The Maximum Entropy Principle applied to a dynamical system proposed by Lorenz

Lorenz has proposed a dynamical system in two versions (I and II) that have both proved very useful as benchmark systems in geophysical fluid dynamics. In version I of the system, used in predictability and data-assimilation studies, the system's state vector is a periodic array of large-scale variables that represents an atmospheric field on a latitude circle. The system is driven by a constant forcing, is linearly damped and has a simple form of advection that causes the system to behave chaotically if the forcing is large enough. The present paper sets out to obtain the statistical properties of version I of Lorenz' system by applying the principle of maximum entropy. The principle of maximum entropy asserts that the system's probability density function should have maximal information entropy, constrained by information on the system's dynamics such as its average energy. Assuming that the system is in a statistically stationary state, the entropy is maximized using the system's average energy and zero averages of the first- and higher order time-derivatives of the energy as constraints. It will be shown that the combination of the energy and its first order time-derivative leads to a rather accurate description of the marginal probability density function of individual variables. If the average second order time-derivative of the energy is used as well, also the correlations between the variables are reproduced. By leaving out the constraint on the average energy - so that no information is used other than statistical stationarity - it is shown that the principle of maximum entropy still yields acceptable results for moderate values of the forcing.