Spatial Diffusion Equation Research Articles

Kinetic Models for Topological Nearest-Neighbor Interactions

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys 163:41–60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

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.

The telegraph equation for cosmic-ray transport with weak adiabatic focusing

Time-dependent solutions of a spatial diffusion equation are often used to describe the transport of solar energetic particles, accelerated in large solar flares. Approximate analytical solutions of the diffusion approximation can complement and guide detailed numerical solutions of the Fokker-Planck equation for the particle distribution function. The accuracy of the diffusion approximation is limited, however, because the signal propagation speed is infinite in the diffusion limit. An improved description of cosmic-ray transport is provided by the telegraph equation, characterised by a finite signal propagation speed. We derive the telegraph equation for the particle density, taking into account adiabatic focusing in a large-scale interplanetary magnetic field in a weak focusing limit. As an illustration, we calculate a propagating pulse solution of the telegraph equation, determine the rise time when the maximum particle intensity is reached at a given distance from the Sun, and compare the results with those obtained in the diffusion approximation. In comparison with the diffusion equation, the telegraph equation predicts an asymmetrical shape of the pulse and a shorter rise time. These potentially significant differences suggest that the more accurate telegraph equation should be used in analysis of the solar energetic particle data, at least to quantify the accuracy of the focused diffusion model.

The inapplicability of spatial diffusion models for solar cosmic rays

view Abstract Citations (4) References (28) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The inapplicability of spatial diffusion models for solar cosmic rays Owens, A. J. ; Gombosi, T. I. Abstract It is noted that because of the rapid temporal evolution of the particle density, the interplanetary propagation of solar cosmic rays cannot be accurately described by a spatial diffusion equation. Using a simple numerical integration scheme, it is shown that the pitch-angle scattering equation gives intensity-time profiles and anisotropies very different from the spatial diffusion results in many cases. Reasons are given for the failure of the diffusion limit, and an explanation is offered for the fact that some temporal profiles of solar particle events look diffusive. Publication: The Astrophysical Journal Pub Date: April 1981 DOI: 10.1086/158812 Bibcode: 1981ApJ...245..328O Keywords: Cosmic Rays; Interplanetary Medium; Particle Diffusion; Solar Corpuscular Radiation; Anisotropic Media; Numerical Integration; Solar Flares; Time Response; Solar Physics full text sources ADS |

Numerical study of solar flare particle propagation in the heliosphere

Numerical solutions are presented for the propagation of solar cosmic rays interplanetary space, including the effects of pitch-angle scattering and adiabatic focusing. The intensity-time profiles can be well fitted by a simple radial spatial diffusion equation with scattering mean-free path λ fit. For low-rigidity particles the radial mean-free path so obtained is significantly larger than the mean-free path calculated from the scattering coefficient due to the inapplicability of the diffusive approximation early in the event. The well-known discrepency between λ fit and the theoretical predictions may be resolved by these calculations.

The interplanetary transport of solar cosmic rays

Numerical solutions are presented for the propagation of solar cosmic rays in interplanetary space, including the effects of pitch-angle scattering and adiabatic focusing. The intensity-time profiles can be well fitted by a simple radial spatial diffusion equation with scattering mean-free path lambda(fit). The radial mean-free path so obtained is significantly larger than the true scattering mean-free path for low-rigidity particles due to both adiabatic focusing and the inapplicability of the diffusive approximation early in the event. The well-known discrepancy between lambda(fit) and the theoretical predictions may be resolved by these calculations.

Cosmic ray propagation in interplanetary space

The validity of the test particle picture and the approximation of static fields, as well as the spatial diffusion approximation, are discussed in a general way before specific technical assumptions are introduced. It is argued that the spatial diffusion equation for the intensity per unit energy has a much wider range of applicability than the kinetic (Fokker‐Planck) equation it is derived from. This gives a strong weight to the phenomenological propagation theory. Its general success (and possible failure at small energies) for the modulation of galactic cosmic rays and solar events is described. Apparent effects like the ‘free boundary’ are given disproportionate weight, since through them the connection to the detailed plasma physics of the solar wind is established. Greatest attention in detail is paid to the pitch angle diffusion theory. A general theory is presented that removes the well‐known secularities of the quasi‐linear approximation. The possible breakdown of any pitch angle diffusion theory at very small energies is perhaps connected to the observed ‘turn up’ of the spectrum at low energies. A first attempt to derive the spatial dependence of the diffusion coefficient in the solar cavity, using such a divergence free scattering theory, is described and compared with recent observations out to 5 AU.