Introduction to Nonequilibrium
Work Theorems -Lecture III
C. Jarzynski
Theoretical Division
Los Alamos National Laboratory
LAUR-06-004
Nonequilibrium work
& equilibrium free energy differences
piston & ideal gas
u
A
e
" #W
B
=e
" #$F
"F = #n p $ #1 ln2
Irreversible process:
1. Prepare in equilibrium (T)
2. Pull out the piston
W = work performed
3. Repeat
A “seeming paradox”
Lua & Grosberg, J Phys Chem (2005)
u
A
For very large piston speeds (u>>vth),
the probability of observing any
piston-particle collisions becomes
negligible, so W=0 almost always.
B
limu"# e
Resolution:
!
$ %W
?
=1 & 2
np
= e$ %'F
The average must be dominated
by very rare realizations.
Q: What is the nature of these dominant realizations?
Convergence of <exp(-βW)>
g(W ) = " (W ) exp(#$ W )
e" # W =
" (W )
!
=
!
"# W
dW
$
W
e
( )
%
% dW g(W )
Ritort, J Stat Mech (2004)
!
typical work values
dominant work values
(rare but important )
What is the nature of the dominant realizations
that contribute the most to <exp(-βW)> ?
Conjugate processes and
macroscopic irreversibility
expansion
compression
u
u
u
u
u
forward process
u
reverse process
Conjugate twins and microscopic reversibility
Trajectories come in
conjugate pairs
For any trajectory that is a solution of Hamilton’s equations
when the parameter is varied from A to B, its conjugate twin
is a solution when the parameter is varied from B to A.
Piston and gas … typical vs dominant realizations
expansion
compression
u
u
Piston and gas: forward vs
reverse 2u
u
u
u
WF "0
u
W R " n p mu 2
Piston and gas … typical vs dominant realizations
expansion
expansion
2u
u
u
Piston and gas: typical vs
dominant
u
u
u
typical
(W
F
" 0)
u
(
dominant W F " #n p mu 2
)
Piston and gas … typical vs dominant realizations
compression
compression
u
u
Piston and gas: typical vs
dominant2u
u
u
u
(
)
dominant W R " 0
u
typical
(W
R
" n p mu 2 )
Duality and exponential averages
The dominant realizations of the forward process are
the conjugate twins of the typical realizations of
the reverse process … and vice-versa.
C.J., PRE (2006)
Piston and gas:
(with u >> vth)
expansion, typical
no collisions
W~0
compression, typical
~np/2 collisions
W ~ +npmu2
expansion, dominant
~np/2 collisions
W ~ - npmu2
compression, dominant
no collisions
W~0
“Causal” and “anti-causal”ensembles
equilibrium at t=0
(causal)
typical
Closely related to irreversibility,
2nd Law of Thermodynamics
dominant
equilibrium at t=τ
(anti-causal)
Evans and Searles, PRE (1996)
Cohen and Berlin, Physica (1960)
x tF
1
pF [" ] =
exp #$ H A (%0F )
ZA
Sketch of proof
γ
!
x
!
!
!
[
pF ["
γ*
]
1
pR [" ] =
exp #$ H B (%0R )
ZB
*
R
t
[
pR
] = exp # W
(
[
[" ]
*
]
F
$ %F )
]
Crooks, J Stat Phys (1998)
exp("#W )
F
=
!
d
$
p
$
exp
% F [ ] ("#W F [$ ]) = exp("#&F ) % d$ pR [$ * ]
The dominant trajectories are those
in the peak region of the integrand.
Convergence of the exponential average
exp("#W )
1 N
$ % exp("#W n )
N n=1
Q: How large must N be in order for this
! approximation to be reasonably good?
A:
N > 1/Pdom
Probability of obtaining
a dominant realization
Estimate this lower bound, for the piston and gas …
First, expansion …
Nc for expansion
Initial conditions:
np/2 particles , vx~2u
W
R
typ
" n p mu
2
"F = #n p $ #1 ln2
typical
Pdom "
dominant
!
n p!
exp[#$m(2u) /2])
(
!
(n /2)! (n /2)!
p
2
np /2
p
[
]
[
]
R
" exp #n p ($mu 2 # ln2) " exp #$ (W typ
+ %F )
!
Now, compression …
Ncand
for gas:
compression
Initial conditions:
Piston
all particles in left
typical vs
half of the box
dominant
W
F
typ
=0
"F = #n p $ #1 ln2
dominant
typical
!
Pdom " 2
#n p
[
F
= exp(#n p ln2) = exp!#$ (W typ
# %F )
]
Number of realizations needed for convergence
[
" exp[# (W
]
$ %F )]
expansion:
F
R
N F > 1/Pdom
" exp # (W typ
+ $F )
compression:
R
N R > 1/Pdom
F
typ
!
More generally, the number of realization required for
convergence
! in the case of the forward process,
grows exponentially in the amount of work dissipated
during the reverse process, and vice-versa.
~ Gore, Ritort, Bustamante, PNAS (2003)
Relation to Second Law
e" #W = e" #$F
W " #F
What is the probability that
the 2nd law will be “violated”
by at least ζ units of energy?
!
!
'
" (
P[W < "F # $ ] =
"e
!
!
"F#$
#&
&F$'
$%

exponentially
rare
dW % (W )
dW #(W ) e ) (&F $' $W )
# ($F%& )
)
+(
%(
dW ' (W ) e% #W = exp(%& /kT)
Compression of dilute classical gas
np interacting
particles,
initial temperature T
Crooks & Jarzynski,
cond-mat/0603116
u~0
A
B
slow compression,
no heat bath
pV = n" #1 = 2E /3
Ideal gas law:
Work:
dW = " p dV
!
dE
"(2E /3V ) dV
(2 / 3
"
%
E1
VB
… integrate:
=$ '
!
E 0 # VA &
!
!
W = E1 " E 0 = #E 0
!
" = (VB /VA )
2/3
#1
Compression of dilute classical gas
Crooks & Jarzynski,
cond-mat/0603116
W = "E 0
u~0
" = (VB /VA )
A
B
!
k = 3n p /2
eq
dE
p
# 0 A ( E 0 )$ (!W % &E 0 )
#
= " dE 0
(#E 0 ) k%1e!% #E 0 & '(W % (E 0 )
$( k )
k+1
" % "W ( + "W / #
=
, - (W )
'
* e
#$( k ) & # )
"(W ) =
!
!
!
e" #W =
" #W
dW
$
W
e
(
)
+
2/3
% VA ( n p
= L = ' * = e" #,F
& VB )
#1