Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.     

Frequency shift of the cesium clock transition due to blackbody radiation

We perform ab initio calculations of the frequency shift induced by a static electric field on the cesium clock hyperfine transition. The calculations are used to find the frequency shifts due to blackbody radiation. Our result [deltanu/E2=-2.26(2)x10(-10) Hz/(V/m)2] is in good agreement with early measurements and ab initio calculations performed in other groups. We present arguments against recent claims that the actual value of the effect might be smaller. The difference (approximately 10%) between ab initio and semiempirical calculations is due to the contribution of the continuum spectrum in the sum over intermediate states.     

An approximate formula for the diffusion coefficient for the periodic Lorentz gas

An approximate stochastic model for the topological dynamics of the periodic triangular Lorentz gas is constructed. The model, together with an extremum principle, is used to find a closed form approximation to the diffusion coefficient as a function of the lattice spacing. This approximation is superior to the popular Machta and Zwanzig result and agrees well with a range of numerical estimates.     

Stochastische Theorie der magnetischen Relaxation

Let a single magnetic dipole be in a constant magnetic field ?₀ and a fluctuating field ?′(t). For this problem the equation of motion is solved exactly. Averaging the magnetic moment over a convenient ensemble relations of relaxation of a macroscopic system of many spins are obtained. In this way a relaxation tensorΦ for magnetisation is derived from the stochastic properties of the fluctuating field which is a generalization ofKubo's oscillator model. For small timesΦ approaches a Gaußian for large times an exponential function. There are two relaxation times which may be expressed by two correlation times of the fluctuating field. The results give a classification of the absorption line shape in Gaußian and Lorentzian type.     

Stochastische theorie der magnetischen relaxation. II

A stochastic theory of magnetic relaxation developped in a former work is expanded. As an application the longitudinal nuclear spin-spin-relaxation in solids and spinlattice-relaxation in liquids are treated. Oscillating relaxation functions are calculated. The results are in good agreement with experimental measurements at CaF₂.