Title :
Equilibrium fluctuations in an N-particle coasting beam: Schottky noise effects
Author :
Bassi, G. ; Ellison, J.A. ; Heinemann, K.
Author_Institution :
Cockcroft Inst., Liverpool
Abstract :
We discuss the longitudinal dynamics of an unbunched beam with a collective effect due to the vacuum chamber and with the discreteness of an N-particle beam (Schottky noise) included. We start with the 2N equations of motion (in angle and energy) with random initial conditions. The 2D phase space density (Klimontovich density) for the N particles is a sum of delta functions and satisfies the Klimontovich equation and the Vlasov equation. An arbitrary function of the energy also satisfies the Vlasov equation and we linearize about a convenient equilibrium density taking the initial conditions to be independent, identically distributed random variables with the equilibrium distribution. The linearized equations can be solved using a Laplace transform in time and a Fourier series in angle. The resultant stochastic process for the phase space density is analyzed and compared with a known result. Work is in progress to study the full nonlinear problem.
Keywords :
Fourier series; Laplace transforms; Vlasov equation; beam handling techniques; fluctuations; functional analysis; particle beam dynamics; random noise; random processes; stochastic processes; 2D phase space density; 2N equations of motion; Fourier series; Klimontovich density; Klimontovich equation; Laplace transform; N-particle coasting beam; Schottky noise effects; Vlasov equation; collective effect; delta functions; equilibrium density; equilibrium fluctuations; linearized equations; longitudinal dynamics; phase space density; random variables; stochastic process; unbunched beam; vacuum chamber; Erbium; Fluctuations; Fourier series; Impedance; Laplace equations; Nonlinear equations; Particle beams; Random variables; Stochastic processes; Zinc;
Conference_Titel :
Particle Accelerator Conference, 2007. PAC. IEEE
Conference_Location :
Albuquerque, NM
Print_ISBN :
978-1-4244-0916-7
DOI :
10.1109/PAC.2007.4440000