Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2010-08-29
Nonlinear Sciences
Chaotic Dynamics
Scientific paper
The evolution of entropy is derived with respect to dynamical systems. For a stochastic system, its relative entropy $D$ evolves in accordance with the second law of thermodynamics; its absolute entropy $H$ may also be so, provided that the stochastic perturbation is additive and the flow of the vector field is nondivergent. For a deterministic system, $dH/dt$ is equal to the mathematical expectation of the divergence of the flow (a result obtained before), and, remarkably, $dD/dt = 0$. That is to say, relative entropy is always conserved. So, for a nonlinear system, though the trajectories of the state variables, say $\ve x$, may appear chaotic in the phase space, say $\Omega$, those of the density function $\rho(\ve x)$ in the new ``phase space'' $L^1(\Omega)$ are not; the corresponding Lyapunov exponent is always zero. This result is expected to have important implications for the ensemble predictions in many applied fields, and may help to analyze chaotic data sets.
No associations
LandOfFree
Remarkable evolutionary laws of absolute and relative entropies with dynamical systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Remarkable evolutionary laws of absolute and relative entropies with dynamical systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Remarkable evolutionary laws of absolute and relative entropies with dynamical systems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-399718