Empirical master equations: Numerics

Abstract

Empirical master equations are linear forecast equations for the probability density distribution in the discrete phase space of observed variables. Their coefficients are determined from data. The choice of the discretization is crucial for the success of the empirical master equations which are used both for prediction and system analysis. The quality of forecasts depends, in particular, on the choice of the time step Dt and of the spatial resolution Dq. The related numerical problems are discussed by working with data from twodimensional dynamical systems. First, simple flow configurations in phase space like pure transport, deformation and rotation are prescribed to provide the data for the master equations. The coefficients of the empirical master equations for these simple flows can be evaluated exactly. The probability density function related to these simple flows is governed by. a Liouville equation which can be solved analytically. These exact solutions are compared to the forecasts obtained from the master equations where mainly uniform Cartesian grids are prescribed. It is found that the numerical diffusion of the predictions decreases with increasing time step in all cases. It is demonstrated that this effect is due to the adjustment of the coefficients of the master equations to the time step. The numerical diffusivity is shown to vary ∼ Dq 2 /Dt in pure transport where analytic solutions of the master equations are available. The master equations provide excellent forecasts for the mean positions of clouds of states in unbounded domains. This forecast quality deteriorates in the presence of boundaries. It is shown, however, that perfect forecasts are possible in all these linear flow situations provided the grid and the time step are adapted to the flow. Second, empirical master equations are derived from the data of the chaotic baker's transformation. It is shown that perfect empirical master equations can be designed in this case provided the grid is uniform. This shows that chaotic systems are not per se difficult to handle by empirical master equations.