power law, which suggests a critical point of time at which the transition takes place. The shifted power law distribution in the supercritical situation is qualitatively in accordance with the practical result. The distribution of the GDPs is derived from the well-known Cobb–Douglas production function. The result presents a change, in supercritical situation, from a shifted power law to the Gaussian distribution. This is a surprising result–the regional GDP distribution of our world will be the Gaussian distribution one day in the future. The discussions based on the changing trend of economic growth suggest it will be true. Therefore, these theoretical attempts may draw a historical picture of our society in the aspects of population and economy.

Lp-maximal hypoelliptic regularity of nonlocal kinetic Fokker–Planck operators

For p∈(1,∞), let u(t,x,v) and f(t,x,v) be in Lp(R×Rd×Rd) and satisfy the following nonlocal kinetic Fokker–Plank equation on R1+2d in the weak sense:∂tu+v⋅∇xu=Δα/2vu+f, where α∈(0,2) and Δvα/2 is the usual fractional Laplacian applied to v-variable. We show that there is a constant C=C(p,α,d)>0 such that for any f(t,x,v)∈Lp(R×Rd×Rd)=Lp(R1+2d),‖Δxα/(2(1+α))u‖p+‖Δvα/2u‖p⩽C‖f‖p, where ‖⋅‖p is the usual Lp-norm in Lp(R1+2d;dz). In fact, in this paper the above inequality is established for a large class of time-dependent non-local kinetic Fokker–Plank equations on R1+2d, with Utv⋅∇x and Lσtνt in place of v⋅∇x and Δvα/2. See Theorem 3.3 for details.

Financial market dynamics: superdiffusive or not?

The behavior of stock market returns over a period of 1–60 d has been investigated for S&P 500 and Nasdaq within the framework of nonextensive Tsallis statistics. Even for such long terms, the distributions of the returns are non-Gaussian. They have fat tails indicating that the stock returns do not follow a random walk model. In this work, a good fit to a Tsallis q-Gaussian distribution is obtained for the distributions of all the returns using the method of Maximum Likelihood Estimate. For all the regions of data considered, the values of the scaling parameter q, estimated from 1 d returns, lie in the range 1.4–1.65. The estimated inverse mean square deviations (beta) show a power law behavior in time with exponent values between −0.91 and −1.1 indicating normal to mildly subdiffusive behavior. Quite often, the dynamics of market return distributions is modelled by a Fokker–Plank (FP) equation either with a linear drift and a nonlinear diffusion term or with just a nonlinear diffusion term. Both of these cases support a q-Gaussian distribution as a solution. The distributions obtained from current estimated parameters are compared with the solutions of the FP equations. For negligible drift term, the inverse mean square deviations (betaFP) from the FP model follow a power law with exponent values between −1.25 and −1.48 indicating superdiffusion. When the drift term is non-negligible, the corresponding betaFP do not follow a power law and become stationary after certain characteristic times that depend on the values of the drift parameter and q. Neither of these behaviors is supported by the results of the empirical fit.

Pricing of basket options in subdiffusive fractional Black–Scholes model

In this paper we generalize the classical multidimensional Black-Scholes model to the subdiffusive case. In the studied model the prices of the underlying assets follow subdiffusive multidimensional geometric Brownian motion. We derive the corresponding fractional Fokker–Plank equation, which describes the probability density function of the asset price. We show that the considered market is arbitrage-free and incomplete. Using the criterion of minimal relative entropy we choose the optimal martingale measure which extends the martingale measure from used in the standard Black–Scholes model. Finally, we derive the subdiffusive Black–Scholes formula for the fair price of basket options and use the approximation methods to compare the classical and subdiffusive prices.

Study on the two-way coupling turbulent model and rheological properties for fiber suspension in the contraction

A two-way coupling turbulent model is derived to understand and predict the interaction between the additive and the carrier flow field. Due to the anisotropic characteristics of the turbulence structure in the contraction, the Reynolds stress mode (RSM) is adopted as the fundamental governing equations. The additional stress item arising from fibers employs the Batchelor's constitutive equation for the dilute and semi dilute suspension and its divergence can have the forms of the additional diffusion term and the additional viscous dissipation term by derivation. Fiber orientation probability distribution is calculated by the Reynolds averaged Fokker–Plank equation. Based on the model above, the influences of fibers concentration on fiber orientation distribution and rheological properties for the suspension in the contraction are discussed by numerical simulation.

Existence of Weak Solutions in Wasserstein Space for a Chemotaxis Model Coupled to Fluid Equations

We consider a coupled system of Keller--Segel-type equations and the incompressible Navier--Stokes equations in spatial dimension two and three. We first establish the existence of a weak solution of a Fokker--Plank equation in the Wasserstein space under the assumption that initial mass is integrable and has finite entropy. As a result, we construct solutions of Keller--Segel--Navier--Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space, in the case that its initial mass is assumed to be bounded and integrable.

Possible Mechanism for Effects Caused by Exposure to Extremely Low Frequency Magnetic Fields

This paper introduces a possible mechanism to explain the changes in neurons following extremely low frequency magnetic fields (ELF-MFs). The model in this paper contains two parts: the physical part and the bio-trigger part. The first part is an electric current equation, which is a generalization of the Nernst–Plank equation with the exposure of ELF-MF, and the second bio-part is a phenomenological model, which is a simple two-state channel with two steady state open (O) state and closed state (C). The two parts are complementary. The first part is utilized to explain changes about synaptic plasticity following ELF-MF exposure at embryonic and postnatal age. The second part is used to simulate the process of three different types of neurons reported by previous papers. According to the results, the present model is feasible for the simulation process. The concept of ion mobility or permeability is emphasized in this paper in part I, and channel state transition probabilities are paid much attention to in part II. In this paper, the ion mobility or permeability is supposed to have close relationships with ion channel transition probabilities, which connects the two parts of the model together.

Stability of TLP tether using a stochastic averaging technique

Tension Leg Platforms (TLPs) are popular deep water platforms for extracting oil and gas. TLPs are connected to the sea bed by pretensioned tethers. If fluctuating tension in tethers is sizable compared to the pretension, many dynamic stability problems may arise which also affects the platform motion. In this paper, stochastic stability of TLP tethers in transverse oscillations is investigated using stochastic averaging technique and the probability flow of the reduced Fokker–Plank equation. Stochastic averaging is used to obtain drift and diffusion coefficients independent of time. Onset of such transverse oscillations may be caused due to regular or random vortex shedding. The tension fluctuation of the tether due to random vertical wave forces on the platform induces multiplicative random excitation in the tether because of the presence of nonlinear drag. As a consequence, dynamic instability may become a significant problem. The tether is modeled as a simply supported uniform beam under initial tension assuming transverse oscillation to take place only in its fundamental natural mode. Thus, the problem is reduced to a single degree of freedom system. Fluid resistance is calculated by Morison's equation consisting of drag and inertia forces. A semi-analytical procedure combined with stochastic averaging technique is employed to obtain the drift and diffusion coefficients of the response in closed form. By examining the probability flow in terms of drift and diffusion coefficients at the two boundaries, the stability of the system is investigated. Stability of tethers of a TLP in 300m water depth, for different pretensions, is studied for sea states of 21m, 16m and 8m significant wave heights represented by PM spectrum. It is shown that for low values of pretension, stochastic stability of tether in transverse direction may take place for large wave heights.

A COMSOL–GEMS interface for modeling coupled reactive-transport geochemical processes

An interface was developed between COMSOL MultiphysicsTM finite element analysis software and (geo)chemical modeling platform, GEMS, for the reactive-transport modeling of (geo)chemical processes in variably saturated porous media. The two standalone software packages are managed from the interface that uses a non-iterative operator splitting technique to couple the transport (COMSOL) and reaction (GEMS) processes. The interface allows modeling media with complex chemistry (e.g. cement) using GEMS thermodynamic database formats. Benchmark comparisons show that the developed interface can be used to predict a variety of reactive-transport processes accurately. The full functionality of the interface was demonstrated to model transport processes, governed by extended Nernst–Plank equation, in Class H Portland cement samples in high pressure and temperature autoclaves simulating systems that are used to store captured carbon dioxide (CO2) in geological reservoirs.

Estimation of the current driven by residual loop voltage in LHCD plasma on EAST Tokamak

The lower hybrid wave current drive (LHCD) is one of the efficient methods of driving the non-inductive current required for Tokamak operating in steady-state. Residual loop voltage exists in Tokamak when the non-inductive current is not fully driven. Residual loop voltage also accelerates the fast electrons generated by the lower hybrid wave (LHW), which can drive extra current and combine with the current driven by the LHW. It is generally difficult to separate these two different components of driven current in the experiment. In this paper, the currents driven by LHCD and residual loop voltage are separated directly by solving the Fokker–Plank equation numerically. The fraction of the current driven by residual loop voltage compared to the current driven by LHW is evaluated on the experimental advanced superconducting tokamak (EAST). The current driven by residual loop voltage is several percent of the currents driven by the LHCD when the residual loop voltage is small, but it increases with the residual loop voltage up to 25% when the residual loop voltage is about 2 V. The hot electrical conductivity is deduced from the net current driven by the residual loop voltage. Its distribution profile is related to the fast electron distribution driven by LHW.

Application of quasi-steady state methods to molecular motor transport on microtubules in fungal hyphae

We consider bidirectional transport of cargo by molecular motors dynein and kinesin that walk along microtubules, and/or diffuse in the cell. The motors compete to transport cargo in opposite directions with respect to microtubule polarity (towards the plus or minus end of the microtubule). In recent work, Gou et al. (2014) used a hierarchical set of models, each consisting of continuum transport equations to track the evolution of motors and their cargo (early endosomes) in the specific case of the fungus Ustilago maydis. We complement their work using a framework of quasi-steady state analysis developed by Newby and Bressloff (2010) and Bressloff and Newby (2013) to reduce the models to an approximating steady state Fokker–Plank equation. This analysis allows us to find analytic approximations to the steady state solutions in many cases where the full models are not easily solved. Consequently, we can make predictions about parameter dependence of the resulting spatial distributions. We also characterize the overall rates of bulk transport and diffusion, and how these are related to state transition parameters, motor speeds, microtubule polarity distribution, and specific assumptions made.

On the Mechanism of Corrosion in Presence of Organic Acids

Theories on the role of organic acids in general, and acetic acid in particular, in carbon steel corrosion were initiated in 1940s with the reports on the significant effect of acetic acid in pipeline corrosion in oil and gas industry1. It was reported that even small amounts of acetic acid (300 ppm) can cause severe corrosion of pipeline steel1. Later on, several studies on the concentration of organic acids in formation water of oil and gas wells have been reported up to 3000 ppm of acetic acid1–4. Despite numerous studies on the mechanism of acetic acid corrosion of pipeline steel, details of the mechanism are still debated. Acetic acid is known as a week acid which is partially dissociate in water. It is also known that weak acids act as a reservoir for hydrogen ions in the corrosion process. This phenomenon known as “buffering effect”, describes the enhanced limiting current in presence of acetic acid compared to the case of strong acids at a similar pH. Additionally, it is suggested that the undissociated acetic acid is also a proton donor and is able to be reduced at the surface of the steel, which is known as “direct reduction” mechanism. The direct reduction mechanism has been used for years5,6. Also, many other studies on the combined effect of acetic acid and carbon dioxide on carbon steel have adopted the same mechanism7–9. However, recent studies suggest that undissociated acetic acid is only involved in the corrosion process as a buffer for hydrogen ions. Tran et al.10 report a series of polarization curves on SS304 stainless steel, showing that the concentration of acetic acid does not affect the charge transfer portion of the cathodic sweeps. Therefore, they have concluded that acetic acid is not involved in charge transfer processes directly and that the buffering effect is the main mechanism. In that study, authors suggest that the same mechanism is valid for carbon steel. In the present study an iron electrode was used instead of carbon steel in order to minimize the effect of impurities such as carbon. Figure 1 presents the experimental data collected in the present work and the comparison with the results of the mathematical simulations. The experimental data are collected using a rotating disk electrode made from pure iron exposed in a two liter, three electrode glass cell in deoxygenated aqueous solution. The mathematical simulation is the solution of Nernst Plank equation for an acetic acid aqueous system. These equations are solved with finite difference method, using Newman’s BANDJ11 algorithm in FORTRAN with a MATLAB user interface. Figure 1 shows a clear linear cathodic charge transfer behavior at both pH values. Additionally the charge transfer part of the cathodic reactions is insensitive to changes in concentration of acetic acid which is the indication that free acetic acid in the system is not directly reduced at the surface of iron. Current results confirm that the proposed mechanism by Tran et al. applies to pure iron which has great similarity to pipeline steel. The mathematical calculations proposed here also support the same conclusion. Additionally, comparison of the calculated charge transfer rates based on these results suggests that the rate of dissolution of iron is slightly retarded in presence of acetic acid.

Nonlinear Instability of the Incoherent State for the Kuramoto–Sakaguchi–Fokker–Plank Equation

We study the nonlinear instability of the incoherent solution to the Kuramoto–Sakaguchi–Fokker–Plank (KSFP) equation in a large coupling strength regime. For our instability analysis, we construct an approximate, exponentially growing perturbation mode using an elementary energy method. This method does not require spectral information from the linearized KSFP equation or an explicit growing solution for the corresponding linear equation. When the distribution function of oscillator’s natural frequencies is either a Dirac measure or a bounded function with a compact support (in a small interval around the origin), the incoherent solution is nonlinearly unstable depending on the relative sizes of the coupling strength and diffusion coefficient.

Noise Treatment in Models of Genetic Switches

Mathematical modeling plays a key role in understanding and characterization of critical chemical processes underlying the dynamics of cellular decision-making. Deterministic modeling of relevant processes in complex genetic networks through a set of biochemical reactions is generally thought to predict an average kinetic behavior. Stochastic kinetic methods, accounting for random fluctuations, become essential to characterize intrinsic heterogeneity due to low copy numbers and noisy environments present in these systems. While in certain regulatory constructs, it has been shown that a fully discrete stochastic treatment of fluctuations through the Chemical Master Equation yields multi-stability when deterministic treatment predicts monostability; deterministic or continuous approaches continues to be the most widespread method for modeling the behavior of cell populations. We explore stochastic variability in cellular dynamics through a varied treatment of intrinsic noise, the randomness arising either at the level of a single reaction, single molecule or as a drift in concentration of the involved species. Our study involves modeling of a genetic toggle switch which is a set of two mutually repressing genes, acting as a cellular “memory device” during cell differentiation process, deciding the final cell fate. We simulate the complex switching dynamics between the multi-attractor states in these systems through direct sampling using continuous stochastic approaches Langevin dynamics and Fokker Plank equation as well as discrete stochastic methods Gillespie's Stochastic Simulation Algorithm and numerical solution of the Chemical Master Equation. The stochastic trajectories obtained through numerical simulations not only unveil the complex dynamics leading to multi-stability, they also help explore rigorously the variations, if any, in the dynamics of transitions. Our results can further be used to compare the statistics of switching events and hence to benchmark the various computational methods available to model stochasticity in such systems.

Investigation of field effects in a solid-state nanopore transistor.

In order to calculate ion currents through solid-state nanopore transistors realistically, we propose a computational model based on the Poisson-Nernst-Plank equation. In the present model, we determine the surface charge density locally on the nanopore by imposing consistency between the ion distribution and the chemical reaction at the surface. The model can consider a non-uniform influence by the gate voltage on the inner surface of the nanopore membrane, which enables us to investigate ion currents depending on the gate geometry such as the thickness and vertical position within the nanopore. We verify the validity of the model by comparing the pH dependence of simulation results with the extant experimental results. We also investigate the transistor behaviour depending on the surface material, pore geometry and gate position. In particular, we propose an optimized system to enhance the on/off ratio of the nanopore transistor.

Determination of ion-exchange kinetic parameters for the conducting nanocomposite polyaniline zirconium titanium phosphate

The kinetics and mechanism for the ion-exchange processes like Mg(II)–H(I), Ca(II)–H(I), Sr(II)–H(I), Ba(II)–H(I), Ni(II)–H(I), Cu(II)–H(I), Mn(II)–H(I) and Zn(II)–H(I) at different temperatures using approximated Nernst–Plank equation under the particle diffusion controlled phenomenon were studied for the polyaniline zirconium titanium phosphate nanocomposite cation-exchanger. Some physical parameters, i.e. fractional attainment of equilibrium U(τ), self-diffusion coefficients (D0), energy of activation (Ea) and entropy of activation (ΔS°) have been estimated. These investigations revealed that the equilibrium is attained faster at higher temperature probably due to availability of thermally enlarged matrix of polyaniline zirconium titanium phosphate (PANI-ZTP) nanocomposite cation exchange material. These results are useful for predicting the ion exchange process occurring on the surface of this cation-exchanger.

Discrete and continuous models for tissue growth and shrinkage

The incorporation of domain growth into stochastic models of biological processes is of increasing interest to mathematical modellers and biologists alike. In many situations, especially in developmental biology, the growth of the underlying tissue domain plays an important role in the redistribution of particles (be they cells or molecules) which may move and react atop the domain. Although such processes have largely been modelled using deterministic, continuum models there is an increasing appetite for individual-based stochastic models which can capture the fine details of the biological movement processes which are being elucidated by modern experimental techniques, and also incorporate the inherent stochasticity of such systems.In this work we study a simple stochastic model of domain growth. From a basic version of this model, Hywood et al. (2013) were able to derive a Fokker–Plank equation (FPE) (in this case an advection–diffusion partial differential equation on a growing domain) which describes the evolution of the probability density of some tracer particles on the domain. We extend their work so that a variety of different domain growth mechanisms can be incorporated and demonstrate a good agreement between the mean tracer density and the solution of the FPE in each case. In addition we incorporate domain shrinkage (via element death) into our individual-level model and demonstrate that we are able to derive coefficients for the FPE in this case as well. For situations in which the drift and diffusion coefficients are not readily available we introduce a numerical coefficient estimation approach and demonstrate the accuracy of this approach by comparing it with situations in which an analytical solution is obtainable.

Evaluation of thermodynamic and kinetic parameters for conducting nanocomposite polypyrrole zirconium titanium phosphate

The kinetics and mechanism for the ion-exchange processes like Mg(II)–H(I), Ca(II)–H(I), Sr(II)–H(I), Ba(II)–H(I), Ni(II)–H(I), Cu(II)–H(I), Mn(II)–H(I) and Zn(II)–H(I) at different temperatures using approximated Nernst–Plank equation under the particle diffusion controlled phenomenon were studied for the polypyrrole zirconium titanium phosphate nanocomposite cation exchanger. TEM proves the formation of the nanocomposite cation exchanger. Some physical parameters, i.e. fractional attainment of equilibrium U(τ), self-diffusion coefficients (D0), energy of activation (Ea) and entropy of activation (ΔS°) have been estimated. These results are useful for predicting the ion exchange process occurring on the surface of this cation-exchanger.

Assessment of spatial relationship between land surface temperature and landuse/cover retrieval from multi-temporal remote sensing data in South Karkheh Sub-basin, Iran

Land surface temperature (LST) is a required input data for modeling in climatological, hydrological, agricultural, and change detection studies. In the current study, remotely sensed thermal infrared data were applied to assess LST in south Karkheh sub-basin, Iran. This research deals with the extraction of LST and land surface emissivity (LSE) and the relationship between vegetation abundance using Landsat 5 TM and Landsat 7 ETM+ for October 1998 and 2002 over the studied area. The landuse/land cover (LULC) map was derived by using maximum likelihood classifier. The variety of LSE was investigated and extracted on the basis of NDVI threshold method, as well as the Plank equation was used to derive LST over the study region. The relationship between LST and different LULC is determined by zonal GIS and regression analysis. The results illustrate that the emissivity lies in the range of 0.860–0.992 and 0.855–0.992 in 1998 and 2002, respectively. The maximum value of LSE equal to ∼0.99 was observed over the dense vegetation, while the minimum LSE value equal to ∼0.880 was found over the sand dune. LST and NDVI analysis exhibits a strong inverse correlation except over water bodies. The strongest coefficient correlation was found over wetland (−0.76 and R2 = 0.78).

Finite element method analysis of Fokker–Plank equation in stationary and evolutionary versions

The problems that often arise in stochastic dynamics can be investigated using the Fokker–Planck (FP) equation. The response of a such systems being subjected to additive and/or multiplicative random noise is represented by probability density function (PDF) that gives the full information about a response random character. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful alternative. In particular the Finite Element Method (FEM) seems to be very effective. A couple of single dynamic linear/non-linear (Duffing and Van Der Pol type) systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation. The resulting PDFs are analyzed and if the analytic results exist mutually compared.