Planck Approach Research Articles

Influence of point-like disorder on the guiding of vortices in a rotating current scheme

Explicit current-dependent expressions for anisotropic longitudinal and transverse (with respect to the current direction) nonlinear magnetoresistivities are represented and analyzed on the basis of a Fokker–Planck approach for two-dimensional single-vortex dynamics in a washboard pinning potential in the presence of point-like disorder. Graphical analysis of the nonlinear resistive responses is presented in the rotating current scheme.

Dynamical evolution of rotating stellar systems: Fokker–Planck and N-body

We have carried out comparative study between direct N-body technique and Fokker–Planck approach for the dynamical evolution of rotating stellar systems. Because of limitations of computing resources, we have restricted to relatively small N ( ∼ 10 , 000 ) systems compared to realistic star clusters. We found that these two different approaches generally give very similar results. Some minor differences are found to exist, but they can be understood because Fokker–Planck approach has to make some simplifying assumptions.

Fokker–Planck approximation of the master equation in molecular biology

The master equation of chemical reactions is solved by first approximating it by the Fokker–Planck equation. Then this equation is discretized in the state space and time by a finite volume method. The difference between the solution of the master equation and the discretized Fokker–Planck equation is analyzed. The solution of the Fokker–Planck equation is compared to the solution of the master equation obtained with Gillespie’s Stochastic Simulation Algorithm (SSA) for problems of interest in the regulation of cell processes. The time dependent and steady state solutions are computed and for equal accuracy in the solutions, the Fokker–Planck approach is more efficient than SSA for low dimensional problems and high accuracy.

Understanding interarrival and interdeparture time statistics from interactions in queuing systems

We discuss the statistics of long queues, in which the interdeparture time statistics is dominated by spatial interactions among the elements in a queue rather than the arrival or exit processes. Based on a Fokker–Planck approach, it is possible to calculate the stationary distance distribution among the elements in a queue as a function of their interaction potential. The results relate to the ones known from Random Matrix Theory. Together with the velocity distribution, one can determine the time-gap distribution as well. This yields an analytical approach to the interdeparture and interarrival time distributions of queuing systems with spatially interacting elements. While these distributions are usually determined from empirical data or from theoretical assumptions about the arrival or exit process, we offer here an alternative interpretation of interdeparture time distributions as an effect of interactions in a queue. This is relevant for the understanding of traffic and production systems and for the optimization of the statistical behavior of some queuing systems.

Different Fokker–Planck approaches to simulate electron transport in plasmas

An evaluation of different approximations of the Rosenbluth potentials is performed. A comparison is made between the Legendre expansion used in the FPI code with spherical Rosenbluth potentials, and an improved version (FPI+) with semi-anisotropic Rosenbluth potentials, and the new ( v, μ) Fokker–Planck code “FPTrans” with fully anisotropic Rosenbluth potentials, for the problem of the relaxation of a bi-Maxwellian ( T ⊥≠ T ∥) electron distribution function, which confirms the validity of the semi-anisotropic approximation, but not of the isotropic one. The improved code FPI+ was also used to compute new propagators for nonlocal electron heat transport.

Analysis of Fokker–Planck approach for foreign exchange market statistics study

In a well-known work (Phys. Rev. Lett. 84 (2000) 5224) it was shown that behaviour of returns for foreign exchange markets in different time scales can be described in terms of Fokker–Planck equation, with Kramers–Moyal coefficients being estimated from the empirical data. In the current paper the authors provide analytical solution for stationary Fokker–Planck equation, which allows explanation of non Gaussian tails of the distribution function. It is also shown that while approximating empirical data one needs to observe some limitations for correct results obtaining.

Fundamental issues on kappa-distributions in space plasmas and interplanetary proton distributions

Numerous in situ observations indicate clearly the presence of nonthermal electron and ion structures as ubiquitous and persistent feature in a variety of astrophysical plasma environments. In particular, the detected suprathermal particle populations are accurately represented by the family of κ-distributions, a power-law in particle speed. After clarifying the characteristics of high-energy tail distributions under various space plasma conditions, different generation mechanisms of energetic particles are introduced where numerical simulations of wave–particle interaction based on a Fokker–Planck approach demonstrate how Landau interaction ultimately leads to κ-like distributions. Because of lack of theoretical justification, the use of the analytical form of κ-functions was frequently criticized. It is shown that these distributions turn out as consequence of an entropy generalization favored by nonextensive thermo-statistics, thus providing the missing link for powerlaw models of suprathermal tails from fundamental physics, along with a physical interpretation of the structure parameter κ. Moreover, with regard to the full nonextensive formalism, compatible also with negative values of κ, it is demonstrated that core–halo distribution structures, as observed for instance under typical interplanetary plasma conditions, are a natural content of the pseudo-additive entropy concept. The significance of the complete κ-distribution family with regard to observed core–halo electron and double-humped ion velocity space characteristics is illuminated, where the observed peak separation scale of interplanetary proton distributions is compatible with a maximum entropy condition.

Fokker–Planck approach to extending the one-flux method of carrier transport in semiconductors to variable energies

The one-flux method of McKelvey, Longini, and Brody [Phys. Rev. 123, 51 (1961)] is extended to arbitrary distributions of the fluxes in energy, based on a Fokker–Planck approach to the Boltzmann equation. The method is used to discuss the backscattering of carriers from a high-field region. It is shown that a key result of the one-flux method, that the backscattering vanishes in the limit of high fields, holds more generally than under the assumptions of the one-flux method.

Influence of Point-Like Disorder on the Hall Resistivity Behavior in an Anisotropic Planar Pinning Potential

Explicit current- and temperature-dependent expressions for anisotropic longitudinal and transverse nonlinear magnetoresistivities are derived and analyzed on the basis of a Fokker–Planck approach for two-dimensional vortex dynamics in a washboard pinning potential in the presence of point-like disorder. Gradually increasing the strength of the point-like pinning (in an experiment this is simply done by irradiation of the sample with different doses of high-energy electrons) this theory predicts a gradual decrease of the anisotropy of the magnetoresistivities. The physics of the transition from the recently discussed new scaling relations for anisotropic Hall resistivities in the absence of point-like pins to the well-known scaling relations for isotropic pinning is elucidated. This is discussed in terms of a gradual isotropization of the guided vortex motion responsible for the existence in a washboard pinning potential of new anisotropic Hall voltages which are odd with respect to magnetic field reversal.

DIFFUSION IN PERIODIC INTERACTING SYSTEMS: QUASI-ELASTIC SPECTRUM IN BISTABLE POTENTIALS

Mechanism of ion transport in periodic interacting systems is investigated by employing the Fokker–Planck approach. This is done through an investigation of the quasielastic peak in the dynamic structure factor S(q, ω). Its half width at half maximum (hwhm) exhibits an oscillatory behavior as a function of the scattering wave-vectors q, reflecting the role of correlations among diffusing particles. All calculations are performed at low temperature and at high friction limit.

Stochastic acceleration by intense laser fields

A stochastic acceleration mechanism which is a direct acceleration mechanism effective for the intense laser case is studied by the Fokker–Planck approach. The Fokker–Planck equation of the electron distribution function is derived from the equation of motion of electrons which interact with filamented laser fields. The Fokker–Planck equation contains nonlinear coefficients and gives an anisotropic distribution in momentum space. The strong directionality of the acceleration is explained. The accelerated electrons tend to be collimated towards the direction of the wave vector. The effective temperature scales as T∝tβ with β≃1.

Fokker–Planck approach to impact ionization distributions in space and time

A Fokker–Planck equation for carrier transport in semiconductors is derived from the Boltzmann transport equation by expanding in Legendre polynomials and assuming the phonon energy exchanged at momentum randomizing collisions is small compared with the mean carrier energy. The method is used to compute impact ionization probability distributions in space and time and the results agree well with those generated by an equivalent Monte Carlo model over a wide range of electric fields from 300 kV/cm to 1 MV/cm.

Simulation approaches to ion channel structure-function relationships.

1. Introduction 4751.1 Ion channels 4751.1.1 Gramicidin 4761.1.2 Helix bundle channels 4771.1.3 K channels 4801.1.4 Porins 4831.1.5 Nicotinic acetylcholine receptor 4831.1.6 Physiological properties 4831.2 Simulations 4841.2.1 Atomistic versus mean-field simulations 4842. Atomistic simulations 4852.1 Modelling of ion-interaction parameters 4852.1.1 Interatomic distances and the problem of ionic radii 4862.1.2 Solvation energy 4872.1.3 Hydration shells and coordination numbers 4892.1.4 Parameters in common use and transferability 4912.1.5 Summary 4912.2 Water in pores versus bulk 4912.2.1 Simple pore models 4942.2.2 gA 4952.2.3 Alm 4962.2.4 LS36 (and LS24) 4962.2.5 Nicotinic receptor M2δ5 4972.2.6 Influenza A M2 4972.2.7 K channels 4972.2.8 nAChR 4982.2.9 Porins 4982.2.10 Relevance 4992.2.11 Problems with simulations 5012.3 Dynamics of ions in pores 5032.3.1 Simple pore models 5032.3.2 Helix bundles 5042.3.3 gA and KcsA 5052.4 Energetics of permeation and ion selectivity 5092.4.1 Potential and free energy profiles 5092.4.2 gA 5102.4.3 α-Helix bundles 5112.4.4 KcsA 5122.4.5 Ion selectivity 5142.4.6 Problems of estimating energetic profiles 5152.5 Conformational changes 5162.5.1 gA 5162.5.2 Alm and LS3 5162.5.3 KcsA 5172.6 Protonation states 5233. Coarse-grained simulations 5243.1 Introduction 5243.1.1 Predicting conductance magnitudes 5253.2 Electro-diffusion: the Nernst–Planck approach 5263.2.1 Calculating the potential profile from Poisson and PB theory 5283.2.2 Calculating the potential profile from BD simulations 5303.2.3 Combining Nernst–Planck and Poisson: PNP 5303.3 Beyond PNP 5323.4 BD simulations 5323.4.1 Basic theory in ion channels 5323.4.2 Incorporating the environment 5333.5 Applications 5353.5.1 Model systems 5353.5.1.1 Solving the Poisson and PB equation for channel-like geometries 5353.5.1.2 Comparing PB, PNP and BD 5363.5.2 Applications to known structures 5373.5.2.1 gA 5373.5.2.2 Porin 5393.5.2.3 LS3 5403.5.2.4 Alm 5423.5.2.5 nAChR 5423.5.2.6 KcsA 5433.6 pKa calculations 5433.7 Selectivity 5443.7.1 Anion/cation selectivity 5453.7.2 Monovalent/divalent ion selectivity 5454. Problems 5464.1 Atomistic simulations 5464.1.1 Problems 5464.1.2 Parameters 5484.2 BD 5494.3 Mean-field simulations 5495. Conclusions 5505.1 Progress 5505.2 The future 5506. Acknowledgements 5517. References 551Ion channels are proteins that form ‘holes’ in membranes through which selected ions move passively down their electrochemical gradients. The ions move quickly, at (nearly) diffusion limited rates (ca. 107 ions s−1 per channel). Ion channels are central to many properties of cell membranes. Traditionally they have been the concern of neuroscientists, as they control the electrical properties of the membranes of excitable cells (neurones, muscle; Hille, 1992). However, it is evident that ion channels are present in many types of cell, not all of which are electrically excitable, from diverse organisms, including plants, bacteria and viruses (where they are involved in functions such as cell homeostasis) in addition to animals. Thus ion channels are of general cell biological importance. They are also of biomedical interest, as several dizeases (‘channelopathies’) have been described which are caused by changes in properties of a specific ion channel (Ashcroft, 2000). Moreover, passive diffusion channels for substances other than ions are common (porins, aquaporins), as are active membrane transport processes coupled to ion gradients or ATP hydrolysis. An understanding of ion channels may also provide a gateway to understanding these processes.

Fokker–Planck approach to the pulse packet propagation in synfire chain

We applied the Fokker–Planck method to the so-called ‘synfire chain’ network model and showed how a synchronous population spike (pulse packet) evolves to a narrow pulse packet (width <1 ms) or fades away, depending on its initial size and width. The results of numerical integration of the Fokker–Planck equation are in good agreement with those of simulations on a network of leaky integrate-and-fire neurons. For a narrow input pulse packet, the integration of the Fokker–Planck equation requires careful numerical treatment. However, we can construct a precise analytical waveform of an output packet, which proves valid for narrow input pulse packets, from the stationary solution to the Fokker–Planck equation and a previously proposed approximate input–output relationship. Our methods enable us also to understand an essential role of the synaptic noise in the evolution, the peculiar temporal evolution of a broader pulse packets, and the irrelevance of the refractory period in determining the waveform of a pulse packet. Furthermore, we elucidate possible functional roles of multiple interactive pulse packets in spatiotemporal information processing, i.e. the association of information and the temporal competition.

Continuities and discontinuities in Planck's Akt der Verzweiflung

Planck's famous communication of December 1900 cannot be understood without taking into account an essential difference between his and Boltzmann's conceptions of the relation between micro- and macro-world. For Boltzmann, macroscopic laws were in principle completely determined by the underlying microscopic dynamics. For Planck, the microscopic dynamics were partly undetermined and had to be supplemented by additional assumptions in order to reach macroscopic laws.Thanks to this leeway, Planck could introduce finite energy-elements and derive his famous black-body law without giving up the continuity and determinism of former dynamical theories. The purpose of this paper is to explain the origins, peculiarities, and evolution of Planck's approach; to show how it ultimately yielded the correct black-body law; and to examine the status of the energy-elements which Planck thereby introduced.

Continuities and discontinuities in Planck's Akt der Verzweiflung

AbstractPlanck's famous communication of December 1900 cannot be understood without taking into account an essential difference between his and Boltzmann's conceptions of the relation between micro‐ and macro‐world. For Boltzmann, macroscopic laws were in principle completely determined by the underlying microscopic dynamics. For Planck, the microscopic dynamics were partly undetermined and had to be supplemented by additional assumptions in order to reach macroscopic laws.Thanks to this leeway, Planck could introduce finite energy‐elements and derive his famous black‐body law without giving up the continuity and determinism of former dynamical theories. The purpose of this paper is to explain the origins, peculiarities, and evolution of Planck's approach; to show how it ultimately yielded the correct black‐body law; and to examine the status of the energy‐elements which Planck thereby introduced.

High field transport in strained Si/GeSi double heterostructure: A Fokker–Planck approach

We report calculations of high electric field transport for the case of a strained Si/GeSi double heterostructure (DHS) considering transport along the Si channel and by applying the analytical Fokker-Planck approach (FPA), where the process is modeled as drift diffusion in energy space. We limit ourselves to electronic transport in the conduction band of the strained Si, where an energy shift between the otherwise degenerate six energy valleys characterizes the band alignment in the DHS. Inter-valley phonon scatterings are considered while intra-valley acoustic phonon scattering is ignored, leading to results valid for high enough temperatures. Our results are compared to previous theoretical work where Monte Carlo simulations were applied. A reasonable agreement between the two approaches is obtained in the high electric field regime.

A test of the Fokker–Planck approach for the description of semiconductor transport properties

We apply the Fokker–Planck approach (FPA), where electronic transport is modeled as a drift-diffusion process in energy space, in the calculation of transport properties of bulk Si for a moderately high electric field along the 〈111〉 direction. The main goal is to test the FPA by comparing the experimental and Monte Carlo data with the FPA results for a well-known semiconductor. As a consequence, a reliable basis for the discussion of the validity and limitations of the FPA would be provided in a realistic model of a well-known semiconductor.

Reaction rates for fluctuating barriers with asymmetric noise

Thermally activated escape over a high barrier fluctuating randomly in time is investigated for asymmetric flipping rates. By means of the Fokker–Planck approach approximate methods developed for symmetric noise are applied to calculate the relaxation eigenvector and ultimate relaxation rate for piecewise linear potentials in the Smoluchowski limit. The ranges of validity of these approaches are discussed. Furthermore, the exact result for weak thermal noise but arbitrary barrier flipping rates is derived and its dependence on the asymmetry studied in detail.

Use of the Fokker–Planck equation in high-field transport problems

The Fokker–Planck drift-diffusion equation is often used to describe Brownian motion of a particle in thermal equilibrium with the surrounding medium. The present paper shows that the equation has the ability to describe charged particle transport far from equilibrium, such as occurs in the presence of a high electric field in a neutral gas or a semiconducting solid. These transport problems are usually tackled by means of the Boltzmann transport equation, but the Fokker–Planck approach is mathematically simpler, and gives insight into the statistics of energy-exchange processes and their thermalization capacity.