Oscillatory Survival Probability and Eigenvalues of the Non-Self-Adjoint Fokker--Planck Operator

Oscillatory Survival Probability and Eigenvalues of the Non-Self-Adjoint Fokker-Planck Operator

We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics dxε = a(xε) dt+ √ 2ε b(xε) dw, which is activated by small noise over the boundary of the domain of attraction D of a stable focus of the drift a(x). The boundary ∂D of the domain is an unstable limit cycle of a(x). The oscillations are explained by a singular perturbation expansion of the spectrum of th...

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of ...

Gyrokinetic Fokker-Planck Collision Operator

Gyrokinetic Fokker-Planck Collision Operator

Gyrokinetic Fokker-Planck collision operator.

The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects i...