Coarse-grained thermodynamics

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. 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

Nevertheless, large-scale phenomena and complicated phase diagrams cannot be investigated within realistic models at the moment, and this is not very likely to change soon. Therefore, theorists have often resorted to coarse-grained models, which capture the features of the substances believed to be essential for the properties of interest. Such models can provide qualitative and semiquantitative insight into the physics of these materials, and hopefully establish general relationships between microscopic and thermodynamic quantities. 

Pagels show that thermodynamic depth is proportional to the difference between the state s thermodynamic entropy (i.e. its coarse grained entropy) and its finegrained entropy, given by fcex volume of points in phase space corresponding to the system s trajectory, where k], is Boltzman s constant. 

Coarse-grained approaches, multiparticle collision dynamics, 90-92 Coarse velocity, linear thermodynamics, regression theorem, 18-20 Coherence spectroscopy, two-pathway excitation ... 

So far, CG approaches offer the most viable route to the molecular modeling of self-organization phenomena in hydrated ionomer membranes. Admittedly, the coarse-grained treatment implies simplifications in structural representation and in interactions, which can be systematically improved with advanced force-matching procedures however, it allows simulating systems with sufficient size and sufficient statishcal 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. 

The efficiency of the CB-MC technique has been used by Maginn et al. (769), who considered the low-occupancy thermodynamics of sorption of alkanes as long as C25 in silicalite. The locations of such long molecules are no longer correctly predicted by considering the end-to-end vector and the chain midpoint. To overcome this problem, a coarse-graining technique was used to describe both the adsorbate and the zeolite, allowing for accurate microscopic characterization. 

Finally, it should be mentioned that a combination of COSMO-RS with tools such as MESODYN [127] or DPD [128] (dissipative particle dynamics) may lead to further progress in the area of the mesoscale modeling of inhomogeneous systems. Such tools are used in academia and industry in order to explore the complexity of the phase behavior of surfactant systems and amphiphilic block-co-polymers. In their coarse-grained 3D description of the long-chain molecules the tools require a thermodynamic kernel... 

Unlike the density of bulk fluids, which is a function of pressure and temperature only (and composition for a mixture), the average density across the interface between a liquid and its vapor, as well as at the liquid/liquid interface, varies as a function of the distance along the interface normal p(z). Like other local thermodynamic quantities[30], it is defined by a coarse-graining procedure The volume of the system is divided into slabs perpendicular to the interface normal, and the density of each slab is computed in the usual way. The thickness of the slabs is chosen to be small enough so that the density does not vary much... 

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