±β

i

Its divergence gives an extra force per unit volume. Inspection of this term

shows that it represents the thermal kinetic energy density, with an equilib-

rium average determined by equipartition:

Πexc (r) = n(r)kB T (r)δ±β . (9.52)

±β

This term is indeed the missing term if we compare Πexc to the pressure

computed from virial and kinetic energy in statistical mechanics (Chapter

17, (17.127) on page 485). It has no in¬‚uence on the force unless there is a

9.4 Conclusion 295

gradient of number density or a gradient of temperature. In addition to the

average contribution, Πexc has a ¬‚uctuating component that adds noise to

the pressure and to the force.

9.4 Conclusion

As we have seen, coarse graining of a Hamiltonian ¬‚uid by spatial averaging

with a weight function, yields the conservation laws, if the macroscopic

quantities are properly de¬ned. However, the equation of motion has an

extra term that can be written as the divergence of an extra pressure term

(9.52). It is related to the local thermal kinetic energy and equals the kinetic

term required to describe pressure in statistical mechanics. With this term

included, and including the local stress tensor derived from the virial of the

local force (9.42), the pressure is a property of the system, determined by

the density of particles and by the interactions between the particles. This

is a manifestation of the local EOS. In ¬‚uid dynamics, where the description

in terms of interacting particles is lost, the EOS is an additional “property”

of the system that enables the determination of local pressure based on

density and temperature (or energy density or entropy density). Note that

local forces between particle pairs, which contribute to the local momentum

¬‚ux density and therefore to the local pressure, cancel in the coarse-grained

force density and do not play a direct role in ¬‚uid forces.

Another important di¬erence between the dynamics of a system of inter-

acting particles and a ¬‚uid continuum is that the coarse-grained dynamical

properties are averages over a ¬nite number of particles and are therefore

¬‚uctuating quantities with limited precision. This introduces “noise” and

will have an in¬‚uence on chaotic features of ¬‚uid dynamics, as turbulence,

but only when the length scale of such features approach molecular size

ranges. For macroscopic length scales the averaging can be done over such a

large number of particles that the ¬‚uctuations become negligible. In the in-

termediate range, where details on an atomic scale are not needed but ¬‚uctu-

ations are not negligible, the term mesoscopic dynamics is used. Mesoscopic

dynamics can be realized either with particles (as Langevin or Brownian dy-

namics with superatomic system description) or with continuum equations,

for example on a grid.

Exercises

9.1 Derive (9.15) from (9.12).

296 Coarse graining from particles to ¬‚uid dynamics

9.2 Derive the second line of (9.50) from the ¬rst line. Note that also

the ¬‚uid velocity is a function of spatial coordinates.

10

Mesoscopic continuum dynamics

10.1 Introduction

The term “mesoscopic” is used for any method that treats nanoscale sys-

tem details (say, 10 to 1000 nm) but averages over atomic details. Systems

treated by mesoscopic methods are typically mixtures (e.g., of polymers

or colloidal particles) that show self-organization on the nanometer scale.

Mesoscopic behavior related to composition and interaction between con-

stituents comes on top of dynamic behavior described by the macroscopic

equations of ¬‚uid dynamics; it is on a level between atoms and continuum

¬‚uids. In mesoscopic dynamics the inherent noise is not negligible, as it is

in macroscopic ¬‚uid dynamics.

Mesoscopic simulations can be realized both with particles and with con-

tinuum equations solved on a grid. In the latter case the continuum variables

are densities of the species occurring in the system. Particle simulations with

“superatoms” using Langevin or Brownian dynamics, as treated in Chap-

ter 8, are already mesoscopic in nature but will not be considered in this

chapter. Also the use of particles to describe continuum equations, as in

dissipative particle dynamics described in Chapter 11, can be categorized as

mesoscopic, but will not be treated in this chapter. Here we consider the

continuum equations for multicomponent mesoscopic systems in the linear

response approximation. The latter means that ¬‚uxes are assumed to be

linearly related to their driving forces. This, in fact, is equivalent to Brown-

ian dynamics in which accelerations are averaged-out and average velocities

are proportional to average, i.e., thermodynamic, forces. The starting point

for mesoscopic dynamics will therefore be the irreversible thermodynamics

in the linear regime, as treated in Chapter 16, Section 16.10.

297

298 Mesoscopic continuum dynamics

10.2 Connection to irreversible thermodynamics

We start with the irreversible entropy production per unit volume σ of

(16.98) on page 446. Replacing the “volume ¬‚ux” J v by the bulk veloc-

ity u we may write

1 1 1 1

σ = Jq · ∇ ’ u · ∇p + I · E ’ J i · (∇μi )p,T . (10.1)

T T T T

i

Here we recognize heat ¬‚ux J q and electric current density I, driven by a

temperature gradient and an electric ¬eld, respectively. The second term

relates to the irreversible process of bulk ¬‚ow caused by a force density, which

is the gradient of the (generalized) pressure tensor including the viscous

stress tensor (see Section 9.2.4 on page 284). The last term is of interest

for the relative di¬usional ¬‚ux of particle species, driven by the gradient of

the thermodynamic potential of that species. Any bulk ¬‚ow J i = ci u, with

all species ¬‚owing with the same average speed, does not contribute to this

term since

ci (∇μi )p,T = 0, (10.2)

i

as a result of the Gibbs“Duhem relation. The term can be written as

1 Ji

σdi¬ = ’ ’ u · [ci (∇μi )p,T ]. (10.3)

T ci

i

The term J i /ci ’ u = ud denotes the average relative velocity of species i

i

with respect to the bulk ¬‚ow velocity, and we may de¬ne the di¬erence ¬‚ux

J d as

i

def

J d = ci ud = J i ’ ci u. (10.4)

i i

It is clear that there are only n’1 independent di¬erence ¬‚uxes for n species,

and the sum may be restricted1 “ eliminating species 0 (the “solvent”) “ to

species 1 to n ’ 1, which yields the equivalent form (see also Chapter 16,

Eq. (16.104)):

1 Ji J0

σdi¬ = ’ ’ · [ci (∇μi )p,T ]. (10.5)

T ci c0

i

Simplifying to a two-component system, with components numbered 0

and 1, the di¬usional entropy production can be written as

1

σdi¬ = ’ [ud c1 (∇)μ1 )p,T + ud c1 (∇μ1 )p,T ], (10.6)

T1 1

1 This is indicated by the prime in the sum.