On Variance-Reduced
Simulations of the
Boltzmann Transport
Equation for
Small-Scale Heat
Transfer
Applications
Nicolas
G.
Hadjiconstant
inou
Gregg A.
Radtke
Lowell L.
Baker
We present and discuss a variance-reduced stochastic particle simulation method for
solving the relaxation-time model of the Boltzmann transport equation. The variance
reduction, achieved by simulating only the deviation from equilibrium, results in a
Department of Mechanical
Engineering,
significant computational efficiency advantage compared with traditional stochastic
Massachusetts Institute
particle methods in the limit of small deviation from equilibrium. More specifically, the
of Technology,
proposed method can efficiently simulate arbitrarily small deviations from equilibrium at
Cambridge, MA
02139
a computational cost that is independent of the deviation from equilibrium, which is in
1 Introduction and
sharp contrast to traditional particle methods. The proposed method is developed and
validated in the context of dilute gases; despite this, it is expected to directly extend to all
Motivation
fields (carriers) for which the relaxation-time approximation is applicable. DOI:
10.1115/1.4002028
tical error decreases with the square root of the number of
samples, this often leads to computationally intractable problems
Particle-mediated energy transport in the transition regime
15.
between the ballistic and diffusive limits has recently received
The present paper describes a particle simulation framework that
significant attention in connection to micro- and nanoscale science
alleviates the above disadvantages to a considerable extent while
and technology 1 . Applications can be found in a variety of
retaining the basic and desirablefeatures of particle methods.
diverse fields such as thin semiconductor films 2,3and
This is achieved by simulating only the deviation from equilibrium,
superlattices 1 , ultrafast processes 4,5, convective heat
a s originally proposed in Ref. 11 . Subsequent work
transfer 6,7, and gas-phase damping 8,10.
16has shown that deviational methods particle methods
For a considerable number of applications, a classical description
simulating the deviation from equilibriumusing the original
using the Boltzmann transport equation represents a good
Boltzmann hard-spherecollision operator require particle
compromise between fidelity and complexity 1 . However, a
cancellation to prevent the number of simulated particles from
numerical solution of the Boltzmann equation remains a formidable
growing in an unbounded manner. A stable deviational method
task due to the complexity associated with the collision operator and
that does not require particle cancellationfor the hard-sphere
the high dimensionality of the distribution function. Both these
gas was first developed in Refs. 17,18; i n that work, it was
features have contributed to the prevalence of particle solution
shown that the growth in the number of simulated particles observed
methods, which are typically able to simulate the collision operator
in the collision-dominated regime when using the traditional
through simple and physically intuitive stochastic processes while
hard-sphere collision operator 16can be overcome using a
employing importance sampling, which reduces computational cost
special but equivalentform of the hard-sphere collision
and memory usage 11 . Another contributing factor to the
operator first derived by Hilbert 19. I n this paper, we exploit
wide usage of particle schemes is their natural treatment of the
this observation to develop a deviational method for simulating the
advection operator, which results in a numerical method that can
Boltzmann equation in the relaxation-time approximation. The
easily handle and accurately capture traveling discontinuities in the
method presented here considers the deviation from a suitably
distribution function 11 . An example of such a particle
defined global equilibrium distribution and appears to be stable
method is direct simulation Monte Carlo DSMC 12,
does not require particle cancellationfor typical deviations
which has become the standard simulation method for dilute gas
from equilibrium.
flow. Methods that are similar in spirit have also been used for
simulating phonon transport in solid state devices 13,14.
One o f the most important disadvantages of particle methods for
2 Background
solving the Boltzmann equation derives from their reliance on
The Boltzmann transport equation is used to describe under
statistical averaging for extracting field quantities from particle data
appropriate
conditionstransport processes in a wide variety of
13,15. I n simulations of processes close to equilibrium,
thermal noise typically exceeds the available signal. When coupled fields 1 , including dilute gas flow 20, phonon 21,
electron 22, neutron 23, and photon transport 24.
with the slow convergence of statistical sampling statisI t may be written as

where f r , c , t is the single-particle distribution function
20, df / dtr , c , t denotes the collision operator, r
= x , y , z is the position vector in physical space, c =
cis the molecular velocity vector, a = ax, ay, azis the
acceleration due to an exterc
r+a·
coll
f
=


1
df dt

coll
Contributed by the Heat Transfer Division of ASME for publication
in the JOURNAL OF H EAT T RANSFER. Manuscript received January 21,
2009; final manuscript received April 12, 2010; published online
August 13, 2010. Assoc. Editor: Kenneth Goodson.
f
+c·
x, cy
f
,
cz
t
Journal of Heat Transfer NOVEMBER 2010, Vol. 132 / 112401-1Copyright © 2010 by ASME
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http://www.asme.org/terms/Terms_Use.cfm
nal field, and t is time. In this paper, we focus on the relaxationtime approximation
 
20,1 df dt
t
coll= 1 f
- floc

= df

2
+ c · f r =0 5 
6 


f
dt
coll
coll
f
t
by simply advecting particles for a time step t , while the
collision step integrates
where flocr , c , t is the local equilibrium distribution function
and is the relaxation time.
1To focus the discussion, we specialize our treatment to the dilute
gas case;however, we hope that this exposition can serve as a
prototype for the development of similar techniques in all fields
where the relaxation-time approximation is applicable. In the interest of simplicity, i n the present paper we assume  c
and that no external forces are present. The first assumption can
be
easily relaxed, as discussed in Sec. 3.1. External fields also require
only relatively straightforward modifications to the algorithm
presented below.
A dilute gas in equilibrium is described by a Maxwell– Boltzmann
distribution, leading to a local equilibrium distribution
floc= nloc3 /
2cloc 3


exp
22
c - uclocloc
by changing the distribution by an amount of df / dtt .
Spatial discretization is introduced by treating collisions as spatially
homogeneous within smallcomputational cells of volume Vcell.
Our approach retains this basic structure although it must be
noted that since computational particles represent the
deviation from equilibrium, they may b e positive or negative,
depending on the sign of the deviation from equilibrium at
the location in phase space where they reside. As in other
particle schemes 12,inthe interest of computational
efficiency, each computational deviaeffof physical deviational particles. Below we discuss the two
tional particle represents an effective number N
main steps in more detail.
3 
which is parametrized by the local number density
nr , t , the local flow velocity uloc= ulocloc=
nlocr , t , and the most probable speed cr , t
=loc2 kBTloc/ m based on the local temperature Tloc
= Tlocr , t . Here, kBis Boltzmann’s constant, and m is the
molecular mass.
e
x
p
c
f
˜
ˆ
d


d
 3.1 Collision Step. The varian
 written as
ˆ


f Within each computational ce
twopart process. This integrati
various quantities, denoted her
time step by sampling the inst
In the first part, we remove
deleting particles with probabi
c
2
0
˜ dft + 


2
In our implementation, this is
acceptancerejection process, w
c .
In the second part, we create a
acceptance-rejection process
 
df dt
coll=
F 1fd
1floc-


7
3 Variance Reduction Formulation
+ t
cct + t t
+ 1
= f

In a recent paper 11 , Baker and Hadjiconstantinou showed
that significant variance reduction can be achieved by simulating
only the deviation fdr , c , t f - feefrom an arbitrary, but
judiciously chosen, underlying equilibrium distribution fr , c , t
.By adopting this approach, it is possible to construct
Monte Carlo
simulation methods 11,16,17,25that can capture arbitrarily
small deviations from equilibrium at a computational cost that is
small and independent of the magnitude of this deviation. This is in
sharp contrast to regular Monte Carlo methods, such as DSMC,
whose computational cost for the same signal-to-noise ratio
increases sharply 15as the deviation from equilibrium
decreases.
4 
max ˆ loct
- F t 
c
ˆ loc
1
1
bound f ˆ loc
c
with c 
1
max
If f ˆ loc
eIn
the work that follows, the underlying equilibrium
distribution fwill be identified with absolute
equilibrium,
feF c =
is a reference equilibriumnumber density and c0
This step can be achieved by the following procedure. Let cc
n03 / 2c0 3
/ m is the most probable molecular speed based on the reference
- F c is negligible for c be a positivevalue
such that fis an L-norm. Furthermore, let c, where · c 
temperature T0. For small deviations from equilibrium, the choice
of an appropriate reference equilibrium is straightforward but
F c from above. Then, repeat Ntimes: 1. Generate
necessary. I f deviations from the reference equilibrium are large,
uniformly distributed, random velocity vectors c
=where n2 kBT00
either due to strong nonlinearity in the problem or an
inappropriate choice of reference equilibrium, the simulation
method becomes less efficient than DSMC due to the large
number of particles required to simulate the deviation from
equilibrium.
1. 2.c - F c R , create a particle with velocity c at sgnf ˆ
d3c d3r
loc
a randomly chosen position within the cell and sign
Vcell
To find N
c - F c . Here, R is a random number
uniformly distributed on 0,1.
2Particle
methods, such as DSMC, typically solve the Boltzmann
equation by applying a splitting scheme,in which molecular motion
is simulated as a series of collisionless advection and collision steps of length t . I n such a scheme, the collisionless
advection step integrates
we note that the number of particles of all velocities and
signsthat should be generated in a cell to obtain the proper
change in the distribution function is
c,
V
1Within the rarefied gas dynamics literature, the relaxation-time approximation
is known as the BGK model 20.
F



c
e
l
l
2
N
o
t
e
t
h
a
t
a
s
y
m
m
e
t
r
N  ef
f
ˆ
 
t
R
Neff

R
3
3
where Vcell
ized algorithm typically provides h
references therein.
i
generated by the above algo
s
112401-2 / Vol. 132, NOVEMBER 2010 Transactions of the ASME
Downloaded
22 Aug 2010 to 18.80.3.157. Redistribution subject to ASME license
t
http://www.asme.org/terms/Terms_Use.cfm
h
e
c
e
l
l
v
o
l
u
m
e
.
T
h
e


e
x
p
e
c
t
e
d


t
o
t
a
l
n
u
m
b
e
r
o
f
p
a
r
t
i
c
l
e
s
u
l
t
i
m
a
t
e
l
y

tant difference, however, i s that only the net number of
-F
deviational particles is sent back into the domain since pairs of
d3c f ˆ
R
loc
positive and negative particles correspond to zero net mass flux
3
and can be cancelled, i.e.,
By equating the above two expressions, we obtain N=8t
nb d
c
cxfdd3c 17
 maxVccellc 3/ ˆ Neffc. 3.2 Advection Step. It can be easily
cx0 xbd3c = cx0
verified that when the
fb
underlying equilibrium distribution is not a function of space or
Equation 16reveals a third contribution to the deviational
time, as is the case here,
population, namely, nb e- F . This contribution is
associated with the molecular flux incident upon the wall due to
the underb is determined from
+c
fd t + fd r 12
e
f · f r= c·
Nc


11 
max8
cc 3



t

cx0

lying equilibrium distribution; n cxnb
e
x0
xn
cxbFd 3c 18
e
b- F , cx
time step is
NF = teff

3b
nb
c dc = x
and thus the advection step for deviational particles is identical to
c
e
that of physical particles.
As shown i n Refs. 16,18, this case can be treated by creating
Boundary condition implementation, however, i s slightly more
deviational particles from the distribution c0. The number of
complex because the mass flux to system boundaries includes
particles per unit wall surface area generated in a
contributions from deviational particles as well as the underlying
b- F d3c 19
equilibrium distribution. Our i mplementation is discussed in more
detail below.
3.2.1 Boundary Condition
Implementation. Here, we extend
previous work on diffuse
boundary conditions to the more
general case of the Maxwell
accommodation model with
accommodation
coefficient . According to this model, a fraction of the
bound the intergrand in Eq. 19from above, cabe a
Let Mmax
molecules impacting the boundary are accommodated
positivevalue such that the integrand is negligible for c
diffusely reflected, while the remaining particles are
1ca, and A be the surface area of the boundary. The requisite
specularly reflected.
number of
times: 1. Generate uniformly distr
representLet ub and cb
c
the boundary velocity and the most probable speed based on the
0. 2.If cxnb ebc - F 
boundary temperature, respectively. Let us assume, without loss of
particle with velocity c and sign sg
generality, that the boundary is
such that c
and cx
1ca
b b
max
parallel to the y - z plane and that the gas lies to the right x
e
0 of the wall. For simplicity, let us also assume that the
boundary does

c

-F

c
.
Ge
ner
ate
d
par
ticl
es
are
ad
ve
cte
d
for
a
ran
do
m
fra
cti
on
of
a
ti
me
ste
p.
not
mo
ve
in
the
dir
ect
ion
nor
ma
l to
its
pla
ne;
i.e.
,
the
x
-co
mp
on
ent
of
uis
zer
o.
Un
der
the
se
co
ndi
tio
ns,
the
Ma
xw
ell
bo
un
dar
y
co
ndi
tio
n
ca
n
be
wri
tte
n

26

as
b
f
,
y, cz+  cx, cy,
x
cx, cz
n
cz, cx
cy
b b
= 1,
f - c c
where in order to simplify the notation we have
suppressed the space and time dependence of the
distribution function. Here,
walls at different temperatures T0 and T1= T0
+

T
wh
ere

T
=

T
bcx, cy, cz=
is determined by mass conservation at the wall, namely,
cb 2-3 / 2

cx
xf
c =- nb
d

exp- c - ub2/ cb 2; the
quantity nb
and a distance L apart in the x
is defined as k = c00/ L , where 0is
reference absolute equilibrium
We start by discussing the variance reduction achieved by the
present method. Validation is discussed in the following section.
x0
0
c
c  d3c 14 
We have performed extensive validations and
4 Results and Discu
performance evaluations of the proposed method using a0 13 
variety of test cases. Here, we present some representative
transient and steady-state results involving heat exchange
between two infinite, parallel
3
c
x
b

13can be written as
fcx, cy, cz= 1- f d- cx,
cy, cz+ nb
dEquation
b
- F cx, cy, cz,
cx
4.1 Relative Statistical Uncertainty. As stated above and as shown
i n Refs. 11,16,18, deviational methods such as the one
presented here exhibit statistical uncertainties that scale with the
local deviation from equilibrium, thus allowing the simulation of
0 15 The algorithm used here for evaluating
nbconsiders deviational particles and the flux of particles due to the arbitrarily low deviations from equilibrium at a cost that is
independent of this deviation. Here, we demonstrate this feature by
underlying equilibstudying the statistical uncertainty of the temperature in a problem
rium distribution feF separately. More specifically, b y
involving heat transfer.
introducing nb= nb e+ nb ddin Eq. 15, we obtain fcx, cy,
Figure 1 shows the relative statistical uncertainty in the
cz= 1- f d- cx, cy, cz+ nb dbc+ nb
temperature T/ T = T/ T0as a function of for
ebcx, cy, cz- F cx, cy, czx, cy, cz, cx0 16 
steady-state heat exchange between the two walls k =1, T0= 273
Using a probabilistic interpretation, this boundary condition may be
K , =1; T
implemented as follows: deviational particles striking the
is defined as the standard deviation in the temperature measured in
boundary are specularly reflected with probability 1or
two computational cells in the middle of the computational dodiffusely reflected with probability . The fraction of deviational
main, each containing approximately N = 950 particles. Our
particles diffusely reflected corresponding to nb dcan be
results are compared with those from a representative
treated using an algorithm similar to DSMC; i.e., particles striking
nondeviational method, namely, DSMC. The DSMC performance
the wall may b e sent back to the computational domain drawn from
is calculated from the theoretical result of Ref. 15obtained
using equilibrium statistical mechanics assuming small
deviation from equilibrium. We also performed DSMC
simulations of the relaxationtime modelto verify that this
theoretical result remains accurate
t
p
h
r
e
i
a
a
t
p
e
p
r
fl
o
u
xal distribution here, c

f
n the equilibrium result 
o
r
Journal
of Heat Transfer NOVEMBER
2010, Vol. 132 / 112401-3
,
x
b
Downloaded 22 Aug 2010 to 18.80.3.157. Redistribution subject to ASME license
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c
x


0


.
O
n
e
i
m
p
o
r
-


0
.
1
.
T
h
e
fi
g
u
r
e
s
h
o
w
s
t
h
a
t
t
h
i
s
i
s
i
n
d
e
e
d
t
h
e
c
a
s
e
,
p
r
o
v
i
d
e
d
t
h
a
t
i
s
410
T
310
/∆ T
e
Fig. 1 The relative s tatistical
uncertainty in temperature, /   T
,asafunctionof e for s teady-state
heat exchange between two parallel
p lates a t d ifferent temperatures
w i t h k = 1 ,   = 1 , a n d T 0= 273 K .
Simulation r esults „ symbols w ith error
bars o n solid line … are p resented and
compared with the t heoretical prediction
† 15 ‡ for DSMC „ dashed line … , which
serves as a canonical case of a
nondeviational method. Stars show actual
DSMC results verifying t hat equilibrium t
heory i s r eliable u p t o a t l east e É
0.3.
210
110
010
-110
2
1
0
-410
T
103
-21
-11
0
0
T
T
0
=
1.5N 20
numerical solution of the linearized 1 relaxa
T
is interpreted as the local temperature value. Figure 1 also shows model of the Boltzmann equation 27and our sim
0
that for 0.3 the relative
statistical unresults for the
heat flux between the two walls, q , for the case = 1 .
compares the heat flux normalized by the free-molecula
ballis-2-, a s a function of k ; here, P0=
equilibrium gas pressure. The agreement between the tw
excellent.
Figure 3 shows a comparison for = 0.826, one of t
for which numerical results for  1 are readily availa
The agreement is again excellent. Figure 4 shows a com
between our simulation results
and an analytical solution 29of the linearized coll
1
Boltzmann equation for oscillatory variation in the boun
tempera-
0.9
0.8
q/q f
m
0.7
0.6
0.5
0.4
0.3
0.2
0 1 2 3 4 5 6 7 8 9 1 0 0.1
1/k
ticvalue,
certainty of the deviational method proposed here remains
q
essentially independent of , i n sharp contrast to “nondeviational”
methods. Moreover, the variance reduction achieved is such that
significant computational savings are expected for 0.1, particularly when considering that the cost savings scale with the square
of the relative statistical uncertainty since statistical noise
decreases with the square root of the number of samples.
4.2 Validation. Figure 2 shows a comparison between the
Fig. 2 Comparison between the
numerical solution o f t he
linearized Boltzmann equation b y
Bassanini et al. † 27 ‡„solid line…
and simulation r esults „ circles …
for t he heat flux between two
parallel, infinite, fully
accommodating walls. Some

=
P
fm
0c0/

numerical solution data f or k > 1
have been transcribed from Ref. †
20 ‡ .
112401-4 / Vol. 132, NOVEMBER
2010 Transactions of the ASME
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q/q f
m
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0 1 2 3 4 5 6 7 8 9 1 0 0.2
1/k
Fig. 3 Comparison between the numerical solution o f t he linearized Boltzmann equation b y
Bassanini et al. † 28 ‡„solid line… and simulation r esults „ circles … for t he heat flux between
two parallel, infinite walls with = 0 .826
valid in small regions close to the wall where the
above condition is not satisfied. Our results show
that sufficiently far from the wall, the agreement
between theory and simulation is excellent.
As shown above, although the computational advantage of the method presented here compared with nondeviational
methods increases as the deviation from equilibrium decreases, the amount of variance reduction achieved is such
that considerable computational savings are obtained even when the deviation from equilibrium is not small. Below,
we present a comparison between our results and DSMC simulations of the relaxation-time model in this latter
regime. Specifically, we simulate an impulsive heating problem where at time t = 0 the temperature of the wall at x
=-L / 2 jumps from T0to T0
+ T ; the gas is initially in equilibrium at temperature T0, and both
/ c01 29; thus, it is
walls are fully accommodating =1.
0.2
not
0.2
0.1
ux 0.1
0
ρρeρ00 0
ec0
-0.1
TTeT
-0.1
-0.5 -0.4 -0.3 -0.2 -0.1 0 -0.2
-0.5 -0.4 -0.3 -0.2 -0.1 0 -0.2
x/L
0
x/L
0.2
0.1
0
0.2
qx0.1
eρ0c3 0
0
-0.1
0
-0.1
-0.2
-0.5 -0.4 -0.3 -0.2 -0.1 0 -0.2
-0.5 -0.4 -0.3 -0.2 -0.1 0
x/L
x/L
Fig. 4 Comparison between the p roposed method „ dots … and the t heoretical r esults
of Ref. † 29‡„solid lines… for L / c=40, k =1, = 1 , and e = 0 .02. Here,
 0= mn0.0
Journal of Heat Transfer NOVEMBER 2010, Vol. 132 / 112401-5
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TT∆T
0.6
0
0.5
0.4
0.3
0.2
0.1
0
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.1
x/L
Fig. 5 I mpulsive heating p roblem for e = 0 .1 and k =10 at t = 0 .02 0, 0 . 1  0, 0 . 2  0,
and 0.4 0. The solid line denotes the p resent method, and stars denote DSMC results.
t = 0.2 0at k = 1 0 requires approximately 40 s.
In contrast, with all discretization parameters the
same and thus similar memory usage,
DSMC requires approximately 150,000 s t o
reach the same
final time t =2 0at k = 0.1 and 65,000 s
for k =10t = 0.20. The CPU times for
other values of can be estimated by noting
that
-2 for small , these times are independent of
for the proposed method, while for DSMC they
scale approximately as .In steady-state
problems—where continuous sampling after
steady state is reached can be performed—the
speedup will in general be smaller because in
the low-variance calculations the time t o
reach steady state becomes an appreciable part
of the total simulation time.
4.3 A Comment on Linear Conditions. The
algorithm described in this paper imposes no
restrictions on the magnitude of
0.8
0.7
TT∆T0
0.6
0.5
0.4
0.3
0.2
0.1
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0
x/L
Fig. 6 I mpulsive heating p roblem for e = 0 .1 and k =1 at t =0.4 0, 1 . 6  0,
and 8 0. The solid line d enotes the p resent method, and stars denote DSMC results.
112401-6 / Vol. 132, NOVEMBER 2010 Transactions of the ASME
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TT∆T0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
x
/
L
Fig. 7 I mpulsive heating p roblem for e = 0 .1 and k =0.1
at t =4 0, 1 2  0, and 40 0. The solid line denotes the p resent
method, and stars denote DSMC results.
 

Alternatively, Eq. 21provides a means of
obtaining bounds for f- F , i.e., max, and thus reducing the
number of rejections if the acceptance-rejection route is followed.
locrejection.
21
5 Conclusions
+

c
ˆ232
, and = Tloc/ T0
0.8
0.7
TT∆T0
0.6
0.5
0.4
0.3
0.2
0.1
We have presented an efficient variance-reduced particle method
for solving the Boltzmann equation in the relaxation-time
approximation. The method combines simplicity with a number of
desirable properties associated with particle methods, such as
robust capture of traveling discontinuities in the distribution
function and efficient collision operator evaluation using
importance sampling 11 , without the high relative
statistical uncertainty associated with traditional particle methods
in low-signal problems.
-0.5
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0
- 1 . This representation can be very useful for improving the computational
efficiency of update Eq. 9 . For example, for isothermal constant
density flows, particles may b e generated from a combination of a normal
distribution and analytic inversion of the cumulative distribution function,
which is significantly more efficient than acceptance-
x
/
L
Fig. 8 I mpulsive heating p roblem for e = 0 .3
and k =1 at t =0.4 0, 1 . 6  0, and 8 0. The solid
line denotes the p resent method and stars denote
DSMC results.
Journal of Heat Transfer NOVEMBER 2010, Vol. 132
/ 112401-7
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In particular, a s shown above, the method presented here can
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volume Greek
accommodation coefficient wdistance
from the wall maxupper bound on the difference
between distributions
dimensionless temperature difference = T /
Tdimensionless temperature = Tloc/
T0-1relaxation time 0
standard deviation probability distribution
function oscillation frequency
dimensionless density = nloc/
n0-1Subscripts and Superscripts
b boundary c
collision d
deviational e
equilibrium
fm free molecular
loc local
bsolute equilibrium
V
0


a

Y
2o
0
 r
k
C.
e
r 
c2
i 1
g
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29Manela, A., and Ha
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1V
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112401-8
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m a 22 Aug 2010 to 18.80.3.157. Redistribution subject to ASME license
Downloaded
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