KTH/CSC
Optimal protocols and optimal
transport in stochastic
termodynamics
KITPC/ITP-CAS Program
Interdisciplinary Applications of Statistical Physics and
Complex Networks
Workshop A – March 14-15 2011
E.A., Carlos Mejia-Monasterio, Paolo Muratore-Ginanneschi [arXiv:1012.2037]
March 15, 2011
Erik Aurell, KTH & Aalto University
1
KTH/CSC
Nonequilbrium
physics of small systems
J. Liphardt et. al., Science 296, 1832, 2002
Contributions by Jarzynski, Bustamante, Cohen, Crooks, Evans, Gawedzki,
Kurchan, Lebowitz, Moriss, Peliti, Ritort, Rondoni, Seifert, Spohn, and many others
September 28, 2010
Erik Aurell, KTH & Aalto University
2
KTH/CSC
e
Fluctuation relations
 W
e
September 28, 2010
 F
“The free energy landscape between
two equilibrium states is well related
to the irreversible work required to
drive the system from one state to the
other”
Erik Aurell, KTH & Aalto University
3
Optimal protocols
KTH/CSC
If you admit for single small systems (the example will follow)

W  dU  Q
 W  dU    Q 
then you can optimize expected dissipated work or released heat
Related to efficiency of the small
system e.g. molecular machines
such as kinesin or ion pumps
Another motivation is the
Xu Zhou, 2008 Nature blogs
variance of JE
S
1

 W i
 F 2 
 2 W
 2 F
as an estimator E
(
e

e
)

(
E
e

e
) S
 

S
September 28, 2010
i 1
Erik Aurell, KTH & Aalto University



4
The stochastic thermodynamics model
KTH/CSC
t   1 V (t , t )  2   t
V (t ; t  ti )  U (t )
~
V (t ; t  t f )  U (t )
(Langevin Equation)
(no control before initial time)
(no control after final time)
t  (V )  t  (t  2   t )
Q  
tf
ti
t  (t  2   t )
~
W  Q  U ( f )  U (i )  U
March 15, 2011
(Stratonovich sense)
tf
W   t   V ( t , t )
ti
Sekimoto Progr. Theor. Phys.180
(1998); Seifert PRL 95 (2005)
Erik Aurell, KTH & Aalto University
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Released heat with initial & final states
KTH/CSC
Pr(i [ xi , xi  dxi ])  i (dxi ) Pr( f  [ x f , x f  dx f ])   f (dx f )
re-writing δQ with the Itô convention gives in expectation:
tf
S (ti )  S (t f )  Q   dt 1 | b |2  1  x  b
ti
b   V ( t , t )
 t S  1 (| b |2  1   b)  1 (b  )S  1  2 S
Optimal control, Bellman equation
S ( xi , ti )
S(x f ,t f )
t m  1  x (bm)  1  2x m
i (dxi )
Density evolution, forward Fokker-Planck
b 
*
March 15, 2011
R( x, t )  1 log m( x, t )
1
2
 x R   x S 
Erik Aurell, KTH & Aalto University
 f ( dx f )
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Optimal control b* depends both on
forward and backward processes
KTH/CSC
t m 
[ t 
1
2
1
2
 x  x ( R  S )m    m
1
 x ( R  S ) x    ]S  
1
2
x
1
4
2
x
2
 x ( R  S )  2   ( R  S )
2
x
1
An ”instantaneous equilibrium” ansatz for the control
b   x R   x
 t m    (vm)  0
*
   21 (S  R)
v   x
March 15, 2011
 t   x
1
2
Erik Aurell, KTH & Aalto University
2
0
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Burgers equation
KTH/CSC
 t   x
1
2
March 15, 2011
Erik Aurell, KTH & Aalto University
2
0
8
Burgers is free motion if no shocks
KTH/CSC
2
d x

0
2
dt
solved by Hopf-Cole transformation if there are
( x  a) 2
x(a)  arg max[ f ( x) 
]
2t
and by Monge-Ampere equation if only initial
and final mass distributions are known
mi (a)
x
det( ) 
a
m f ( x)
March 15, 2011
Erik Aurell, KTH & Aalto University
9
KTH/CSC
Burgers’ equation with initial and final
densities is well-known in Cosmology
Frisch et al Nature (2002), 417 260; Brenier et al MNRAS (2003), 346 501
x, t f
a, ti
...but here we see that it comes up also in mesoscopics.
Monge-Ampere equation and Hopf-Cole transformation
can be combined into a minimization of quadratic cost
1
t
( x  a)
March 15, 2011
2
(with average over initial or final state) is
minimal released heat by a small system
Erik Aurell, KTH & Aalto University
10
The quadratic penalty term means MongeAmpere-Kantorovich optimal transport
KTH/CSC
Expected generated heat between initial and final states has one
entropy change term, and one ”Burgers term” (released heat):
S   R  2
Q*  Si  S f 
1

 
f
  i   t ( x  a) 2
This quadratic penalty term can be minimized by discretization,
and looking for minimal transport cost.
Similarly for minimal expected work done on the small system.
W
*
March 15, 2011


t
~
( x  a)  U  U  1 log m f  log mi
2
Erik Aurell, KTH & Aalto University
11
The examples of Schmiedl & Seifert
KTH/CSC
T. Schmiedl & U. Seifert ”Optimal Finite-time processes in
stochastic thermodynamics”, Phys Rev Lett 98 (2007): 108301
Initial state in equilibrium. Final state is not fixed: final control is.
U ( xi ) 
1
2
xi
2
2
R( xi , ti )   xi  Const.
~
U (x f ) 
1
2
c
2
xf  
2
2
R( x f , t f )   x f  q  Const.
r
2
Optimizing over r and q in ”Burgers formula” for the work gives
ct
q

ct  2
March 15, 2011
(Seifert’s ”protocol jump formula”)
Erik Aurell, KTH & Aalto University
12
KTH/CSC
e
More complicated optimal transport to
optimize protocols in stochastics
 W
e
 F
Estimating free energy
differences using Jarzynski’s
equation has statistical
fluctuations – which can be
minimized in the same way
as for heat and work above

J. Liphardt et. al., Science 296, 1832, 2002
e
 t 
March 15, 2011
1
6
2  W
…with some auxiliary field
 t m  1  x  [( x )m]  0
 x  2  2x  2  x   x log m  0
2
Erik Aurell, KTH & Aalto University
13
Conclusions and open problems
KTH/CSC
We can solve the problems of optimal protocols in the
nonequilibrium physics of small systems
The solutions are in terms of optimal (deterministic) transport.
For released heat or dissipated work, the optimal transport
problem is Burgers equation and mass transport by the Burgers
Field. Very efficient methods have been worked out in Cosmology.
What do shocks and caustics in the optimal control problem
mean for stochastic thermodynamics?
Does any of this generalize to other systems e.g. jump processes?
September 28, 2010
Erik Aurell, KTH & Aalto University
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KTH/CSC
Thanks to
Carlos Meija-Monasteiro
Paolo Muratore-Ginanneschi
Ralf Eichhorn
Stefano Bo
March 15, 2011
Erik Aurell, KTH & Aalto University
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