# Non Equilibrium Statistical Mechanics and Lyapunov Instability

Mechanics

Jul 18, 2012 (5 years and 10 months ago)

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Lecture Notes on:

NonEquilibrium Statistical Mechanics
and Lyapunov Instability

by
Denis J. Evans and Debra J. Searles
Research School of Chemistry
Canberra, ANU, ACT 2601
Australia
presented at
on
“CHAOS AND IRREVERSIBILITY
(Classical aspects)”
held at
Bolyai College, Eotvos University
Budapest, August 31 - September 6, 1997
Liouville Equation for N-particle distribution function

=−

•≡−
ft
t
ftift
(,)
[
˙
(,)](,)
ΓΓ
ΓΓ
ΓΓΓΓΓΓL
(1)
Equation of motion of phase function
dA
dt
A
iLA
()
˙
()
()
ΓΓ
ΓΓ
ΓΓ
ΓΓ
ΓΓ=•

(2)
So,
iLiiiL=•

=

•−=

•≡
˙
...,
˙
...,
˙
()ΓΓ
ΓΓΓΓ
ΓΓ
ΓΓ
ΓΓΓΓLLΛ
(3)
and since,
df
dtt
ff=

+•

=−[
˙
]ΓΓ
ΓΓ
Λ
(4)
Λis called the phase space compression factor.The formal solution of the equations
of motion,
2
ftitf
it
n
f
n
n
(,)exp[](,)
()
!
(,)ΓΓΓΓΓΓ=−=

=

L
L
00
0
(5)
and
AtiLtA
iLt
n
A
n
n
(())exp[](())
()
!
(())ΓΓΓΓΓΓ=+=
=

00
0
(6)
q
p
Γ(0)
Γ(t)
3
Response theory
Consider an initial equilibrium ensemble:
f
H
dH
(,)
exp[()]
exp[()]
ΓΓ
ΓΓ
ΓΓΓΓ
0
0
0
=

β
β
(7)
ftiLtf(,)exp[()](,)ΓΓΓΓ=−+Λ0
(8)
Now employ a Dyson decomposition
exp[()]
exp[]exp[()]exp[()]
−+
=−−−+−−

iLt
iLtdsiLsiLts
t
Λ
ΛΛ
0
(9)
Substitute recursively,
exp[()]
exp[]
exp[]exp[()]
exp[]exp[()]exp[()]
.....
−+
=−
−−−−
+−−−−−

∫∫
iLt
iLt
dsiLsiLts
dsdsiLsiLssiLts
t
st
Λ
Λ
ΛΛ
11
0
1
12
0
212
0
1
1
(10)
4

Heat Q, is removed by the thermostat to ensure the possibility of a
nonequilibrium steady state. J is called the dissipative flux. The
momenta appearing in the equations of motion are peculiar.
Equations of motion
dqi
dt
=
pi
m
+
C
iF
e
dpi
dt
=F
i
+DiFe
−αp
i
α is chosen to keep the peculiar kinetic energy, K, constant:
Gaussian Thermostat
dQ
dt
=−
2Kα
=−
J

F
e
F
e
(t)
time, t
f(Γ,0)=
exp[

β
H0
(Γ)

exp[

β
H
0
(Γ)
Initial equilibrium distribution:
f(Γ,t)=exp[−iL(Γ)t]f(Γ,0)
Time dependent nonequilibrium distribution
J(t)
]
]
5
exp[()]
exp[]
()exp[]
()()exp[]
.....
exp[()]exp[]
−+
=−
−−−
+−−−

=−−−

∫∫

iLt
iLt
dssiLt
dsdsssiLt
dssiLt
t
st
t
Λ
Λ
ΛΛ
Λ
1
0
1
12
0
2
0
1
0
1
(11)
Substituting into the equation for the distribution function gives,
ftdssHt
t
(,)exp[()]exp[()]ΓΓ=−−−

Λ
0
0
β
(12)
For isokinetic equations of motion,
˙
˙
q
p
F
pFFp
i
i
ie
iiiei
m
C
D
=+
=+−α
(13)
6
From equations of motion,
dH
dt
dH
dt
dH
dt
K
e
000
2
=+
=−−JF().ΓΓα
(14)
and
Λ=+31NOα()
(15)
This leads to the so-called Kawasakiexpression for the nonequilibrium distribution
function,
ftdssf
t
e
(,)exp[()](,)ΓΓΓΓ=−−•

βJF
0
0
(16)
We can use this to compute averages,
<>=
=−−•

∫∫
BtdftB
dBdssf
t
e
()(,)()
()exp[()](,)
ΓΓΓΓΓΓ
ΓΓΓΓΓΓβJF
0
0
(17)
7
dBtdtdBtft
dBtf
e
e
<>=−−•
=−•

()/()()(,)
()()(,)
β
β
ΓΓΓΓΓΓ
ΓΓΓΓ
JF
JF00
(18)
Yielding the Transient Time Correlation Functionexpression for an average,
<>=−•<>

BtdsBs
e
t
()()()βFJ
00
(19)
In the small field limit we can linearise both Kawasaki and TTCF giving, theLinear
Response formula
lim()()()
F
e
t
eq
e
BtdsBs

<>=−•<>

0
0
0βFJ
(20)
8
Green-Kubo Relations for linear thermal Transport
Coefficients
1Self Diffusion coefficient
Ddst
iieq
=<•>

1
3
0
0
vv()()
(21)
2Thermal Conductivity
λ=<•>

V
kT
dst
B
QQeq
3
0
2
0
JJ()()
(22)
3Shear Viscosity
η=<>

V
kT
dsPPt
B
xyxyeq
0
0()()
(23)
4Bulk Viscosity
ηV
B
eq
VkT
dspVpVptVtpV=<−<>−<>>

1
00
0
[()()][()()]
(24)
9
NEMD Algorithms for Navier-Stokes transport
coefficients.
SLLODalgorithm for shear viscosity
˙
˙
q
p
i
pFip
i
i
i
iiyii
m
y
p
=+
=−−
γ
γα
, which is equivalent to:
˙˙
()q
F
i
i
i
i
m
ty=+γδ(25)
satisfies AIΓand the dissipative flux is PxyV. The shear viscosity, η, is computed as,
η
γ
γ
=−
→→∞

limlim
()
0
0
1
t
t
xy
t
dsPs
SLLODalgorithm for viscous flow
˙
˙
q
p
qu
pFpup
i
i
i
iiii
m
=+•
=−•−
∇∇
∇∇α
(26)
satisfies AIΓand the dissipative flux is PV.
Colour Conductivityalgorithm for self diffusion
10
˙
˙
(/)
q
p
pFipi
i
i
iiiciix
m
cFcJ
=
=−−−αρ
(27)
where
J
V
cx
cxii
i
N
=
=

1
1
˙
and
(/)/pi
iicx
i
N
B
cJmNkT−=
=

ρ
2
1
3
(28)
satisfies AIΓand the dissipative flux is JxV. The self diffusion coefficient,D,
D
t
dsJs
F
Ft
t
cx
c
=
→→∞

limlim
()
0
0
1
βρ
Evans Heat flow algorithm
11
˙
˙
()
,
q
p
pFF
FqFFqFp
i
i
iiiQ
ijijQ
j
N
jkjkQ
jk
N
i
m
EE
N
=
=−−
+•−•−
==
∑∑
1
2
1
2
11
α
(29)
where
E
p
m
N
i
i
N
ij
ij
N
=+
=
∑∑
{}/
,
2
1
2
1
2
Φ
,
satisfies AIΓand the dissipative flux is J
Q
V,
where J
Q
is the heat flux vector,
J
p
qF
p
Q
ii
i
ijij
i
ij
N
V
E
mm
=−•
∑∑
1
2
,
.
The thermal conductivity, λ, can be computed,
λ=
→→∞

limlim
()
Ft
t
Qx
Qx
Tt
dsJs
F
0
0
1
(30)
12
Note: NEMD algorithms and Green Kubo relations are also known for thermal and
mutual diffusion (Soret and Dufour effects) in non-ideal binary mixtures, and for the 12
or so viscosity coefficients of nematic liquid crystals.
13
Newton's Constitutive Relation
for Shear Flow
x
y
z
h
u
x(y)=γy
u
top
Strain rate, γ = utop/h, Shear stress, -Pxy
= F
D/A
Viscosity, η = Shear stress/strain rate
In a Newtonian fluid η is independent of γ.
Drag force on top surface, F
D
=iPxy
A = -iηγA, where η is the coefficient of shear viscosity.
Pxy is the force in the x-direction across a unit area whose normal is parallel to the y-axis.
Viscous heating = work done =
force x velocity = F
D
.utop= Pxy.A.γ.h = P
xy.γ.V
A
rea = A
Volume = V
14
Shearing Periodic BoundaryConditions
L
y
m
i
n
i
mum
i
mage
cell of particle 1
Ld
x
(t)
Ld
x
(t)
relative velocity of cell layers =
γL
primitive cell
part
i
c
l
e 1
part
i
c
l
e
2
Legend
Lees-Edwards Periodic Boundar
y
Conditions
velocity =
γL
15
Lees-Edwards periodic boundary conditions are non-autonomous. The coordinates in
the primitive cell are insufficientto calculate trajectories or thermophysical phase
variables. For example the pressure tensor can be written.
P
V
pp
m
yFxxyydt
V
pp
m
xFxy
xy
xiyi
ijxijijijx
j
ji
N
i
N
xiyi
ijyijijij
j
ji
N
i
N
=−

=−
[]

=

=
=

=
∑∑
∑∑
11
2
11
2
11
11
(,,,,())
(,)
The motion of a unit cell of N particles under SLLOD dynamics but employing Lees-
Edwards periodic boundary conditions, is identicalto the motion one would observe
for an infinite periodic array of particles evolving under SLLOD but without reference
to the boundary conditions. If the initial infinite system is periodic, SLLOD dynamics
will preserve that symmetry forever [1]. Lees-Edwards periodic boundary conditions
are the natural generalisation of periodic boundary conditions to shear flow.
16
SLLODalgorithm for shear viscosity
˙
˙
q
p
i
pFip
i
i
i
iiyii
m
y
p
=+
=−−
γ
γα
, which is equivalent to:
˙˙
()q
F
i
i
i
i
m
ty=+γδ(25)
x
x
Probability of
vx(y)
Probability of
vx(y)
y
y
Equilibrium at t = 0
-
Local Equilibrium at t = 0
+
with strain rate γ.
t = 0
-
t = 0
+
<
v
x(y)> = iγy
SLLOD changes an equilibrium distribution at t = 0
-
,
to a
local
equilibrium distribution.
The SLLOD equations of motion (25) are equivalent to Newton’s equations for t > 0
+,
with a linear shift applied to the initial x-velocities of the particles.
17