∞

w(r) 4πr2 dr = 1. (9.26)

0

This condition implies that the integral of the number density over a large

volume approximates the number of particles within that volume. The

weight function is not prescribed in detail, but it should present a smoothing

over space, and preferably (but not necessarily) be positive and monotoni-

cally decreasing with r. A useful and practical example is the 3D Gaussian

function

√ ’3 r2

w(r) = (σ 2π) exp ’ 2 . (9.27)

2σ

Note on the symmetry of weight functions We have made the weight function

w(r) a function of the distance only, and therefore the weight function is perfectly

symmetric in space, and invariant for rotation. This is not a necessary condition,

and we could take a weight function w(r) that is not rotationally invariant, but

still of high, e.g., cubic, symmetry, such as a product function

w(r) = w1 (x)w1 (y)w1 (z), (9.28)

where w1 (x) is a symmetric function in x. Product functions have the advantage

that their Fourier transforms are a product of the Fourier transforms of the one-

dimensional weight functions. Normalization according to (9.26) is not valid for

product functions in general, but must be replaced by the normalization of each of

the 1D functions:

+∞

w1 (x) dx = 1. (9.29)

’∞

Simple one-dimensional weight functions are listed below:

290 Coarse graining from particles to ¬‚uid dynamics

(i) Constant weight

w1 (x) = 1/(2a) for |x| ¤ a

= 0 for |x| > a. (9.30)

The Fourier transform of this function is a sinc function, sin ka/(ka).

(ii) Triangular weight

w1 (x) = a’2 (a ’ |x|) for |x| ¤ a

= 0 for |x| > a. (9.31)

This function is in fact a convolution of the previous function with it-

self, and therefore its Fourier transform is the square of a sinc function

[2 sin( 1 ka)/(ka)]2 .

2

(iii) Sinc function

1 sin(πx/a)

w1 (x) . (9.32)

a πx/a

This function has a band-limited Fourier transform that is constant up to

|k| = π/a and zero for larger |k|.

(iv) Normal distribution

x2

1

√ exp ’ 2

w1 (x) = . (9.33)

2σ

σ 2π

The Fourier transform of this function is a Gaussian function of k, propor-

tional to exp(’ 1 σ 2 k 2 ). The 3D product function is a Gaussian function

2

of the distance r. In fact, the Gaussian function is the only 1D function

that yields a fully isotropic 3D product function, and is therefore a preferred

weight function.

9.3.1 De¬nitions

We now de¬ne the following averaged quantities:

(i) Number density

def

w(r ’ r i ).

n(r) = (9.34)

i

(ii) Mass density

def

mi w(r ’ r i ).

ρ(r) = (9.35)

i

(iii) Mass ¬‚ux density or momentum density

def

mi v i w(r ’ r i ).

J (r) = (9.36)

i

9.3 Coarse graining in space 291

(iv) Fluid velocity

J (r)

def

u(r) = . (9.37)

ρ(r)

This de¬nition is only valid if ρ di¬ers from zero. The ¬‚uid velocity

is undetermined for regions of space where both the mass density and

the mass ¬‚ux density are zero, e.g., outside the region to which the

particles are con¬ned.

(v) Force per unit volume

def

F i w(r ’ r i ),

f (r) = (9.38)

i

where F i is the force acting on particle i. This force consists of an

internal contribution due to interactions between the particles of the

system, and an external contribution due to external sources.

(vi) Stress tensor and pressure The de¬nitions of the stress tensor σ, the

pressure, and the viscous stress tensor, are discussed below.

(vii) Momentum ¬‚ux density tensor

def

Π±β (r) = ’σ±β (r) + mi vi± viβ w(r ’ r i ). (9.39)

i

Note that the de¬nition of Π uses the weighted particle velocities and

not the ¬‚uid velocities as in (9.10). With the present de¬nition linear

momentum is conserved, but Newton™s equation for the acceleration

has extra terms (see below).

(viii) Temperature

’ u(r))2 w(r ’ r i )

i mi (v i

def

T (r) = . (9.40)

3kB n(r)

Temperature is only de¬ned for regions where the number density

di¬ers from zero. It is assumed that all degrees of freedom behave

classically so that the classical equipartition theorem applies. For

hard quantum degrees of freedom or for holonomic constraints cor-

rections must be made.

9.3.2 Stress tensor and pressure

The coarse-grained stress tensor should be de¬ned such that its divergence

equals the internal force per unit volume (see (9.7)). As is elaborated in

292 Coarse graining from particles to ¬‚uid dynamics

Chapter 17 in connection with locality of the virial, there is no unique so-

lution, because any divergence-free tensor can be added to the stress tensor

without changing the force derived from it.

For forces between point particles, the stress tensor is localized on force

lines that begin and end on the particles, but are further arbitrary in shape.

Scho¬eld and Henderson (1982) have suggested the following realization of

the stress tensor:

σ±β = ’ δ(r ’ r c ) dxcβ ,

int

Fi± (9.41)

C0i

i