The maximum entropy principle applied to a dynamical system proposed by Lorenz
Abstract
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 dataassimilation studies, the system's state vector is a periodic array of largescale 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 timederivatives of the energy as constraints. It will be shown that the combination of the energy and its first order timederivative leads to a rather accurate description of the marginal probability density function of individual variables. If the average second order timederivative 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.
 Publication:

European Physical Journal B
 Pub Date:
 January 2014
 DOI:
 10.1140/epjb/e2013406812
 Bibcode:
 2014EPJB...87....7V
 Keywords:

 Statistical and Nonlinear Physics