Fokker-Planck Approximation Research Articles

A Fokker–Planck based kinetic model for diatomic rarefied gas flows

A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

Influences of correlated colored-noises on logistic model for tree growth

By virtue of Liouville Theorem and unified colored-noise approximation approach, an approximate Fokker-Planck equation for a tree growth Logistic model subjected to cross-correlated colored noises is derived, and the steady-state probability distribution (SPD) function is obtained. The steady-state properties of the Logistic model are analyzed. We find the following: (1) the position of peak of SPD moves toward left side as D increases while the position of the peak moves toward the contrary direction with Q increasing; (2) the peak of SPD becomes narrow and grows in height as |λ| increases, and for the case of λ >0, the position of peak moves toward right as D increases, but it is opposite for the case of λQ increases.

Logical stochastic resonance in triple-well potential systems driven by colored noise

In this work, the logic stochastic resonance (LSR) phenomenon in a class of stochastic triple-well potential systems is investigated. Approximate Fokker-Planck equation is first obtained by using decoupling approximation. Then, we show that LSR can be successfully induced by additive or multiplicative Gaussian colored noise in some cases. In the absence of internal noise, LSR implementation seems impossible for a = 0 (The parameter a characterizes the depth of the potential well) since the two side wells are so deep that the particle cannot hop over the barrier into the middle well when the input signal is 0. With the increasing of a, the optimal noise band to yield flexible logic gates appears and moves to higher level of noise as the correlation time of noise increases. Compared with the Gaussian white noise, the reliable region in the parameter plane of potential depth parameter a and additive noise strength D first expands and then shrinks with increasing noise color. Furthermore, the effects of multiplicative Gaussian colored noise on LSR are investigated. It was found that the flexible and reliable logic behavior can be yielded for a = 0 due to the fact that the multiplicative Gaussian colored noise strongly affects the shape of the potential function. With the increasing of a, i.e., a = 0.25, multiplicative Gaussian white noise cannot yield desired logic behavior. Fortunately, LSR can also be expected by adjusting the correlation time of Gaussian colored noise. It can also be observed that the reliable region in the parameter plane of potential depth parameter a and multiplicative noise strength Q is small for the case of Gaussian white noise and it becomes larger with the increasing of noise color.

Universality classes of transport in time-dependent random potentials

The growth of the average kinetic energy of classical particles is studied for potentials that are random both in space and time. Such potentials are relevant for recent experiments in optics and in atom optics. It is found that for small velocities uniform acceleration takes place, and at a later stage fluctuations of the potential are encountered, resulting in a regime of anomalous diffusion. This regime was studied in the framework of the Fokker-Planck approximation. The diffusion coefficient in velocity was expressed in terms of the average power spectral density, which is the Fourier transform of the potential correlation function. This enabled to establish a scaling form for the Fokker-Planck equation and to compute the large and small velocity limits of the diffusion coefficient. A classification of the random potentials into universality classes, characterized by the form of the diffusion coefficient in the limit of large and small velocity, was performed. It was shown that one-dimensional systems exhibit a large variety of universality classes, contrary to systems in higher dimensions, where only one universality class is possible. The relation to Chirikov resonances, which are central in the theory of chaos, was demonstrated. The general theory was applied and numerically tested for specific physically relevant examples.

Imaging through turbid media by polarized light

A theoretical approach to the calculation of the image of an inhomogeneity in transillumination of tissue-like media using polarized radiation is developed in the approximation of the basic polarization modes. The method is based on the solution of the nonstationary transfer equation in the Fokker-Planck approximation. An expression for the edge spread function of the image of a totally absorbing half-plane is derived. A generalization to the image profile of inhomogeneity that represents a finite-width stripe is presented. The spatial resolution (sharpness) and contrast of image are analyzed at various time intervals of detection. The image parameters are compared for the polarization-difference technique and imaging in unpolarized light. It is demonstrated that the imaging using the difference of linear polarizations is similar to the imaging using the pulsed unpolarized radiation with a detection interval of about z/2c (z is the thickness of the sample and c is the velocity of light). The results are in agreement with the experimental data on the differential polarization transillumination of tissue-like media.

SUPERGIANT FAST X-RAY TRANSIENTS: SWIFT RESULTS AND THEORETICAL PROGRESS

We present an overview of our Supergiant Fast X-ray Transients (SFXT) project, by highlighting the unique observational contribution Swift is giving to this exciting new field. Since 2007 Swift has been detecting outbursts from these fast transients with the BAT and following them intensively for days with the XRT so that we now have a firm estimate of the time SFXTs spend in each intensity phase. We summarize some preliminary results on our XSPEC model that numerically solves the bulk-motion radiative transfer equation with a strong magnetic field, in the Fokker-Planck approximation.

Time-delay effect of a stochastic genotype selection model

The time-delay effect of a genotype selection model driven by noise is investigated. The approximate delay Fokker-Planck equation is obtained based on the small time delay method and the stochastic equivalent rule. The explicit expression of the steady state probability distribution is derived, and the effect of the time delay on the steady sate properties of the genotype selection model driven by noise is discussed. It is found that the time delay can induce the transition from monostable state to bistable state. The time delay is helpful to select one type of genes from another type of mixing genes. The theoretical predictions are found to be in basic agreement with numerical results.

Colored noise induced switch in the gene transcriptional regulatory system

The colored noise induced switch in the gene transcriptional regulatory system is investigated. The approximate Fokker-Planck equation is obtained based on the Novikov theorem and the Fox approach. The explicit expressions of the steady state probability distribution, the mean value, and the mean first passage time are derived. After the numerical computations, these results show that the TF-A monomer concentration switches from the off position to the on position with the self-correlation time of the multiplicative noise increasing. The TF-A monomer concentration switches from the on position to the off position with the self-correlation time of the additive noise increasing. With the two kinds of the self-correlation time increasing, the mean first passage time becomes large, namely, the TF-A monomer concentration switch becomes difficult. The theoretical predictions are found to be in basic agreement with numerical results.

Fokker–Planck model for computational studies of monatomic rarefied gas flows

In this study, we propose a non-linear continuous stochastic velocity process for simulations of monatomic gas flows. The model equation is derived from a Fokker–Planck approximation of the Boltzmann equation. By introducing a cubic non-linear drift term, the model leads to the correct Prandtl number of 2/3 for monatomic gas, which is crucial to study heat transport phenomena. Moreover, a highly accurate scheme to evolve the computational particles in velocity- and physical space is devised. An important property of this integration scheme is that it ensures energy conservation and honours the tortuosity of particle trajectories. Especially in situations with small to moderate Knudsen numbers, this allows to proceed with much larger time steps than with direct simulation Monte Carlo (DSMC), i.e. the mean collision time not necessarily has to be resolved, and thus leads to more efficient simulations. Another computational advantage is that no direct collisions have to be calculated in the proposed algorithm. For validation, different micro-channel flow test cases in the near continuum and transitional regimes were considered. Detailed comparisons with DSMC for Knudsen numbers between 0.07 and 2 reveal that the new solution algorithm based on the Fokker–Planck approximation for the collision operator can accurately predict molecular stresses and heat flux and thus also gas velocity and temperature profiles. Moreover, for the Knudsen Paradox, it is shown that good agreement with DSMC is achieved up to a Knudsen number of about 5.

Noise-Induced Modulation of the Relaxation Kinetics around a Non-Equilibrium Steady State of Non-Linear Chemical Reaction Networks

Stochastic effects from correlated noise non-trivially modulate the kinetics of non-linear chemical reaction networks. This is especially important in systems where reactions are confined to small volumes and reactants are delivered in bursts. We characterise how the two noise sources confinement and burst modulate the relaxation kinetics of a non-linear reaction network around a non-equilibrium steady state. We find that the lifetimes of species change with burst input and confinement. Confinement increases the lifetimes of all species that are involved in any non-linear reaction as a reactant. Burst monotonically increases or decreases lifetimes. Competition between burst-induced and confinement-induced modulation may hence lead to a non-monotonic modulation. We quantify lifetime as the integral of the time autocorrelation function (ACF) of concentration fluctuations around a non-equilibrium steady state of the reaction network. Furthermore, we look at the first and second derivatives of the ACF, each of which is affected in opposite ways by burst and confinement. This allows discriminating between these two noise sources. We analytically derive the ACF from the linear Fokker–Planck approximation of the chemical master equation in order to establish a baseline for the burst-induced modulation at low confinement. Effects of higher confinement are then studied using a partial-propensity stochastic simulation algorithm. The results presented here may help understand the mechanisms that deviate stochastic kinetics from its deterministic counterpart. In addition, they may be instrumental when using fluorescence-lifetime imaging microscopy (FLIM) or fluorescence-correlation spectroscopy (FCS) to measure confinement and burst in systems with known reaction rates, or, alternatively, to correct for the effects of confinement and burst when experimentally measuring reaction rates.

Large fluctuations and fixation in evolutionary games

We study large fluctuations in evolutionary games belonging to the coordination andanti-coordination classes. The dynamics of these games, modeling cooperationdilemmas, is characterized by a coexistence fixed point separating two absorbingstates. We are particularly interested in the problem of fixation that refers to thepossibility that a few mutants take over the entire population. Here, the fixationphenomenon is induced by large fluctuations and is investigated by a semiclassical WKB(Wentzel–Kramers–Brillouin) theory generalized to treat stochastic systems possessingmultiple absorbing states. Importantly, this method allows us to analyze the combinedinfluence of selection and random fluctuations on the evolutionary dynamics beyond theweak selection limit often considered in previous works. We accurately compute,including pre-exponential factors, the probability distribution function in thelong-lived coexistence state and the mean fixation time necessary for a few mutants totake over the entire population in anti-coordination games, and also the fixationprobability in the coordination class. Our analytical results compare excellently withextensive numerical simulations. Furthermore, we demonstrate that our treatmentis superior to the Fokker–Planck approximation when the selection intensity isfinite.

Reduced multi-scale kinetic models for the relativistic electron transport in solid targets: Effects related to secondary electrons

AbstractA reduced mathematical model for the transport of high current relativistic electron beams in a dense collisional plasma is developed. Based on the hypothesis that the density of relativistic electrons is much less than the plasma density and their energy is much higher than the plasma temperature, a model with two energy scales is proposed, where the beam and plasma electrons are considered as two coupled sub-systems, which exchange the energy and particles due to collisions. The process of energy exchange is described in the Fokker-Planck approximation, where the pitch angle electron-ion and electron-electron collisions dominate. The process of particle exchange between populations, leading to the production of secondary energetic electrons, is described with a Boltzmann term. The electron-electron collisions with small impact parameters make an important contribution in the overall dynamics of the beam electrons.

The mean first-passage time for a cancer development system driven by colored cross-correlated noises

In this paper, the mean first-passage times for a cancer development system driven by colored cross-correlated noises are investigated. Based on the Novikov theorem and the Fox approach, the approximate Fokker-Planck equation and the explicit expressions of the mean first-passage time are derived. Numerical results show that: if the coupling strength between the two noises is negative, the mean first-passage time is a decreasing function of the two noise intensities, but an increasing function of the correlation time; if the coupling strength between the two noises is positive, then the value of the monotonic mean first-passage time versus the additive noise intensity depends on the transition direction. And in addition, the mean first-passage time is a non-monotonic function of the multiplicative noise intensity, but a decreasing function of the correlation time.

Modeling firing-rate dynamics: From spiking to firing-rate networks

Event Abstract Back to Event Modeling firing-rate dynamics: From spiking to firing-rate networks Evan S. Schaffer1* and L. F. Abbott1 1 Columbia University, United States Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models have the obvious shortcoming of using a single time constant (typically a membrane or synaptic time constant) to describe all changes in rate, and they require neurons in a network to fire asynchronously. This is likely to be violated in many cases; in fact, transient synchronization of subgroups of neurons may be an important mechanism for generating rapid behavioral responses. To address this issue without losing the advantages associated with a simple firing-rate description, we have developed a form of firing-rate model based on an approximate Fokker-Planck analysis. A Fokker-Planck equation can be used to describe the distribution of membrane potentials for a population of neurons receiving noisy input. However, most methods for approximating solutions to this type of equation lead to considerably more complex equations than are practical for large networks. For example, there is no closed-form solution describing a population of Integrate-and-Fire (IAF) neurons receiving arbitrary time-varying input. A linear approximation to the response can be calculated, yielding impressively high accuracy, but it involves cumbersome equations (e.g. Brunel & Hakim, 1999; Mattia & Del Giudice, 2002; Ostojic et al., 2009). We show here that in a variant of this model, the Quadratic IAF, the fully nonlinear rate response can be approximated in a surprisingly simple form. Importantly, this approximate solution makes no assumptions about the shape, amplitude, or continuity of the input current. With an understanding of how dynamic external inputs drive firing rates for both asynchronous and synchronous populations of neurons, we study how units described in this way can be linked to describe the firing-rate dynamics of spiking networks with various patterns of synaptic connectivity. We find that the novel firing-rate model captures the time-varying firing-rates of the spiking network across a wide range of parameters. This holds equally well in parameter ranges where the asynchronous state is stable, and where highly synchronized firing occurs. Furthermore, the model also reproduces the dynamics of transient synchronization, which can be quite complicated. Finally, we show that the rich firing dynamics of a network of both excitatory and inhibitory neurons can be well approximated by a coupled E-I rate network. The simplicity of the model we have derived makes it highly amenable to use as the basis for network models. This will hopefully make tractable the study of the dynamics of large networks. Conference: Computational and Systems Neuroscience 2010, Salt Lake City, UT, United States, 25 Feb - 2 Mar, 2010. Presentation Type: Poster Presentation Topic: Poster session II Citation: Schaffer ES and Abbott LF (2010). Modeling firing-rate dynamics: From spiking to firing-rate networks. Front. Neurosci. Conference Abstract: Computational and Systems Neuroscience 2010. doi: 10.3389/conf.fnins.2010.03.00269 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 05 Mar 2010; Published Online: 05 Mar 2010. * Correspondence: Evan S Schaffer, Columbia University, New York, United States, ess2129@columbia.edu Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Evan S Schaffer L. F Abbott Google Evan S Schaffer L. F Abbott Google Scholar Evan S Schaffer L. F Abbott PubMed Evan S Schaffer L. F Abbott Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Comparison of SRIM, MCNPX and GEANT simulations with experimental data for thick Al absorbers

Proton computerized tomography deals with relatively thick targets like the human head or trunk. In this case precise analytical calculation of the proton final energy is a rather complicated task, thus the Monte Carlo simulation stands out as a solution. We used the GEANT4.8.2 code to calculate the proton final energy spectra after passing a thick Al absorber and compared it with the same conditions of the experimental data. The ICRU49, Ziegler85 and Ziegler2000 models from the low energy extension pack were used. The results were also compared with the SRIM2008 and MCNPX2.4 simulations, and with solutions of the Boltzmann transport equation in the Fokker-Planck approximation.

Pure Multiplicative Noises Induced Population Extinction in an Anti-tumor Model under Immune Surveillance

The dynamical characters of a theoretical anti-tumor model under immune surveillance subjected to a pure multiplicative noise are investigated. The effects of pure multiplicative noise on the stationary probability distribution (SPD) and the mean first passage time (MFPT) are analysed based on the approximate Fokker-Planck equation of the system in detail. For the anti-tumor model, with the multiplicative noise intensity D increasing, the tumor population move towards to extinction and the extinction rate can be enhanced. Numerical simulations are carried out to check the approximate theoretical results. Reasonably good agreement is obtained.

Numerical computation of isotropic Compton scattering

Compton scattering is involved in many astrophysical situations. It is well known and has been studied in detail for the past fifty years. Exact formulae for the different cross sections are often complex, and essentially asymptotic expressions have been used in the past. Numerical capabilities have now developed to a point where they enable the direct use of exact formulae in sophisticated codes that deal with all kinds of interactions in plasmas. Although the numerical computation of the Compton cross section is simple in principle, its practical evaluation is often prone to accuracy issues. These can be severe in some astrophysical situations but are often not addressed properly. In this paper we investigate numerical issues related to the computation of the Compton scattering contribution to the time evolution of interacting photon and particle populations. An exact form of the isotropic Compton cross section free of numerical cancellations is derived. Its accuracy is investigated and compared to other formulae. Then, several methods to solve the kinetic equations using this cross section are studied. The regimes where existing cross sections can be evaluated numerically are given. We find that the cross section derived here allows for accurate and fast numerical evaluation for any photon and electron energy. The most efficient way to solve the kinetic equations is a method combining a direct integration of the cross section over the photon and particle distributions and a Fokker-Planck approximation. Expressions describing this combination are given.

Cosmological hydrogen recombination: influence of resonance and electron scattering

In this paper we consider the effects of resonance and electron scattering on the escape of Lyman alpha photons during cosmological hydrogen recombination. We pay particular attention to the influence of atomic recoil, Doppler boosting and Doppler broadening using a Fokker-Planck approximation of the redistribution function describing the scattering of photons on the Lyman alpha resonance of moving hydrogen atoms. We extend the computations of our recent paper on the influence of the 3d/3s-1s two-photon channels on the dynamics of hydrogen recombination, simultaneously including the full time-dependence of the problem, the thermodynamic corrections factor, leading to a frequency-dependent asymmetry between the emission and absorption profile, and the quantum-mechanical corrections related to the two-photon nature of the 3d/3s-1s emission and absorption process on the exact shape of the Lyman alpha emission profile. We show here that due to the redistribution of photons over frequency hydrogen recombination is sped up by DN_e/N_e~-0.6% at z~900. For the CMB temperature and polarization power spectra this results in |DC_l/C_l|~0.5%-1% at l >~ 1500, and therefore will be important for the analysis of future CMB data in the context of the PLANCK Surveyor, SPT and ACT. The main contribution to this correction is coming from the atomic recoil effect (DN_e/N_e~-1.2% at z~900), while Doppler boosting and Doppler broadening partially cancel this correction, again slowing hydrogen recombination down by DN_e/N_e~0.6% at z~900. The influence of electron scattering close to the maximum of the Thomson visibility function at z~1100 can be neglected. (abridged)

Comparison of light scattering models for diffuse optical tomography

We reconstruct images of the absorption and the scattering coefficients for diffuse optical tomography using five different models for light propagation in tissues: (1) the radiative transport equation, (2) the delta-Eddington approximation, (3) the Fokker-Planck approximation, (4) the Fokker-Planck-Eddington approximation and (5) the generalized Fokker-Planck-Eddington approximation. The last four models listed are approximations of the radiative transport equation that take into account forward-peaked scattering analytically. Using simulated data from the numerical solution of radiative transport equation, we solve the inverse problem for the absorption and scattering coefficients using the transport-backtransport method. Through comparison of the numerical results, we show that all of these light scattering models produce good image reconstructions. In addition, we show that these approximations afford considerable computational savings over solving the radiative transport equation. However, all of the models exhibit significant "cross-talk" between absorption and scattering coefficient images. Among the approximations, we have found that the generalized Fokker-Planck-Eddington equation produced the best image reconstructions in comparison with the image reconstructions produced by the radiative transport equation.

Generalized Pearson distributions for charged particles interacting with an electric and/or a magnetic field

The linear Boltzmann equation for elastic and/or inelastic scattering is applied to derive the distribution function of a spatially homogeneous system of charged particles spreading in a host medium of two-level atoms and subjected to external electric and/or magnetic fields. We construct a Fokker–Planck approximation to the kinetic equations and derive the most general class of distributions for the given problem by discussing in detail some physically meaningful cases. The equivalence with the transport theory of electrons in a phonon background is also discussed.