Coarse-Grained Variables and Models

We start with a microscopic polymer model, whose state is specifiedby a point in 6N-dimensional phase space, z e f with z = (ri. r pj. p ), a short notation for the positions and momenta of all N particles. The model is described by the microscopic Hamiltonian H(z) with inter- and intramolecular interactions. The coarse-grained model eliminates some of the (huge number of) microscopic degrees of freedom. The level of detail that is retained is specified by the choice of coarsegrained variables x = (xi. ) with 

Instead of the full, microscopic distribution q(z), the coarse-grained model is already specified by the reduced probability distribution p(x) = (6(x—fl(z))). Knowledge of p(x) allows to calculate averages of quantities a(II(z)) via (a(n)) = Jpdza(n(z))Q(z) = Jdxa(x)p(x). Instead of p(x), coarse-grained models are often described by the so alled potentials of mean force l/mf T In p(x) 

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It is still an open problem to extend the above analysis to models (such as studied by Ivanov et al. [122]) where an attractive interaction between the effective monomers is also present, so that variable solvent quality is implicitly modeled. Clearly it will require a major effort to extend the techniques described for short alkanes (Sects. 4.1 and 4.2) to coarse-grained off-lattice models for stiff chains in explicit solvent. 

FIGURE 26.11 The schematics of coarse-graining idea realized by the top-down thermodynamically consistent DPD model. The Voronoi cell represents a fragment of continuum fluid. This fragment can be treated as a dissipative particle with variable mass, volume, temperature and entropy. The thermodynamically consistent DPD particles interact with forces dependent not only on their positions and velocities but on the current thermodynamic states of interacting particles as well. 

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