Non-equilibrium geosystems: analyzing Lyapunov concepts on pattern formation



Abstract 1-
Introduction[1] In geomorphology literature, equilibrium refers to researches of Davis and
Gilbert who presented steady-state and dynamics equilibrium. To analyze chaotic
geosystems and pattern formation in non-equlibrium conditions, the nonlinear
thermodynamics concepts is needed. In this article, have been analyzed
geosystems responses in non-equilibrium states by Lyapunov theory.
Self-organization of dissipative systems leads to form regular pattern in
non-equilibrium which can be a sign to forecast geosystem evolutions.   Non-Equilibrium Geosystems:  The second law of thermodynamics asserts that if a spontaneous reaction
occurs, the reaction moves towards an irreversible state of equilibrium and in
the process, becomes increasingly random or disordered. It is this increasing
disorder or entropy of a system that forces a spontaneous reaction to persist
but, once a system attains maximum entropy or equilibrium, the spontaneous
reaction ceases to continue. Non-equilibrium systems
are maintained in a state away from thermodynamic equilibrium by the steady
injection and transport of energy. Most interesting to us is systems displaying
regular or nearly regular spatial structures same ripple marks and river
pattern in geosystems. A particular
nonequilibrium system can be thought of as occupying a point in a
three-dimensional parameter space with axes labeled by three dimensionless
parameters R, Γ, and N. The parameter R is some
dimensionless parameter like the Rayleigh number that measures the strength of
driving compared to dissipation. For many systems, driving a system further
from equilibrium by increasing R to larger values leads to chaos and then to
ever-more complicated spatiotemporal states for which there is ever finer
spatial structure and ever faster temporal dynamics.   Lyapunov theory and regular pattern An n-dimensional (where the number of dimensions equals the number of
components) system has n Lyapunov exponents, which determine the rate of
convergence or divergence of initially similar system states in the system
phase space, and thus the sensitivity to perturbations or to variations in
initial conditions. The system is not, and
cannot be, chaotic unless there is at least one positive Lyapunov exponent.
Because an unstable system has at least one λ > 0, dynamic instability is
tantamount to a chaotic system. Deterministic chaos is a property of some
nonlinear systems whereby even simple deterministic systems can produce
complex, pseudorandom patterns, independently of stochastic forcing or
environmental heterogeneity. In chaotic systems complexity and unpredictability
are inherent in system dynamics. Such systems are strongly sensitive to initial
conditions, in that initially similar states diverge exponentially, on average,
and become increasingly different over time. Chaotic systems are also sensitive
to perturbations of all magnitudes. The Kolmogorov (K-)
entropy of a nonlinear system measures its 'chaoticity', because K-entropy is
equal to the sum of the positive Lyapunov exponents. In real landscapes,
measured entropy can be due to deterministic complexity, or to 'colored noise',
the combination of randomness and deterministic order. Culling (1988b) was
apparently the first to suggest exploiting the relationship between K-entropy
(estimated using standard statistical or information theoretic entropy
measures) and chaos in geomorphic systems. There are three forms of entropy referred to in geomorphology.
Thermodynamic entropy is a measure of the amount of thermal energy unavailable
to do work, or the disorder in a closed system. Statistical (information
theoretic) entropy measures the loss of information in a transmission, or the
degree of disorder in a statistical distribution. Kolmogorov (K-) entropy
measures the expansion of a system's phase space (the n-dimensional space
defining all possible system states or combinations of values of the n
components).   Conclusion Many geomorphic
systems show clear evidence of chaotic dynamics and deterministic complexity.
These phenomena cause many nonlinear dynamical systems to behave unpredictably
(at certain scales), to exhibit extraordinary sensitivity to initial
conditions, and to show complicated, pseudorandom patterns even in the absence
of environmental heterogeneity and stochastic forcing. The positive λ reflect
the K-entropy or 'chaoticity', and the rate of disorganization the negative λ
give the rate of organization. If an open, dissipative geomorphic system is to
organize itself, there must be at least one positive Lyapunov exponent, but the
sum of λ must be negative. The sum of the diagonal elements of the system
interaction matrix is equal to the sum of real parts of the complex
eigenvalues, and to the Lyapunov exponents.