Behavioral Equivalences

Beahvioral equivalences serve to establish in which cases two reactive (possible concurrent) systems offer similar interaction capabilities relatively to other systems representing their operating environment. Behavioral equivalences have been mainly developed in the context of process algebras, mathematically rigorous languages that have been used for describing and verifying properties of concurrent communicating systems. By relying on the so called structural operational semantics (SOS), labelled transition systems, are associated to each term of a process algebra. Behavioral equivalences are used to abstract from unwanted details and identify those labelled transition systems that react “similarly” to external experiments. Due to the large number of properties which may be relevant in the analysis of concurrent systems, many different theories of equivalences have been proposed in the literature. The main contenders consider those systems equivalent that (i) perform the same sequences of actions, or (ii) perform the same sequences of actions and after each sequence are ready to accept the same sets of actions, or (iii) perform the same sequences of actions and after each sequence exhibit, recursively, the same behavior. This approach leads to many different equivalences that preserve significantly different properties of systems

Statistics of finite scale local Lyapunov exponents in fully developed homogeneous isotropic turbulence

The present work analyzes the statistics of finite scale local Lyapunov exponents of pairs of fluid particles trajectories in fully developed incompressible homogeneous isotropic turbulence. According to the hypothesis of fully developed chaos, this statistics is here analyzed assuming that the entropy associated to the fluid kinematic state is maximum. The distribution of the local Lyapunov exponents results to be an unsymmetrical uniform function in a proper interval of variation. From this PDF, we determine the relationship between average and maximum Lyapunov exponents, and the longitudinal velocity correlation function. This link, which in turn leads to the closure of von K\`arm\`an-Howarth and Corrsin equations, agrees with results of previous works, supporting the proposed PDF calculation, at least for the purposes of the energy cascade main effect estimation. Furthermore, through the property that the Lyapunov vectors tend to align the direction of the maximum growth rate of trajectories distance, we obtain the link between maximum and average Lyapunov exponents in line with the previous results. To validate the proposed theoretical results, we present different numerical simulations whose results justify the hypotheses of the present analysis.Comment: Research article. arXiv admin note: text overlap with arXiv:1706.0097

Refinement of a previous hypothesis of the Lyapunov analysis of isotropic turbulence

The purpose of this brief comunication is to improve a hypothesis of the previous work of the author (de Divitiis, Theor Comput Fluid Dyn, doi:10.1007/s00162-010-0211-9) dealing with the finite--scale Lyapunov analysis of isotropic turbulence. There, the analytical expression of the structure function of the longitudinal velocity difference $\Delta u_r$ is derived through a statistical analysis of the Fourier transformed Navier-Stokes equations, and by means of considerations regarding the scales of the velocity fluctuations, which arise from the Kolmogorov theory. Due to these latter considerations, this Lyapunov analysis seems to need some of the results of the Kolmogorov theory. This work proposes a more rigorous demonstration which leads to the same structure function, without using the Kolmogorov scale. This proof assumes that pair and triple longitudinal correlations are sufficient to determine the statistics of $\Delta u_r$, and adopts a reasonable canonical decomposition of the velocity difference in terms of proper stochastic variables which are adequate to describe the mechanism of kinetic energy cascade.Comment: 6 page