Program on « NONEQUILIBRIUM STEADY STATES »
Institut Henri Poincaré
10 September - 12 October 2007
Pierre GASPARD
Center for Nonlinear Phenomena and Complex Systems,
Université Libre de Bruxelles,
Brussels, Belgium
1)
TRANSPORT AND THE ESCAPE RATE FORMALISM
2)
HYDRODYNAMIC MODES AND NONEQUILIBRIUM STEADY STATES
3)
AB INITIO DERIVATION OF ENTROPY PRODUCTION
4)
TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS
TRANSPORT AND
THE ESCAPE RATE FORMALISM
Pierre GASPARD
Brussels, Belgium
J. R. Dorfman, College Park
G. Nicolis, Brussels
S. A. Rice, Chicago
F. Baras, Dijon
• INTRODUCTION: IRREVERSIBLE PROCESSES
AND THE BREAKING OF TIME-REVERSAL SYMMETRY
• TRANSPORT COEFFICIENTS & THEIR HELFAND MOMENT
• ESCAPE OF HELFAND MOMENTS & FRACTAL REPELLER
• CHAOS-TRANSPORT FORMULAE
• CONCLUSIONS
NONEQUILIBRIUM SYSTEMS
diffusion
between two reservoirs
electric conduction
molecular motor: FoF1-ATPase
K. Kinosita and coworkers (2001): F1-ATPase + filament/bead
C. Voss and N. Kruse (1996): NO2/H2/Pt reaction
diameter 20 nm
001
101
011
111

IRREVERSIBLE PROCESSES
viscosity, heat conductivity, electric conductivity,…
density of particles: n
Example: diffusion
∂t n = D 2 n
diffusion equation:
diffusion coefficient: Green-Kubo formula
D


v x (0)v x (t) dt
0
entropy density:
s = n ln (n0/n)
entropy current:
js = -D (n) ln (n0/en)
entropy source:
ss = D (n)2/n ≥ 0
balance equation for entropy:
∂t s + .js = ss ≥ 0
Second law of thermodynamics: entropy S 
dS deS di S


dt
dt
dt
with
di S
0
dt

 s dr
space
entropy flow
deS
di S  0

entropy production
HAMILTONIAN DYNAMICS
A system of particles evolves in time according to Hamilton’s equations:
dra
H

dt
pa
Hamiltonian function:

Time-reversal symmetry:
dpa
H
dt
ra
N
p2a
H 
 U(r1,r2 ,...,rN )
2ma
a1
(r1,p1,r2,p2,...,rN ,pN )  (r1,-p1,r2,-p2,...,rN ,-pN )

Determinism: Cauchy’s
theorem asserts the unicity of the trajectory issued from
initial conditions in the phase space M of the positions ra and momenta pa of the


particles:
  (r1,p1,r2,p2,...,rN ,pN )  M
dimM = 2Nd
Flow: one-dimensional Abelian group of time evolution:
  t (0 )  M

Liouville’s theorem: Hamiltonian dynamics preserves the phase-space volumes:
d  dr1 dp1 dr2 dp2 ... drN dpN

LIOUVILLE’S EQUATION: STATISTICAL ENSEMBLES
Liouville’s equation: time evolution of the probability density p(,t)
local conservation of probability in the phase space: continuity equation
Ýp)  0
 p  div(
t
Liouville’s equation for Hamiltonian systems: Liouville’s theorem
Ýp)  -p div(
Ý) - 
Ý grad p  H, p  Lˆ p
t p  -div(

=0
H 
H  
ˆ
L  H,   



r

p

p

r
a
a
a 
a1  a
N
Liouvillian operator:

Time-independent systems:
pt  e p0  Pˆ t p0
 operator:
Frobenius-Perron
pt   Pˆ t p0   p0 -t 
Lˆ t

Statistical average 
of a physical observable A():
At

A(t 0
) p0 0  d0 

A() p0 -t  d 
 A() p  d
t
Time-reversal symmetry: induced by the symmetry of Hamiltonian dynamics
TIME-REVERSAL SYMMETRY (r,v) = (r,-v)
Newton’s fundamental equation of motion for atoms or molecules composing matter
is time-reversal symmetric.
d 2r
m 2  F(r)
Phase space:
velocity v
dt
trajectory 1 = (trajectory 2)

position r
0
time reversal 
trajectory 2 = (trajectory 1)
BREAKING OF TIME-REVERSAL SYMMETRY
Selecting the initial condition typically breaks the time-reversal symmetry.
Phase space:
d 2r
m 2  F(r)
dt
velocity v
This trajectory is selected by the initial condition.
*

initial condition
position r
0
time reversal 
The time-reversed trajectory is not selected by the initial condition
if it is distinct from the selected trajectory.
HARMONIC OSCILLATOR
All the trajectories are time-reversal symmetric in the harmonic oscillator.
Phase space:
d 2r
m 2  -kr
dt
velocity v

position r
0
time reversal 
self-reversed trajectories
FREE PARTICLE
Almost all of the trajectories are distinct from their time reversal.
Phase space:
d 2r
m 2 0
dt
velocity v

self-reversed trajectories at zero velocity
position r
0
time reversal 
PENDULUM
Phase space:
The oscillating trajectories are time-reversal symmetric
while the rotating trajectories are not.
d 2f
g

sin f
velocity v
2
dt

unstable direction
stable direction
•
0
self-reversed
trajectories
•
time reversal 
angle f
STATISTICAL MECHANICS
weighting each trajectory with a probability
-> invariant probability distribution
Phase space:
velocity v
position
0
time reversal 
e.g. nonequilibrium steady state between two reservoirs:
breaking of time-reversal symmetry.
STATISTICAL EQUILIBRIUM
The time-reversal symmetry is restored
e.g. by ergodicity (detailed balance).
Phase space:
velocity v
position r
0
time reversal 
BREAKING OF TIME-REVERSAL SYMMETRY (r,v) = (r,-v)
Newton’s equation of mechanics is time-reversal symmetric
if the Hamiltonian H is even in the momenta.
Liouville equation of statistical mechanics,
ruling the time evolution of the probability density p
is also time-reversal symmetric.
p  rÝp  vÝp


0
t
r
v
p
 H, p  Lˆ p
t
The solution of an equation may have a lower
symmetry than the equation itself
(spontaneous symmetry breaking).
Typical Newtonian trajectories T are different
from their time-reversal image T :
T ≠ T
Irreversible behavior is obtained by weighting differently
the trajectories T and their time-reversal image T with a probability measure.
Spontaneous symmetry breaking: relaxation modes of an autonomous system
Explicit symmetry breaking: nonequilibrium steady state by the boundary conditions
DYNAMICAL INSTABILITY
The possibility to predict the future of the system depends on the stability or
instability of the trajectories of Hamilton’s equations.
Most systems are not integrable and presents the property of sensitivity to initial
conditions according to which two nearby trajectories tend to separate at an
exponential rate.
i (t)
1
i  lim
ln
t  t
i (0)
1  max  2  3  ...  0  ...  2 f -1  2 f
Lyapunov exponents:
Spectrum of Lyapunov exponents:
Pairing rule for Hamiltonian systems (symplectic character):  i ,-i i1
f

 i 0
2f
Liouville’s theorem:

i1
Prediction limited by the Lyapunov time:
 1

t  tLyap 
ln final
max
initial

A statistical description is required beyond the Lyapunov time.

CHAOTIC BEHAVIOR IN MOLECULAR DYNAMICS
Hard-sphere gas:
intercollisional time t
diameter d
mean free path l
Perturbation on the velocity angle:

l n
 n   0     0 e  t
d 
t  nt
Estimation of the largest Lyapunov exponent: (Krylov 1940’s)
 
1
t
l

ln
 1010 sec-1 (air in the room)
d
CHAOTIC BEHAVIOR IN MOLECULAR DYNAMICS (cont’d)
Hard-sphere gas: spectrum of Lyapunov exponents
(dynamical system of 33 hard spheres of unit diameter and mass
at unit temperature and density 0.001)

STATISTICAL AVERAGE: PROBABILITY MEASURE
Ergodicity (Boltzmann 1871, 1884): time average = phase-space average
1
lim
T  T
0
T
 A(  ) dt   A()   d 
t
0
0
A  A 0
0
stationary probability density representing the invariant probability measure m

Spectrum of unitary time evolution:
t
iGˆ t
ˆ
pt  U p0  e p0
Ergodicity:
ˆ  0
The stationary probability density is unique: G
0

The eigenvalue z  0 is non-degenerate.


Gˆ  iLˆ
DYNAMICAL RANDOMNESS / TEMPORAL DISORDER
Brownian motion >
< deterministic chaos by Rössler
How random is a fluctuating process?
A process is random if there are many possible paths.
Ex: coin tossing
The longer the time interval, the larger the number of possible paths.
Typically, they multiple exponentially in time: tree of possible paths:
…
…
t = 4Dt
t = 3Dt
t = 2Dt
t = Dt
(w = 0 or 1)
0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111
000, 001, 010, 011, 100, 101, 110, 111
00, 01, 10, 11
0, 1
Hence, the path probabilities decay exponentially:
m(w0 w1 w2 … wn-1) ~ exp( -h Dt n )
The decay rate h is a measure of dynamical randomness / temporal disorder.
h is the so-called entropy per unit time.
DYNAMICAL RANDOMNESS AND ENTROPIES PER UNIT TIME
Partition of the phase space into domains: coarse-graining P  {C1,C2,...,CM }
Stroboscopic observation of the system at sampling time t:
kt   Cw

Path or history: succession of coarse-grained states
w  w0w1w2 wn-1
 path or history:
Multiple-time probability to observe a given
m( w ) = m( w0w1w2 wn-1 ) = m Cw 0 -tCw1 


Entropy per unit time:
1
1
h(P )  lim -  m( w ) ln m( w ) = lim m( w 0w1w 2

n 
n

nt w
nt w 0w1w 2 w n-1

Kolmogorov-Sinai entropy per unit time:
m( w 0w1w 2
 -(n-1)tCw n-1 
w n-1 ) ln m( w 0w1w 2
hKS  Sup h(P )
closed systems: Pesin’s theorem:

(k  0,1,2,...,n -1)
w n-1 )  exp(-hnt ) 
P
hKS   i
i 0
1
( w 0w1w 2
w n-1 )

1
exp(  i t )
 0
w n-1 )
DYNAMICAL RANDOMNESS IN STATISTICAL MECHANICS
Typically a stochastic process such as Brownian motion is much more
random than a chaotic system: its Kolmogorov-Sinai entropy per unit time is infinite.
Partition into cells of size , sampling time t
Brownian motion:
h() 
D
2
Birth-and-death processes, probabilistic cellular automata: h(t )  ln
1
t
Boltzmann-Lorentz equation for a gas of hard spheres of diameter s and mass m
 T and density n:
at temperature

h()  4 n s
2
with
  DtD3vD2  * 100
skBT
2
kBT 399kBT
ln
m
ns 2 m
m
Deterministic theory (Dorfman & van Beijeren):


hKS 
 i  4 n 2s 2
i 0
kBT
m
ln
3.9
ns 3
Chaos is a principle of order in nonequilibrium statistical mechanics.
ESCAPE-RATE FORMALISM: DIFFUSION
escape of a particle
out of the diffusive media
Ex: neutron in a reactor
• diffusion coefficient D = lim t∞ (1/2t) < (xt - x0 )2 >
diffusion equation: ∂t p(x, t) ≈ D ∂x2 p(x, t)
absorbing boundary conditions: p (-L/2, t ) = p (+L /2, t ) = 0
solution: p(x, t) ~ exp-g t) cos( x / L )
escape rate: g≈ D ( / L )2
P. Gaspard & G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693; P. Gaspard & F. Baras, Phys. Rev. E 51 (1995) 5332
ESCAPE-RATE FORMALISM:
THE TRANSPORT COEFFICIENTS & THEIR HELFAND MOMENT
Transport coefficients:
Green-Kubo formula:



J 0( ) J t( ) dt
( )
dG
J ( ) 
dt
microscopic current:
0
1
(Gt( ) - G0( ) )2
Einstein formula:   lim
t  2t


Transport property:
Helfand moment: G
( )
t
G
( )
0

t
J
( )
t'
0

moment:
)
G(D 
 xa
self-diffusion:
shear viscosity:
bulk viscosity: 
  
 43 
heat conductivity

electric
 conductivity:
N
1
G( ) 
x a pay

VkBT a1
N
1
( )
G 
x a pax

VkBT a1
N
1
G ( ) 
 x a (E a - E a )
2
VkBT a1
G
( )
1

VkBT
N
eZ x
a
a1
a
dt'
ESCAPE-RATE FORMALISM:
ESCAPE OF THE HELFAND MOMENT
Diffusion of a Brownian particle
-

2
 Gt( )  
Shear viscosity

2
1
(Gt( ) - G0( ) )2
t  2t
p
2 p
 2
g  Gt( )
t
g
  lim
Einstein formula:
diffusive equation:

absorbing boundary conditions:
p(g   /2,t)  0
 jg j 
   a j exp -g j t  sin 
solution of diffusive equation: p(g,t)
 
2 
 
j1

2
 
escape rate:
g  g1    
for   
 

j 2
g j   
  
ESCAPE-RATE FORMALISM:
ESCAPE RATE & PROBABILITY MEASURE
stretching factors:
w   (w 0w1w 2
n-1
w n-1 )   w
k
k 0
 0  s
1  -s
-g nt
(
w
)

e

-1
escape rate:
w
invariant probability measure:

average Lyapunov exponent:

KS entropy per unit time:
m(w ) 
(w )
-1
 (w )
-1
w
1
 m(w ) ln (w )
n  nt
w
  lim
h  lim n 
1
m(w ) ln m(w )   - g

nt w

ESCAPE-RATE FORMALISM:
ESCAPE-RATE FORMULA
stretching factors:
w   (w 0w1w 2
w n-1 )
Ruelle topological pressure:
1
-

P( )  lim
ln  (w )
n  nt
w
escape rate:
g  -P(1)
Lyapunov exponent:
  (1)  -P'(1)
generalized fractal dimensions:

w
m(w )

(w)
q
(q-1)d q
1
(w )  (w )
Pq  (1- q) dq  -q g
-1
escape-rate formula (f = 2):
g   - hKS  (1- d1)
escape-rate formula (f > 2):
g   i - h
KS   i (1- d1,i )
i 0
closed system: Pesin’s identity:

hKS   i
i 0
i 0
ESCAPE-RATE FORMALISM
CHAOS-TRANSPORT FORMULA
Combining the result from transport theory with
the escape-rate formula from dynamical systems theory,
we obtain the chaos-transport relationship

 2
 2
  lim   
i - hKS 
 lim    i (1- di )



 ,V  
 ,V  
i 0
i 0



large-deviation dynamical relationship

transport
dynamical
instability
∑i i+
g
dynamical
randomness
hKS
Out of equilibrium, the system has less dynamical randomness
than possible by its dynamical instability.
P. Gaspard & G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693; J. R. Dorfman, & P. Gaspard, Phys. Rev. E 51 (1995) 28
ESCAPE-RATE FORMALISM: DIFFUSION
• Helfand moment for diffusion: Gt = xi
diffusion coefficient  = lim t∞ (1/2t) < (Gt - G0 )2 >
diffusion equation: ∂t p(x, t) ≈ D ∂x2 p(x, t)
absorbing boundary conditions: p (-L/2, t ) = p (+L /2, t ) = 0
solution: p(x, t) ~ exp-g t) cos( x / L )
escape rate: g≈ D ( / L )2
• dynamical systems theory
escape rate (leading Pollicott-Ruelle resonance): g- hKS 1- dI
chaos-transport relationship:
D = lim L∞ ( L /  )2 1- dI (L)]
P. Gaspard & G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693; P. Gaspard & F. Baras, Phys. Rev. E 51 (1995) 5332
ESCAPE-RATE FORMALISM: VISCOSITY
• Helfand moment for viscosity: Gt = ∑i xi pyi /(VkBT) 1/2
viscosity coefficient  = lim t∞ (1/2t) < (Gt - G0 )2 >
diffusivity equation for the Helfand moment: ∂t p(g, t) ≈  ∂g2 p(g, t)
absorbing boundary conditions: p (-/2, t ) = p (+/2, t ) = 0
solution: p(g, t) ~ exp-g t) cos( g /  )
escape rate: g≈  ( /  )2
• dynamical systems theory
escape rate (leading Pollicott-Ruelle resonance): g∑i i - hKS 1- dI 
chaos-transport relationship:
= lim∞ ( /  )2 (∑i i - hKS )
J. R. Dorfman, & P. Gaspard, Phys. Rev. E 51 (1995) 28; S. Viscardy & P. Gaspard, Phys. Rev. E 68 (2003) 041205.
CONCLUSIONS
Breaking of time-reversal symmetry in the statistical description
Escape-rate formalism: nonequilibrium transients
fractal repeller
diffusion D :


D  /L  g  
 i - hKS 


0
 i
L
(1990)
viscosity  :
 /  


 g  
 i - hKS 


0
 i
L
(1995)
2
2

2f
Hamiltonian systems: Liouville theorem:

i
0
g   i - hKS  -  i - hKS
i 0
i1

thermostated systems: no Liouville theorem
volume contraction rate:

2f
s  - i  -  i -  i  -  i - hKS
 i1
 0
 0
 0
Pesin’s identity on the attractor: hKS   i
i
http://homepages.ulb.ac.be/~gaspard
i 0
i
i
i 0