Simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings

Stochastic differential equations are investigated which reduce in the deterministic limit to the canonical equations of motion of a Hamiltonian system with one degree of freedom. For example, stochastic differential equations of this type describe synchrotron oscillations of particles in storage rings under the influence of external fluctuating electromagnetic fields. In the first part of the article new numerical integration algorithms are proposed which take into account the symplectic structure of the deterministic Hamiltonian system. It is demonstrated that in the case of small white noise the algorithm is more efficient than conventional schemes for the integration of stochastic differential equations. In the second part the algorithms are applied to synchrotron oscillations. Analytical approximations for the expectation value of the squared longitudinal phase difference between the particle and the reference particle on the design orbit are derived. These approximations are tested by comparison with numerical results which are obtained by use of the symplectic integration algorithms.