Coarse-grained modeling

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[1][2] 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,[1][2] nucleic acids,[3][4] lipid membranes,[2][5] carbohydrates[6] or water.[7] 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.[1]

The coarse-grained modeling originates from work by Michael Levitt and Ariel Warshel in 1970s.[8][9][10] Coarse-grained models are presently often used as components of multiscale modeling protocols in combination with reconstruction tools[11] (from coarse-grained to atomistic representation) and atomistic resolution models.[1] Atomistic resolution models alone are presently not efficient enough to handle large system sizes and simulation timescales.[1][2]

Coarse graining and fine graining in statistical mechanics addresses the subject of entropy ${\displaystyle S}$, and thus the second law of thermodynamics. One has to realise that the concept of temperature ${\displaystyle T}$ 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 ${\displaystyle S}$ 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[12] . 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

${\displaystyle {\frac {d}{dt}}(\Delta q\Delta p)=0}$

states that a phase space volume ${\displaystyle \Gamma }$ (spanned by ${\displaystyle q}$ and ${\displaystyle p}$, here in one spatial dimension) remains constant in the course of time, no matter where the point ${\displaystyle q,p}$ contained in ${\displaystyle \Delta q\Delta p}$ moves. This is a consideration in classical mechanics. In order to relate this view to macroscopic physics one surrounds each point ${\displaystyle q,p}$ e.g. with a sphere of some fixed volume - a procedure called coarse graining. The trajectory of this sphere in phase space then covers also other points and hence its volume in phase space grows. The entropy ${\displaystyle S}$ 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 ${\displaystyle E}$, 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 ${\displaystyle \rho (q_{i},p_{i},t)}$ describing the motion of the point ${\displaystyle q_{i},p_{i}}$ with phase space element ${\displaystyle \Delta q_{i}\Delta p_{i}}$. In the case of equilibrium or steady motion the equation of continuity implies that the probability density ${\displaystyle \rho }$ is independent of time ${\displaystyle t}$. We take ${\displaystyle \rho _{i}=\rho (q_{i},p_{i})}$ as nonzero only inside the phase space volume ${\displaystyle V_{\Gamma }}$. One then defines the entropy ${\displaystyle S}$ by the relation

${\displaystyle S=-\Sigma _{i}\rho _{i}\ln \rho _{i},}$ where ${\displaystyle \Sigma _{i}=1\rho _{i}.}$

Then,by maximisation for a given energy ${\displaystyle E}$, i.e. linking ${\displaystyle \delta S=0}$ with ${\displaystyle \delta }$ of the other sum equal to zero via a Lagrange multiplier ${\displaystyle \lambda }$, one obatins

${\displaystyle V_{\Gamma }=e^{(\lambda +1)}={\frac {1}{\rho }}}$ and ${\displaystyle S=\ln V_{\Gamma }}$.

This is again a consideration in classical mechanics.

In quantum mechanics the phase space becomes a space of states, and the probability density ${\displaystyle \rho }$ with a subspace ${\displaystyle \Gamma }$ of dimension or number of states ${\displaystyle N_{\Gamma }}$ specified by a projection operator ${\displaystyle P_{\Gamma }}$. Then the entropy ${\displaystyle S}$ is (obtained as above)

${\displaystyle S=-Tr\rho \ln \rho =\ln N_{\gamma },}$

and is described as fine grained or von Neumann entropy. If ${\displaystyle N_{\Gamma }=1}$, the entropy vanishes and the system is said to be in a pure state. The microcanonical ensemble is again a large number of noninteracting copies of the given system and ${\displaystyle S}$, energy ${\displaystyle E}$ etc. become ensemble averages.

Now consider interaction of a given system with another one - or in ensemble terminology - the given system and the large number of replicas all immersed in a big one called a heat bath characterised by ${\displaystyle \rho }$. Since the systems interact only via the heat bath, the individual systems of the ensemble can have different energies ${\displaystyle E_{i},E_{j},...}$ depending on which energy state ${\displaystyle E_{i},E_{j},...}$ they are in. This interaction is described as entanglement and the ensemble as canonical ensemble (the macrocanonical ensemble permits also exchange of particles).

The interaction of the ensemble elements via the heat bath leads to temperature ${\displaystyle T}$, as we now show.[13] Considering two elements with energies ${\displaystyle E_{i},E_{j}}$, the probability of finding these in the heat bath is proportional to ${\displaystyle \rho (E_{i})\rho (E_{j})}$, and this is proportional to ${\displaystyle \rho (E_{i}+E_{j})}$ if we consider the binary system as a system in the same heat bath defined by the function ${\displaystyle \rho }$. It follows that ${\displaystyle \rho (E)\propto e^{-\mu E}}$, where ${\displaystyle \mu }$ is a constant. Normalisation then implies

${\displaystyle \rho (E_{i})={\frac {e^{-\mu E_{i}}}{\Sigma _{j}e^{-\mu E_{j}}}},\Sigma _{i}\rho (E_{i})=1.}$

Then in terms of ensemble averages

${\displaystyle {\bar {S}}=-{{\bar {\ln }}\rho }}$, and ${\displaystyle \mu \equiv {\frac {1}{T}},\;k_{B}=1,}$

or by comparison with the second law of thermodynamics. ${\displaystyle {\bar {S}}}$ is now the entanglement entropy or fine grained von Neumann entropy. This is zero if the system is in a pure state, and is nonzero when in a mixed (entangled) case.

The classical coarse grained thermal entropy is not always the same as the (mostly smaller) quantum mechanical fine grained entropy. The difference is called information. An example of coarse graining is provided by Brownian motion[14]