H. Kagermann

On the motion of nonspherical particles in a turbulent flow

Noninteracting rigid spheroids are considered which are suspended in a homogeneous turbulent flow. A renormalized expansion is used to derive a kinetic equation whose coefficients are determined by the Lagrangian correlation function of the turbulent velocity field and by the translational diffusion coefficient D and the orientational relaxation frequency ωu of the spheroid. For isotropic turbulence, coupled integral relations for D and ωu are obtained, applying the moment method. These are solved numerically for different correlation frequencies of the energy spectrum. The effect of the renormalization and the resulting coupling between the translational and orientational relaxation are discussed.

Particle motion in stochastic force fields

Usually Fokker-Planck or master equations are used as kinetic equations for the problem of particle motion in stochastic force fields. With the Stratonovich calculus also a kinetic equation is derived which formally looks like a Fokker-Planck equation, but differs in some essential aspect. Moreover, we get an operator equation which is more suitable to derive moment equations than the kinetic equations. As a starting point for the derivation of the Fokker-Planck and master equations it is common to use the Chapman-Kolmogorov equation or the Liouville equation with Markov assumptions. Because this is not necessary for our derivation with the Stratonovich calculus we also give derivations for the former kinetic equations without starting from the Chapman-Kolmogorov equation and show how the Markov assumption could be taken into account at the end of the derivation. In particular we study the motion of charged particles in a stochastic magnetic field. This example shows that significant differences between the various kinetic equations always should be expected in the case of a nonvanishing correlation time of the stochastic force field and nonlinear fluctuation equations.

Memory effects in the motion of nonspherical Brownian particles

A system of nonlinear generalized Langevin equations is stated to describe the motion of noninteracting nonspherical Brownian particles. The coupling of translational and rotational degrees of freedom is taken into account via anisotropic translational and rotational friction coefficients, a Magnus force term and a systematic torque. A kinetic equation is derived with time- dependent coefficient functions, which are explicitly calculated for the one-mode case. As an example the frequency dependence of the coefficient of flow birefringence is discussed.

Nonspherical Brownian particles. Kinetic description and application to flow birefringence

For a system of noninteracting spheroidal Brownian particles a Fokker-Planck equation is derived which takes into account a hydrodynamical coupling between translational and orientational degrees of freedom. With the moment method, transport relaxation equations are obtained from which an expression for the coefficient of flow birefringence is inferred.

Stochastic equations arising from test particle problems

Dynamical functions depending on the state of one marked test particle of a classical many-body system are considered. The time evolution is described by differential equations whose coefficients are random and in addition depend on the initial state of the test-particle. To remove this dependence a weak-coupling approximation is used. Due to the finite correlation time of the driving stochastic process different equations for the test-particle propagator are obtained, if a one-time description is used. It is shown that this ambiguity is characteristic for weakly coupled systems and vanishes only in the weak-coupling limit. The generator of the resulting Markovian process consists of the differentiations with respect to the velocity- and position variables up to second order.

The influence of random temperature modulation on the convective instability

Benard convection is considered under the influence of randomly modulated surface temperature. Using Galerkins method the linearized stochastic dynamical equations for the Fourier modes are stated. The stability is decided by means of the corresponding moment equations and discussed for the two-mode case analytically. It is shown that the onset of instability is always advanced and depends significantly on the Prandtl number and the strength of the temperature fluctuations.

Stochastic equations arising from test particle problems: III. Stability of randomly driven oscillators

Abstract As a model for resonant wave-wave interaction three nonlinearly coupled oscillators in a harmonic lattice are considered. Using a stochastic description the influence of the nonresonant oscillators on the energy transfer is studied in the weak coupling limit. Conditions for amplitude and energy instability and the stationary solutions are discussed.

Stochastic equations arising from test particle problems: II. Application to the harmonic lattice and the electron plasma

Abstract To compare the different kinetic equations derived in a previous paper for weakly coupled systems, the results are applied to coupled harmonic oscillators and a one-component plasma in a magnetic field. Using the harmonic interaction as an example, it is demonstrated that reasonable results can be inferred only from the kinetic equation, which is characterized by an additional time average. Applying this equation to an electron plasma in a magnetic field B, the Balescu-Lenard equation is recovered for B = 0, but a modification is obtained for B ≠. For strong fields the diffusion coefficient is discussed.

Anomalous plasma diffusion across a strong magnetic field

Abstract Moment equations, appropriate for the calculation of time-dependent transport coefficients, are derived with the aid of the generalized Stratonovich method. The starting point is a kinetic equation for the two-time, one-particle probability density including coefficients, which are determined for a stable and homogeneous plasma in a self-consistent manner. In particular, the spatial test particle diffusion across a strong uniform magnetic field is considered in the linit t → ∞ including the influence of finite gyroradius. The diffusion coefficient is a sum over wavenumbers k . The part of the sum with k z = 0 (where z is the direction of the magnetic field) results in an anomalous diffusion coefficient for nonthermal plasmas, which is a generalization of the known results for thermal systems. Further the observation is recovered that for thermal systems the part with nonvanishing k z does not influence the anomalous diffusion. However, an effect may be found if the k z - values of the force-correlation are bounded above, or the electron temperature is high compared to the ion temperature.