Slow relaxation dynamics transport

MD simulations with a constant energy is nothing but Hamiltonian dynamics. Recent accumulation of MD simulations will certainly contribute to our further understanding of Hamiltonian systems, especially in higher dimensions. The purpose of this section is to sketch briefly how the slow relaxation process emerges in the Hamiltonian dynamics, and especially to show that transport properties of phase-space trajectories reflect various underlying invariant structures. 

In this chapter we have discussed the origin of slow relaxation observed MD simulations for liquids or supercooled liquids in the light of the theory of dynamical systems. We have introduced several established scenarios that explain anomalous transports and sticky motions in phase space, and then we examined... 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

There are two hypothetical limiting cases of interest. In one, an infinitely slow cooling rate maintains thermodynamic equilibrium to the ideal glass, and the equilibrium formalism is applicable. In the other a fluid in equilibrium (at its fictive temperature) is quenched infinitely fast to a temperature low enough so that no molecular transport occurs. In this case, what were dynamic fluctuations in time becomes static fluctuations in space. The most elementary treatment of this glass is then as a thermodynamic system with one additional parameter, the fictive temperature. In an actual experiment, of course, relaxations take place and the state of the system is dependent upon its entire thermal history and requires many parameters for its definition. Detailed discussion of the use of irreversible thermodynamics for the study of relaxation processes in liquids and glasses is contained in reviews by Davies (1956, 1960). 

Dynamic properties of i.s.e.s. differ greatly for various electrode types and constructions. When the capacitance of analyte/active surface interface is the only cause of response delay, then relaxation time (or time constant of first-order step-response characteristic) is in the order of milliseconds. When the transport of ions across the dynamic Prandtl layer to the surface of the i.s.e. is the main factor (i.e., this transport is the slowest process of equilibrium reinstallation), for a mixing velocity of about lOcm/s a relaxation time of several seconds occurs. This is typical of solid-membrane electrodes with the exception of glass ones. On the other hand, the limited rate of the exchange process in the liquid membrane, the small diffusion flux of the tested ions into the membrane, the slow dynamics for the creation of diffusion potential and the solubility of the active component of the membrane in the testing solution are the main reasons for the slow response of liquid ion-exchanger electrodes (time constants 10-30 s or even more). 

FIGURE 5.3 Modeling of transient water flux data for Nation 117. (a) The relaxation of the experimental outlet vapor pressure (open circle) for Nafion 117 in LE mode at 50°C, flow chamber volume V = 0.125 L, flow rate V = 0.1 L min membrane area A = 2 cm, and saturation vapor pressure = 12336.7 Pa. Plotted for comparison are model simulations for a slow transport coefficient (dash dot), fast transport coefficient (dash), and a concentration-dependent transport coefficient (gray), (b) Water concentration profiles calculated in the model at different time. (Reprinted from Electrochem. Commun. 13, Rinaldo, S. G. et al. Vaporization exchange model for dynamic water sorption in Nafion Transient solution, 5-7, Figures 1 and 2, Copyright (2011) Elsevier. With permission.)... 

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