Stochastic continuity equation and related processes

Bassi, Gabriele; Bazzani, Armando; Mais, H.; Turchetti, G.

doi:10.1016/j.physa.2004.08.083

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…To derive a diffusion equation for the action variables one considers the stochastic Liouville equation 37,38 for the distribution function ρ(θ, I, t)…”

Section: Diffusion In Hamiltonian Systemsmentioning

confidence: 99%

Diffusion in stochastically perturbed Hamiltonian systems with applications to the recent LHC dynamic aperture experiments

Bazzani

Mazzarisi

Giovannozzi

et al. 2019

Nonlinear Dynamics and Collective Effects in Particle Beam Physics

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In this paper, we review diffusion phenomena in stochastically perturbed Hamiltonian systems, with the aim of defining the framework to use Nekhoroshev-like estimates as prototypes for the form of the diffusion coefficient. A discussion of the features of this framework is carried out. More importantly, the results of numerical simulations based on the proposed models are compared against the experimental data from recent measurements performed at the CERN Large Hadron Collider (LHC) of the extent of phase space where bounded motions occur. The main conclusions are presented and discussed in detail together with future steps.

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“…To derive a diffusion equation for the action variables one considers the stochastic Liouville equation 37,38 for the distribution function ρ(θ, I, t)…”

Section: Diffusion In Hamiltonian Systemsmentioning

confidence: 99%

Diffusion in stochastically perturbed Hamiltonian systems with applications to the recent LHC dynamic aperture experiments

Bazzani

Mazzarisi

Giovannozzi

et al. 2019

Nonlinear Dynamics and Collective Effects in Particle Beam Physics

Self Cite

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“…As a consequence ρ = ρ ξ satisfies the Fokker-Planck equation [16] corresponding to the Stratonovich interpretation of the stochastic Hamilton's equations. This equation, which can be derived by computing the average via a Dyson expansion of the evolution operator for the fluctuating densityρ [17], explicitly reads ∂ρ ∂t…”

Section: The Hamiltonian With Noisementioning

confidence: 99%

“…where x is a point in the phase plane and ξ(t) is a white noise, the stochastic Liouville equation reads [17] ∂ρ ∂t (x, t; ξ)…”

Section: Numerical Schemes For a Generic Hamiltonian With White Noisementioning

confidence: 99%

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Hamiltonian dynamics with a weak noise and the echo effect for the rotator model

Turchetti¹,

Bassi²,

Bazzani³

et al. 2006

J. Phys. A: Math. Gen.

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We analyze the effect of a weak noise on the Hamiltonian transport from the analytical and numerical viewpoint. A solvable model, the noisy rotator, is proposed to illustrate the basic phenomena. In the absence of noise, the phase space evolution is a shear flow, whose angular correlations decay following a power law, which depends on the smoothness of the initial action distribution. If the action has a fluctuating component, given by a Wiener process, then the angular correlations decay exponentially according to e − 2 t 3 /6 or faster, where is the noise amplitude. The echo effect is well suited to investigate the competition between the decorrelation due to filamentation and noise. The noisy rotator model allows an exhaustive analytical investigation of the process for a wide class of initial conditions and a generic disturbance. The echo time is proportional to the delay τ of the disturbance and its amplitude is proportional to λτ where λ is the amplitude of the disturbance. The noise reduces the echo amplitude by e −c 2 τ 3 , where c depends on the Fourier components of the initial angular distribution, and of the disturbance applied at time τ . The analytical results, derived in the limit λ → 0, τ → ∞, with λτ finite and sufficiently small to justify a first order expansion, are checked numerically. For more realistic models the analytical procedure would provide qualitative results and scaling laws. Quantitative results are obtained by solving the Fokker-Planck equation with a numerical scheme based on splitting: back propagation and biquadratic interpolation for the integrable part, implicit finite difference scheme for the noise component. The application to a noisy pendulum describing the longitudinal dynamics in a particle accelerator is considered, and we determine the value of the noise amplitude , below which the echo cannot be detected.

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Stochastic continuity equation and related processes

Cited by 4 publications

References 7 publications

Diffusion in stochastically perturbed Hamiltonian systems with applications to the recent LHC dynamic aperture experiments

Diffusion in stochastically perturbed Hamiltonian systems with applications to the recent LHC dynamic aperture experiments

Hamiltonian dynamics with a weak noise and the echo effect for the rotator model

Symbolic Computations

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