The maximum entropy principle applied to a dynamical system proposed by Lorenz
The maximum entropy principle applied to a dynamical system proposed by Lorenz
复制标题
洛伦兹提出的应用于动力系统的最大熵原理
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
C. Severijns
中科院分区：
文献类型：
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作者：
W. Verkley;C. Severijns
摘要
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 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.