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G. D. Lythe

Publications and source records attributed to G. D. Lythe.

8 recordsLinked to original sources
arXiv1998

Stochastic Stokes' drift

Classical Stokes' drift is the small time-averaged drift velocity of suspended non-diffusing particles in a fluid due to the presence of a wave. We consider the effect of adding diffusion to the motion of the particles, and show in particular that a non-zero time-averaged drift velocity exists in general even when the classical Stokes' drift is zero. Our results are obtained from a general procedure for calculating ensemble-averaged Lagrangian mean velocities for motion that is close to Brownian, and are verified by numerical simulations in the case of sinusoidal forcing.

physics.class-ph↗
arXiv1998

Stochastic PDEs: domain formation in dynamic transitions

Spatiotemporal evolution in the real Ginzburg-Landau equation is studied with space-time noise and a slowly increasing critical parameter. Analytical estimates for the characteristic size of the domains formed in a slow sweep through the critical point agree with the results of finite difference solution of the stochastic PDEs.

cond-mat.stat-mech↗
arXiv1997

Dynamics controlled by additive noise

Analysis is presented of a system whose dynamics are dramatically simplified by tiny amounts of additive noise. The dynamics divide naturally into two phases. In the slower phase, trajectories are close to an invariant manifold; this allows small random disturbances to exert a controlling influence. A map is derived which provides an accurate description of the trajectories.

adap-org↗
arXiv1997

Noise and dynamic transitions

A parabolic stochastic PDE is studied analytically and numerically, when a bifurcation parameter is slowly increased through its critical value. The aim is to understand the effect of noise on delayed bifurcations in systems with spatial degrees of freedom. Realisations of the nonautonomous stochastic PDE remain near the unstable configuration for a long time after the bifurcation parameter passes through its critical value, then jump to a new configuration. The effect of the nonlinearity is to freeze in the spatial structure formed from the noise near the critical value.

adap-org↗
arXiv1997

A noise-controlled dynamic bifurcation

We consider a slow passage through a point of loss of stability. If the passage is sufficiently slow, the dynamics are controlled by additive random disturbances, even if they are extremely small. We derive expressions for the `exit value' distribution when the parameter is explicitly a function of time and the dynamics are controlled by additive Gaussian noise. We derive a new expression for the small correction introduced if the noise is coloured (exponentially correlated). There is good agreement with results obtained from simulation of sample paths of the appropriate stochastic differential equations. Multiplicative noise does not produce noise-controlled dynamics in this fashion.

adap-org↗
arXiv1997

Stochastic calculus: application to dynamic bifurcations and threshold crossings

For the dynamic pitchfork bifurcation in the presence of white noise, the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter. The threshold crossing problem used, for example, to model the firing of a single cortical neuron is considered, concentrating on quantities that may be experimentally measurable but have so far received little attention. Expressions for the statistics of pre-threshold excursions, occupation density and last crossing time of zero are compared with results from numerical generation of paths.

adap-org↗
arXiv1997

Noise and slow-fast dynamics in a three-wave resonance problem

Recent research on the dynamics of certain fluid dynamical instabilities shows that when there is a slow invariant manifold subject to fast timescale instability the dynamics are extremely sensitive to noise. The behaviour of such systems can be described in terms of a one-dimensional map, and previous work has shown how the effect of noise can be modelled by a simple adjustment to the map. Here we undertake an in depth investigation of a particular set of equations, using the methods of stochastic integration. We confirm the prediction of the earlier studies that the noise becomes important when mu|log(epsilon)| = O(1), where mu is the small timescale ratio and εis the noise level. In addition, we present detailed information about the statistics of the solution when the noise is a dominant effect; the analytical results show excellent agreement with numerical simulations.

adap-org↗