Nonequilibrium Working Group: Fundamental Science

In order to go from the microscopic degrees of freedom at the shortest spatiotemporal scales to the nonlinear coherent structures and relevant physical variables which control the macroscopic dynamics, a variety of closely related theoretical and numerical methods will be employed. The strategy is summarized below. The ultimate product of this approach will be a bridge from a microscopic theory to the significant macroscopic variables that describe the relaxation from nonequilibrium phases in realistic experimental situations. For example, a predictive transport equation will give enormously more physical meaning to the representation of nonequilibrium processes in practical computer codes.

More detailed descriptions of the problems to be tackled will be made available soon. Examples of some of the equations may be found here.

Step I

The first priority will be a reliable characterization of the spectrum and statistics of fluctuations intrinsic to the microtheory. Practical methods for accounting for both mean fields and fluctuations in a systematic way have been developed only recently by several of the participants in this project. Fluctuation-dissipation relations are automatically satisfied in this self-consistent treatment, and one obtains a well-defined set of equations for the hierarchy of correlations in a moment expansion. Solving these equations will require high spatiotemporal resolution.

Step II

The high resolution modeling will be used to identify nonlinear coherent structures in the mean fields and the significant physical variables so that a systematic coarse-graining or thinning of degrees of freedom can be carried out. This will focus the computational effort efficiently on the collective modes of greatest interest in long-time prediction. Techniques for analytically and numerically thinning degrees of freedom include homogenization, static and dynamical renormalization group (RG) methods, and nonlinear slaving methods. This systematic incorporation of small scale fluctuations in a self-consistent RG framework holds the greatest promise of extending the predictive modeling of nonequilibrium systems.

Step III

The result of the thinning process is the determination of an effective theory for the collective degrees of freedom interacting with the correct noise, which is generally nonlocal and colored. At this stage, new algorithms will be developed to solve the resulting stochastic PDEs.

Step IV

This next level of theory at the mesoscale will be compared to the underlying microtheory for numerical control and verification as well as validating experiments. Finally, either a direct comparison with experimental data can be carried out, or further thinning of degrees of freedom may be necessary in order to reach the macroscales of interest.

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