Statistical thermodynamics coarse graining

As discussed in Section 6.5.3, coarse-grained molecular modeling approaches offer the most viable route to the molecular modeling of hydrated ionomer membranes. The coarse-grained treatment implies simplification in interactions, which can be systematically improved with advanced forcematching procedures, but allows simulations of systems with sufficient size and sufficient statistical sampling. Structural correlations, thermodynamic properties, and transport parameters can be studied. 

Thus, Prigogine and Petrosky (PP) introduced the model of a Large Poincare system (EPS). As stated above, the latter is, in fact, a large system, to which the operation of Thermodynamic limit is applied. Clearly, there exists no real system satisfying strictly the definition of a EPS This infinite system is an idealization, on which, by the way, all of statistical mechanics is based. One should thus be more specific about the statement The irreversible processes... cannot be interpreted as approximations of the fundamental laws (statement 1). Quite explicitly, the approximations that are avoided in the PP theory are (a) the arbitrary coarse-graining and (b) the restriction to small parameters. 

In such a situation computer simulations can be enormously helpful. One of their main advantages is that they do not just produce statistical averages of tremendously coarse-grained thermodynamic observables, e.g., energy or pressure. Rather, a simulation often has complete knowledge of the microstate of a system, sometimes even as a function of time. If one knows where all ions are and if one has ways to average, one has access to basically all kinds of correlation functions. This section discusses two of them. 

Abstract We review recent work on scale-bridging modeling approaches applied to aqueous electrolytes and polyelectrolytes, connecting the local quantum chemical details to classical statistical and thermodynamics properties. We discuss solvation and pairing of ions in water, ways to include solvent degrees of freedom in effective ion-ion interactions, and coarse-grained simulations of polyelectrolytes including dielectric boundary effects. 

To imderstand systems consisting of many elements with the aid of thermodynamics and mechanics is the objective of statistical thermod5mamics. The advent of single macromolecule experimental methods emphasizes the importance of the study of mesoscopic systems and fluctuations. Macroscopic and mesoscopic observables (henceforth simply called macro-observables) are interpreted as sums of many microscopic quantities or coarse-grained stochastic quantities. Thus, elucidating macroscopic or mesoscopic behavior in terms of the properties of elements inevitably becomes statistical. The laws of large numbers and the large deviation principle (8) become the key mathematical tools. 

After defining the structural representation of the coarse-grained system and interaction terms, a simulation protocol has to be devised. This protocol involves an equilibration or thermalization phase, as well as production runs. In a production run, the system trajectory is simulated under well-defined thermodynamic conditions, over a specified period of time, in order to generate a statistically meaningful ensemble of uncorrelated system configurations. In the final step, these configurations are analyzed using RDF, and density maps that provide direct information on size, shape, and distribution of phase domains in the composite medium. 

Multiscale Modeling and Coarse Graining of Polymer Dynamics Simulations Guided by Statistical Beyond-Equilibrium Thermodynamics... 

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