# Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature Page: 6 of 35

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4

of motion are Hamilton's equations for this conserved

energy functional plus random and dissipative forces due

to fluctuations.

The classical ensemble must obey the constraint that

the position and momenta of the atoms are consistent

with the mean displacement and momentum fields. Let

the displacement of atom p be u =, - x,o where x~o

is its equilibrium position. The displacement of mesh

node j is a weighted average of the atomic displacements

uj = fP up, (1)

where fj, is a weighting function, related to the micro-

scopic analog of FEM interpolating functions below. An

analogous relation applies to the momenta p,. Since the

nodal displacements are fewer or equal to the atomic po-

sitions in number, fixing the nodal displacements and

momenta does not (necessarily) determine the atomic

coordinates entirely. Some subspace of phase space re-

mains, corresponding to degrees of freedom that are miss-

ing from the mesh. We define the CG energy as the aver-

age energy of the canonical ensemble on this constrained

phase space:

E(uk, nk) ( HMD )uknk (2)

Idx[dp, HMD E HMD A/Z, (3)

Z(uk, uk) Jdxdp, EOHMD A, (4)

A H 6 I U7h- EN U ~f9N) 6 (n7 - (5)

where )3 1/(kT) is the inverse temperature, Z is the

partition function and 6(u) is a three-dimensional delta

function. The delta functions enforce the mean field con-

straint (1). Note that Latin indices, j, k, .. ., denote mesh

nodes and Greek indices, , v, ..., denote atoms. The en-

ergy (3) is computed below (Eq. (27)).

When the mesh nodes and the atomic sites are identi-

cal, fj,, =j, , and the CGMD equations of motion agree

with the atomistic equations of motion.32 As the mesh

size increases some short-wavelength degrees of freedom

are not supported by the coarse mesh. These degrees of

freedom are not neglected entirely, because their thermo-

dynamic average effect has been retained. This approx-

imation is expected to be good provided the system is

initially in thermal equilibrium, and changes to the sys-

tem would only produce adiabatic changes in the missing

degrees of freedom. In particular, the relaxation time of

those degrees of freedom should be fast compared to the

driving forces in the CG region. As long as this condition

is satisfied, the long wavelength modes may be driven out

of equilibrium without problems.33

We have written the CG energy as an internal energy,

a function of the entropy, S, rather than the tempera-

ture. This is designed for systems in which the short

wavelength modes change adiabatically. This is a goodapproximation, for example, when long wavelength elas-

tic waves propagate through a covalent solid in the linear

regime at finite temperature.34,35 In other systems, the

short wavelength modes may be in contact with a heat

bath, so that their evolution is isothermal rather than

isentropic. For example, the electron gas in metals can

act as a heat bath. Then the Helmholz free energy,F(uk, nk) -kT log Z,

(6)

should be used rather than the internal energy. In this

case, the ensemble average behavior of the CG collective

modes is exactly the same as that of the corresponding

averaged atomic modes in the underlying atomistic sys-

tem:13F (7)

(Uj.. - -ug) = dujdu. (ux. ujj e

__ ,V. ... f., XIdu dp, u.1 up e-HMD (8)

which follows from plugging in the expression for F (6)

and (4) into (7) and integrating the delta functions.36

This equation shows the equivalence of all unnormalized

correlation functions, but since the partition functions

(zero point functions) are identical, the normalized cor-

relation functions are the same, as well. The emergence of

the canonical distribution in other cases requires a treat-

ment of thermal relaxation processes (cf. Section V). It

should be noted that even in the isothermal ensemble the

faithfulness of correlation functions applies only to equal-

time correlation functions at equilibrium, and consider-

ation of dissipative processes are needed to reproduce

interesting correlation functions such as the autocorrela-

tion function (, (0)2, (t)) associated with the fluctation-

dissipation theorem.

To end this section, we note that the definition of the

CGMD energy may appear to neglect the well-known

quantum mechanical contributions to lattice dynamics.

Phonons are bosons, after all, and they should obey

Bose-Einstein statistics. The definition of the CGMD

energy (3) is clearly a classical expression based on Boltz-

mann statistics. To what extent can it be expected to be

valid? The reason the classical expression is valid for

most of the conceivable applications of CGMD is that

the Bose-Einstein distribution most strongly affects the

lowest states; i.e., exactly the states that are retained ex-

plicitly in the CGMD Hamiltonian. The higher energy

states have low occupation in equilibrium, and are not

affected significantly by strong quantum effects such as

Bose condensation. The CGMD Hamiltonian is there-

fore expected to be a good description of the coarse-

grained system. It may be necessary to use a path in-

tegral, or other quantum mechanical version, of MD to

treat the retained degrees of freedom at sufficiently low

temperature,37 but the internal degrees of freedom are

described well by Eq. (3).

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Rudd, R E & Broughton, J Q. Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature, article, May 30, 2005; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc877423/m1/6/: accessed May 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.