No abstract

Abstract. Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of Wiener processes in these fully implicit schemes are substituted by some truncated random variables. A number of symplectic integrators is constructed. Special attention is paid to systems with separable Hamiltonians. Some results of numerical experiments are presented. They demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones.

We present a new method for isothermal rigid body simulations using the quaternion representation and Langevin dynamics. It can be combined with the traditional Langevin or gradient (Brownian) dynamics for the translational degrees of freedom to correctly sample the canonical distribution in a simulation of rigid molecules. We propose simple, quasisymplectic second-order numerical integrators and test their performance on the TIP4P model of water. We also investigate the optimal choice of thermostat parameters.

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantage of this approach is its simplicity and universality. It works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus and we study only smooth test functions. However we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.

Phase variation of surface structures occurs in diverse bacterial species due to stochastic, high frequency, reversible mutations. Multiple genes of Campylobacter jejuni are subject to phase variable gene expression due to mutations in polyC/G tracts. A modal length of nine repeats was detected for polyC/G tracts within C. jejuni genomes. Switching rates for these tracts were measured using chromosomally-located reporter constructs and high rates were observed for cj1139 (G8) and cj0031 (G9). Alteration of the cj1139 tract from G8 to G11 increased mutability 10-fold and changed the mutational pattern from predominantly insertions to mainly deletions. Using a multiplex PCR, major changes were detected in ‘on/off’ status for some phase variable genes during passage of C. jejuni in chickens. Utilization of observed switching rates in a stochastic, theoretical model of phase variation demonstrated links between mutability and genetic diversity but could not replicate observed population diversity. We propose that modal repeat numbers have evolved in C. jejuni genomes due to molecular drivers associated with the mutational patterns of these polyC/G repeats, rather than by selection for particular switching rates, and that factors other than mutational drift are responsible for generating genetic diversity during host colonization by this bacterial pathogen.

Abstract. Hamiltonian systems with additive noise possess the property of preserving symplectic structure. Numerical methods with the same property are constructed for such systems. Special attention is paid to systems with separable Hamiltonians and to second-order differential equations with additive noise. Some numerical tests are presented. Key words.Hamiltonian systems with additive noise, symplectic integration, mean-square methods for stochastic differential equations

A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDEs) in which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. The proposed special balanced scheme is explicit and its mean-square order of convergence is 1/2. Some numerical tests are presented.Key words. SDEs with nonglobally Lipschitz coefficients, numerical integration of SDEs in the mean-square sense, balanced methods, fully implicit methods, strong convergence, almost sure convergence

S ta b le and fast sem i-im p licit in teg r a tio n o f th e sto ch a stic L andau-L ifshitz eq u a tio n J .H . M e n tin k 1, M .V . T retyak ov2, A . F a so lin o 1, M .I. K a tsn e lso n 1, T h . R a sin g 1 A b s t r a c t . We propose new sem i-im plicit num erical m ethods for th e integ ratio n of th e sto ch astic L andau-L ifshitz equation w ith b u ilt-in angular m om entum conservation. T he perform ance of th e proposed in teg rato rs is te ste d on th e 1D H eisenberg chain. For th is system , our schem es show b e tte r stab ility properties and allow us to use considerably larger tim e steps th a n sta n d a rd explicit m ethods. A t th e sam e tim e, these sem i-im plicit schem es are also of com parable accuracy to and co m putationally m uch cheaper th a n th e sta n d a rd m idpoint im plicit m ethod. T he results are of key im p o rtan ce for atom istic spin dynam ics sim ulations and th e stu d y of spin dynam ics beyond th e m acro spin approxim ation.