The Dissipation Theorem and
Nonequilibrium Properties
Debra J. Searles (Bernhardt)
School of Biomolecular and Physical Sciences
Queensland Micro and Nano-technology Centre
Griffith University
Brisbane, Qld 4111
AUSTRALIA
Collaborators
Denis J. Evans
Stephen R. Williams
ANU
Sarah J. Brookes
Griffith
Davood Ajloo
Pouria Dasmeh
DUBS
The Dissipation Theorem leads to
the result:
Fe
t
B(t) = B(0) + "0 B(s)!(0) ds
Phase Variable
Dissipation Function
Numerical Consequences
Fe
!=
“MD study of the
nanorheology of ndodecane
confined between
planar surfaces”
~2.5 nm
" Pxy (t)
#
=
" % Pxy (t)f($,0)d$
#
MD: Cui, McCabe, Cummings, Cochran, JCP, 118, 8941 (2003)
Expt: Hu, Carson, Granick, PRL 66, 2758 & Granick, Science, 253, 1374 (1991)
Plan
Part A: The Dissipation Function
- Background
- Definition
Part B: Why is the Dissipation Function Useful?
- Example - Le Chatelier’s principle
Part C: The Dissipation Theorem
- Derivation
- Theoretical implications
-
Steady state
Determination of equilibrium distribution
- Numerical implications
Part D: Summary
A. The Dissipation Function
• The time-integral of the dissipation function:
– Logarithm of the probability of observing sets of trajectories
and their time-reversed conjugate trajectories
– Related to ‘relative entropy’ or difference in ‘surprise’ from
information theory
• Defined in order to generalise the Fluctuation
Theorem
A. The Dissipation Function
The Fluctuation Theorem
p(Jt = A)
= e!AFe V"
p(Jt = !A)
t
Jt = !0 ds J(s)
• Originally constant NVE
• Eqns of motion:
Fe
q! i = pi / m + Ci iFe
p! i = Fi + Di iFe ! "pi
Isoenergetic constraint
+
- +
- + +
+
A. The Dissipation Function
The Fluctuation Theorem
p(Jc,t = A)
= eAFe V"
p(Jc,t = !A)
1
0.8
Fe
0.6
+
p(Jc,tPr(!
= A)
t ) 0.4
- +
- + -
0.2
0
-0.2
-2
-1.5
-1
-0.5
0
!t
0.5
1
1.5
A
2
+
+
A. The Dissipation Function
The Fluctuation Theorem
• Generalise
– Time reversible, deterministic dynamics
– Every trajectory will have a conjugate related by a timereversible symmetry
– The trajectory and its conjugate will have time-averaged
dissipative fluxes that are equal in magnitude and opposite in
sign
– Ratio of observing sets of conjugate trajectories
Fe
q! i = pi / m + Ci iFe
p! i = Fi + Di iFe ! Si"pi
!
! ! +! +
+ ! + + !
A. The Dissipation Function
d!(t)
!(t)
!
d!
d!
!*
**
!
d!(t)
(t)
If the dynamics is reversible ! = iSt ! * .
(
)
d!
d!
t
= exp " $0 #(s) ds = e"# t (! ) =
d!(t)
d! *
!=
" !
i#
"#
t
! t = "0 !(s) ds
A. The Dissipation Function
Now consider the relative
probability of observing the phase
volumes dΓ and dΓ*:
p(d!)
*
p(d! )
=
f ( !, 0 ) d!
(
)
d!(t)
d!
d! *
d! * (t)
f ! * ,0 d! *
f ( !,0 ) "# t (! )
=
e
f ( !(t),0 )
$ e%t (! )
! t (") = ln
f ( ", 0 )
# $t
f ( "(t),0 )
f(Γ,0)
f(Γ,t)
*ergodic
consistency
! t (") = #JtVFe$
A. The Dissipation Function
Fluctuation Theorem for the
Dissipation Function
The fluctuation theorem for any dissipation function, Ω,
is:
p(! t = A)
= eA
p(! t = "A)
t
! t = "0 ds !(s)
Using the FT, it is simply to show the Second Law
Inequality:
!t " 0
Evans & Searles, Ad. Phys. 51, 1529-1585 (2002)
Sevick, Prabhakar, Williams & Searles, Ann. Rev. Phys. Chem. 59, 603-633 (2008)
Part B
Why is the Dissipation Function
Useful?
Why is the Dissipation Function
Useful?
•
For nonequilibrium dynamics the dissipation function takes on a
form resembling the rate of extensive entropy production
(-JVFe/(kT)) and can be shown to be equal to it at small fields
!=
"
kB
+ O(Fe2 )
•
It satisfies a Fluctuation Theorem - in small systems it gives a
probability of observing positive and negative values
•
Its time-averaged value is always positive in a nonequilibrium
system (Second Law Inequality)
•
It is the central argument in the Dissipation Theorem
B. Why is the DF interesting?
Example: Le Chatelier’s Principle
If a chemical system at equilibrium
experiences a change in concentration,
temperature, volume, or total pressure, then
the equilibrium shifts to partially counter-act
the imposed change.
Folded Protein ! Unfolded Protein
q = !H > 0
Increase in temperature favours the forward reaction shifts the equilibrium to the right
Folded Protein ! Unfolded Protein
B. Why is the DF interesting?
The Dissipation Function for
Temperature Change
Defined as:
f ( ", 0 )
! t (") = ln
# $t
f ( "(t),0 )
Need to know
• the initial distribution
- Nosé-Hoover extended canonical distribution
• dynamics which allows a temperature change
- Nosé-Hoover thermostatted dynamics
- temperature change must be a time-symmetric
protocol - step function
B. Why is the DF interesting?
Dissipation Function for
Temperature Change
Substitute for f and Λ and rearrange:
(
)
! t (") = (#1 $ # 2 ) H0 ("(t))$H0 ("(0))
(!1 " ! 2 ) (H0 (#(t)) " H0 (#(0))) $ 0
!1 > ! 2
T2 > T1
H0 (!(t)) " H0 (!(0)) = #H $ 0
B. Why is the DF interesting?
Dissipation Function for Temperature
Change
But also satisfy the Fluctuation Theorem
(
)
p( H0 (!(t))"H0 (!(0)) = A)
(
)
p( H0 (!(t))"H0 (!(0)) = "A)
= e(#1 "#2 )A
Why is the Dissipation Function
Useful?
•
For nonequilibrium dynamics the dissipation function takes on a
form resembling the rate of extensive entropy production
(-JVFe/(kT)) and can be shown to be equal to it at small fields
!=
"
kB
+ O(Fe2 )
•
It satisfies a Fluctuation Theorem - in small systems it gives a
probability of observing positive and negative values
•
Its time-averaged value is always positive in a nonequilibrium
system (Second Law Inequality)
•
It is the central argument in the Dissipation Theorem
Part C
The Dissipation Theorem
! t (") = ln
f ( ", 0 )
# $t
f ( "(t),0 )
C. The Dissipation Theorem
Derivation
Fe
• The evolution of the phase space density in its
streaming form is:
df(!,t) # " !
"&
= % + !(!) ( f(!,t) = )*(!)f(!,t)
$ "t
dt
"! '
• The solution to this is:
f(!(t),t) = e
t
" $0 #(!(s)) ds
f(!,0) = e"# t (! )f(!,0)
C. The Dissipation Theorem
f(!(t),t) = e"# t (! )f(!,0)
Fe
Remember - the definition of the dissipation function is
f ( !, 0 ) "# t (! )
e
$ e%t (! )
f ( !(t),0 )
So:
f(!(t),t) = e"# t (! )f(!,0) = e$t (! )f(!(t),0)
True for any Γ, so transform Γ(t) → Γ(0)
f(!(0),t) = e
" t (!(# t))
f(!(0),0) = e $
0
"(!(s))ds
#t
f(!(0),0)
C. The Dissipation Theorem
Response of phase variables
Fe
We can use the distribution function to evaluate
0
#(!(s))ds
"
$t
B(t) = " B(!)f(!,t)d! = " B(!)e
f(!,0)d!
By differentiation and integration
B(t) = B(0) +
t
"0
B(s)!(0) ds
Note that the ensemble average is wrt to the initial distribution.
C. The Dissipation Theorem
Comparison with past work..
0
"(!(s))ds
f(!(0),t) = e $# t
f(!(0),0)
Fe
t
B(t) = B(0) + "0 B(s)!(0) ds
• Kawasaki - adiabatic (unthermostatted)
• Evans and Morriss - homogeneously
thermostatted nonequilibrium dynamics
(Gaussian isokinetic)
• This is more general
Evans, Searles, Williams, JCP, 128 014504(2008); 249901 (2008)
C. The Dissipation Theorem
Theoretical Consequences
Fe
• Nonlinear response for arbitrary dynamics,
(inc. boundary thermostatting, part of the
system subject to field etc.)
• Application to systems that are relaxing
towards equilibrium (so dynamics is just the
equilibrium dynamics)
C. The Dissipation Theorem
Numerical Consequences
Fe
!=
“MD study of the
nanorheology of ndodecane
confined between
planar surfaces”
~2.5 nm
" Pxy (t)
#
=
" % Pxy (t)f($,0)d$
#
MD: Cui, McCabe, Cummings, Cochran, JCP, 118, 8941 (2003)
Expt: Hu, Carson, Granick, PRL 66, 2758 & Granick, Science, 253, 1374 (1991)
C. The Dissipation Theorem
Numerical Consequences
Fe
B(t) = " B(t)f(!,0) d!
B(t) ! B(t)
Odd B
=
B(t;F
)f(",0)
d"
!
B(t;
0)f(",0)
d"
#
#
e
eq
only!
0
#(!(s))ds
"
$t
B(t) = " B(!)f(!,t)d! = " B(!)e
f(!,0)d!
t
B(t) = B(0) + "0 B(s)!(0) ds
C. The Dissipation Theorem
Numerical Consequences
Fe thermostatted walls
• Colour diffusion between
Fe
Wall particles:
Fluid particles:
q! i = pi / m
p! i = Fi + ciFe
!=
- - +- +
+ - + + -
Npart
" cipxiFe / kB Twall
i=Nwall +1
q! i = pi / m
p! i = Fi + Fwi ! "pi
N
#i=1wall Fi ipi
"= N
#i=1wall pi ipi
C. The Dissipation Theorem
Numerical Consequences
Fe
Considered various properties and fields colour current and fluid pressure
B(t) = " B(t)f(!,0) d! # B(t)
eq
Expect best at
high fields
0
#(!(s))ds
"
$t
B(t) = " B(!)f(!,t)d! = " B(!)e
f(!,0)d!
Expect best at
low fields
t
B(t) = B(0) + "0 B(s)!(0) ds
C. The Dissipation Theorem
Numerical Consequences
Fe Twall=1.0
Fe = 0.001; n=0.8;
B(t) = " B(t)f(!,0) d!
t
B(t) = B(0) + "0 B(s)!(0) ds
6.298
0.0006
0.0004
6.297
0.0002
p
J
c
6.296
fluid
0
6.295
-0.0002
6.294
-0.0004
6.293
-0.0006
0
1
2
3
4
Time
5
6
7
8
0
1
2
3
4
t
5
6
7
8
C. The Dissipation Theorem
Numerical Consequences
Fe Twall=1.0
Fe = 1.0; n=0.8;
B(t) = " B(t)f(!,0) d!
t
B(t) = B(0) + "0 B(s)!(0) ds
6.36
0.08
6.34
0.06
6.32
0.04
J
6.3
c
p
fluid
0.02
6.28
6.26
0
6.24
-0.02
0
1
2
3
4
t
5
6
7
8
6.22
0
1
2
3
4
t
5
6
7
8
C. The Dissipation Theorem
Numerical Consequences
Fe conductivities
The Fe=0.001 and Fe=0.01
match (error bars similar size and smaller
than the blue squares)
0.036
0.08
0.07
Fe = 0.01 response
0.06
J /F
c
0.035
Fe = 0.001 response
0.034
Fe = 1.0 response
0.05
J /F
c
e
0.033
e
0.04
0.032
0.03
0.031
0.02
0.03
0.01
0.029
0
0.028
0
1
2
3
4
t
5
6
7
8
2
3
4
5
t
6
7
8
Part D
Summary and Future Work
Conclusions and Future …
•
•
•
•
•
•
•
The dissipation function is of central importance in nonequilibrium
statistical mechanics - appears in the fluctuation theorem, second
law inequality and the dissipation theorem
The dissipation theorem gives a relation for the nonlinear response of
phase functions
The dissipation theorem shows how a distribution function changes
due to application/change/removal of a field
The dissipation theorem provides a practical route to properties of
inhomogeneously thermostatted systems (e.g. wall thermostatted or
remotely thermostatted) at small fields
Like to use this approach to study more practical problems
Relationship between formation of a steady state and correlations explore more widely
Other numerical approaches to obtaining accurate results more
efficiently
Acknowledgements
• Australian Research Council for their
support through a Discovery Grant
• Collaborators
• DUBS, GU and ANUSF for computing
time and support