Coarse-grained modeling, coarse-grained models, aim at simulating the behaviour of complex systems using their coarse-grained (simplified) representation. Coarse-grained models are widely used for molecular modeling of biomolecules at various granularity levels.
A wide range of coarse-grained models have been proposed. They are usually dedicated to computational modeling of specific molecules: proteins, nucleic acids, lipid membranes, carbohydrates or water. In these models, molecules are represented not by individual atoms, but by "pseudo-atoms" approximating groups of atoms, such as whole amino acid residue. By decreasing the degrees of freedom much longer simulation times can be studied at the expense of molecular detail. Coarse-grained models have found practical applications in molecular dynamics simulations. Another case of interest is the simplification of a given discrete-state system, as very often descriptions of the same system at different levels of detail are possible. An example is given by the chemomechanical dynamics of a molecular machine, such as Kinesin.
The coarse-grained modeling originates from work by Michael Levitt and Ariel Warshel in 1970s. Coarse-grained models are presently often used as components of multiscale modeling protocols in combination with reconstruction tools (from coarse-grained to atomistic representation) and atomistic resolution models. Atomistic resolution models alone are presently not efficient enough to handle large system sizes and simulation timescales.
Coarse graining and fine graining in statistical mechanics addresses the subject of entropy , and thus the second law of thermodynamics. One has to realise that the concept of temperature cannot be attributed to an arbitrarily microscopic particle since this does not radiate thermally like a macroscopic or "black body". However, one can attribute a nonzero entropy to an object with as few as two states like a "bit" (and nothing else). The entropies of the two cases are called thermal entropy and von Neumann entropy respectively. They are also distinguished by the terms coarse grained and fine grained respectively. This latter distinction is related to the aspect spelled out above and is elaborated on below.
The Liouville theorem (sometimes also called Liouville equation)
states that a phase space volume (spanned by and , here in one spatial dimension) remains constant in the course of time, no matter where the point contained in moves. This is a consideration in classical mechanics. In order to relate this view to macroscopic physics one surrounds each point e.g. with a sphere of some fixed volume - a procedure called coarse graining which lumps together points or states of similar behaviour. The trajectory of this sphere in phase space then covers also other points and hence its volume in phase space grows. The entropy associated with this consideration, whether zero or not, is called coarse grained entropy or thermal entropy. A large number of such systems, i.e. the one under consideration together with many copies, is called an ensemble. If these systems do not interact with each other or anything else, and each has the same energy , the ensemble is called a microcanonical ensemble. Each replica system appears with the same probability, and temperature does not enter.
Now suppose we define a probability density describing the motion of the point with phase space element . In the case of equilibrium or steady motion the equation of continuity implies that the probability density is independent of time . We take as nonzero only inside the phase space volume . One then defines the entropy by the relation where Then,by maximisation for a given energy , i.e. linking with of the other sum equal to zero via a Lagrange multiplier , one obtains (as in the case of a lattice of spins or with a bit at each lattice point) and , the volume of being proportional to the exponential of S. This is again a consideration in classical mechanics.
In quantum mechanics the phase space becomes a space of states, and the probability density an operator with a subspace of states of dimension or number of states specified by a projection operator . Then the entropy is (obtained as above)
and is described as fine grained or von Neumann entropy. If , the entropy vanishes and the system is said to be in a pure state. Here the exponential of S is proportional to the number of states. The microcanonical ensemble is again a large number of noninteracting copies of the given system and , energy etc. become ensemble averages.