Dynamical systems time asymmetry

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Mossbauer spectroscopy should also be mentioned here as a very promising method for combining the structural and dynamic studies of biomolecular systems. The asymmetry of Mossbauer spectra caused by the anisotropy of vibrations of Mossbauer atoms allowed—for example, to find that the mean square amplitude of vibrations of Fe atoms normal to the plane of porphyrin ring (which are responsible for many important biological functions of hemoproteins) is about five times larger than in the... 

Such an attitude to equilibrium thermodynamics - the science which revealed irreversibility of the evolution of isolated systems and asymmetry of natural processes with respect to time - is related to some circumstances that require a thorough analysis. Here we will emphasize only one of them which is the most important for imderstanding further text. It lies in the fact that the most important notion of thermodynamics, i.e. equilibrium, became interpreted exclusively as the state of rest (absence of any forces and flows in the thermodynamic system) and equilibrium processes - as those identical to reversible ones. These one-sided interpretations ignored the Galileo principle of relativity, the third law of Newton and the Boltzmann probabilistic interpretations of entropy that allow dynamic interpretations of equilibria and irreversible interpretations of equilibrium processes. 

Numerical simulations were performed in the authors group in order to better understand experimental observations [77, 80]. They are based on minimal assumptions about the system a random distribution of TLS s in space around the molecule, an interaction decreasing like the inverse cubic distance between molecule and TLS, and a broad spread of asymmetries and jumping times due to tunneling dynamics and disorder. No correlation was assumed between jumping rates, asymmetries and distances from the molecule. Simulated lineshapes and correlation functions ate qualitatively similar to experimental data (Fig. 13). They confirm that the isolation of a single TLS in the correlation function is possible, even when several... 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

On the other hand, a step decrease in feed hydrogen resulted in a relatively very rapid and monotonic decline to the final steady-state ethylene concentration. It should be noted that the sum of all hydraulic and mixing lags for this system is of the order of 75 s and the diffusional relaxation time (R /Dg) is much smaller than one second. Hence, the extremely slow response observed in the step-up experiment and its asymmetry compared to the step-down result suggest that non-linear dynamics of the gas phase-catalyst surface interaction play a major role in unsteady reactor behavior. 

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