Most Probable Histories for Nonlinear Dynamics: Tracking Climate Transitions

The eeective action provides an appropriate cost function to determine most probable (or optimal) histories for nonlinear dynamics with strong noise. In such strong-coupling problems, a nonperturbative technique is required to calculate the eeective action. We have proposed a Rayleigh-Ritz variational approximation, which employs simple moment-closures or intuitive guesses of the statistics to calculate the eeective action. We consider here an application to climate dynamics, within a simple \bimodal" Langevin model similar to that proposed by C. Nicolis & G. Nicolis Tellus 33 225 (1981)]. Capturing climate state transitions even in this simple model is known to present a serious problem for standard methods of data assimilation. In contrast, it is shown that the effective action for the climate history is already well-approximated by a one-moment closure and that the optimal, minimizing history robustly tracks climate change, even with large observation errors. Furthermore, the Hessian of the eeective action provides the ensemble variance as a realistic measure of conndence level in the predicted optimal history. 1. The Problem and the Model It is a truism of statistical mechanics that most-probable states are obtained as the extremizers of suitable thermodynamic functions. For example, according to the Boltzmann principle, the most probable state of a system in thermodynamic equilibrium is that which maximizes the entropy. Onsager showed that there are similar variational principles as well to determine most-probable histories, or time-sequences of states 1]. Thus, in thermodynamic equilibrium, the most probable sequence of uctuations is the one that minimizes the Onsager-Machlup action, which represents a time-integrated dissipation or entropy-production Date: May 27, 2000. We thank the organizers for their invitation to participate in the Brussels meeting , \Nonlinear Science: Dynamics and Stochasticity," in honor of the 60th birthday of Gr egoire Nicolis. The innuence of Prof. Nicolis upon our subject is evident and we are pleased to ooer this contribution as a testament to his work.