Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0401027

On the master equation approach to kinetic theory: linear and nonlinear Fokker--Planck equations

Michael K. -H. Kiessling·Carlo Lancellotti
View original source ↗Read source document ↗Export metadata ↓

Abstract

We discuss the relationship between kinetic equations of the Fokker-Planck type (two linear and one non-linear) and the Kolmogorov (a.k.a. master) equations of certain N-body diffusion processes, in the context of Kac's "propagation of chaos" limit. The linear Fokker-Planck equations are well-known, but here they are derived as a limit N->infty of a simple linear diffusion equation on (3N-C)-dimensional N-velocity spheres of radius sqrt(N) (with C=1 or 4 depending on whether the system conserves energy only or energy and momentum). In this case, a spectral gap separating the zero eigenvalue from the positive spectrum of the Laplacian remains as N->infty,so that the exponential approach to equilibrium of the master evolution is passed on to the limiting Fokker-Planck evolution in R^3. The non-linear Fokker-Planck equation is known as Landau's equation in the plasma physics literature. Its N-particle master equation, originally introduced (in the 1950s) by Balescu and Prigogine (BP), is studied here on the (3N-4)-dimensional N-velocity sphere. It is shown that the BP master equation represents a superposition of diffusion processes on certain two-dimensional sub-manifolds of R^{3N} determined by the conservation laws for two-particle collisions. The initial value problem for the BP master equation is proved to be well-posed and its solutions are shown to decay exponentially fast to equilibrium. However, the first non-zero eigenvalue of the BP operator is shown to vanish in the limit N->infty. This indicates that the exponentially fast approach to equilibrium may not be passed from the finite-N master equation on to Landau's nonlinear kinetic equation.

Explore related subjects

math-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael K. -H. Kiessling, Carlo Lancellotti. 2004-09-27. On the master equation approach to kinetic theory: linear and nonlinear Fokker--Planck equations. https://doi.org/10.1081/tt-200053929

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Local conservation laws, symmetries and Hamiltonian structure of three-dimensional incompressible Euler flow

All smooth local conservation laws, symmetries and adjoint symmetries of three-dimensional homogeneous incompressible Euler flow are classified in the primitive velocity--pressure variables. Explicit dependence on time and position and arbitrary finite orders of mixed time and spatial derivatives are allowed. Up to trivial currents, the conservation laws consist of incompressibility, linear and angular momentum, kinetic energy, helicity and generalised time-dependent momentum. Every local adjoint symmetry is represented by a multiplier of these currents. Every finite-order local symmetry is equivalent on solutions to a classical point symmetry, resolving a conjecture of Olver. An explicit local first-order Hamiltonian structure of the primitive system is also established in the geometric theory of partial differential equations. It maps the energy multiplier to time translation; its kernel on local adjoint symmetries is spanned by the helicity and incompressibility multipliers. The proof first excludes higher-order symmetries by comparing their responses to transverse and harmonic characteristic directions in compatible derivative data. The Hamiltonian map and its kernel then determine all local adjoint symmetries from the symmetry classification. Exact multiplier identities and the correspondence between multipliers and current classes complete the conservation-law classification.

math-ph↗
arXiv2026

Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies

We construct generalized Frobenius structures on Hurwitz spaces of arbitrary genus using extended primary differentials. Through contour integrals involving the Cauchy kernel, we derive explicit dispersionless Lax representations for the principal hierarchies of these structures and their almost-dual counterparts. We also construct solutions to the open WDVV equations for both the original and almost-dual structures, and derive explicit principal hierarchies for the resulting rank-one extensions. Finally, we show that these constructions admit Shramchenko deformations.

math-ph↗
arXiv2026

The dynamics through the first two Lagrange necks in the small-mass restricted three-body problem

We study the planar circular restricted three-body problem (PCR3BP) for sufficiently small mass ratios, from low energies to slightly above the second Lagrange value. We show that retrograde orbits around the primaries and Lyapunov orbits at the necks form the binding of a finite energy foliation. We construct these foliations using the theory of pseudoholomorphic curves in symplectizations developed by Hofer, Wysocki and Zehnder. Slightly above the second Lagrange value, the dynamics is transverse to families of disks bounded by the retrograde orbits, and cylinders connecting the retrograde orbits to the Lyapunov orbits, forming a $3-2-3-2$ foliation. The proof uses the rotating Kepler problem and Hill's lunar problem to control the singular limit as the mass ratio $μ\to 0$. As a consequence of the foliation, both Lyapunov orbits have infinitely many homoclinic orbits.

math-ph↗