KTH/CSC
Erik Aurell, KTH & Aalto University 1March 15, 2011
Optimal protocols and optimal transport in stochastic
termodynamics
KITPC/ITP-CAS ProgramInterdisciplinary Applications of Statistical Physics and
Complex NetworksWorkshop A – March 14-15 2011
E.A., Carlos Mejia-Monasterio, Paolo Muratore-Ginanneschi [arXiv:1012.2037]
KTH/CSC
Erik Aurell, KTH & Aalto University 2September 28, 2010
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
KTH/CSC
Erik Aurell, KTH & Aalto University 3September 28, 2010
“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”
FW ee
Fluctuation relations
KTH/CSC
Erik Aurell, KTH & Aalto University 4September 28, 2010
Optimal protocolsIf you admit for single small systems (the example will follow)
QdUW
then you can optimize expected dissipated work or released heat
QdUW
Xu Zhou, 2008 Nature blogs
Related to efficiency of the smallsystem e.g. molecular machinessuch as kinesin or ion pumps
Another motivation is thevariance of JEas an estimator SeeEee
SE FW
S
i
FW i
)()(1 22
1
2
KTH/CSC
Erik Aurell, KTH & Aalto University 5March 15, 2011
tttt V 2),(1
The stochastic thermodynamics model
(Langevin Equation)
)();( tit UttV (no control before initial time)
)(~
);( tft UttV (no control after final time)
)2()( tttt V (Stratonovich sense)
f
i
t
t tttQ )2( ),( t
t
t tt
f
i
VW
UUUQW if )()(~ Sekimoto Progr. Theor. Phys.180
(1998); Seifert PRL 95 (2005)
KTH/CSC
Erik Aurell, KTH & Aalto University 6March 15, 2011
Released heat with initial & final states
f
i
t
t xfi bbdtQtStS 121 ||)()(
)(]),[Pr( iiiiii dxdxxx )(]),[Pr( ffffff dxdxxx
re-writing δQ with the Itô convention gives in expectation:
Density evolution, forward Fokker-Planck
Optimal control, Bellman equation
)( ii dx )( ff dx
SRb xx 21*
),(log),( 1 txmtxR
SSbbbSt211121 )()|(|
),( ttVb
),( ff txS),( ii txS
mbmm xxt211 )(
KTH/CSC
Erik Aurell, KTH & Aalto University 7March 15, 2011
)()(])([ 22
12
4121
21 SRSRSSR xxxxxt
mmSRm xxxt21
21 )(
)(21 RS
Optimal control b* depends both on forward and backward processes
An ”instantaneous equilibrium” ansatz for the control xxRb * 0)( vmmt
xv 0
2
21 xt
KTH/CSC
Erik Aurell, KTH & Aalto University 9March 15, 2011
02
2
dt
xdBurgers is free motion if no shocks
)(
)()det(
xm
am
a
x
f
i
]2
)()(max[arg)(
2
t
axxax f
solved by Hopf-Cole transformation if there are
and by Monge-Ampere equation if only initial and final mass distributions are known
KTH/CSC
Erik Aurell, KTH & Aalto University 10March 15, 2011
Burgers’ equation with initial and finaldensities is well-known in Cosmology
Frisch et al Nature (2002), 417 260; Brenier et al MNRAS (2003), 346 501
ita, ftx,
(with average over initial or final state) is minimal released heat by a small system
21 )( axt
...but here we see that it comes up also in mesoscopics.Monge-Ampere equation and Hopf-Cole transformationcan be combined into a minimization of quadratic cost
KTH/CSC
Erik Aurell, KTH & Aalto University 11March 15, 2011
21* )( axSSQ tiffi
Expected generated heat between initial and final states has one entropy change term, and one ”Burgers term” (released heat):
ift mmUUaxW loglog~
)( 12*
The quadratic penalty term means Monge-Ampere-Kantorovich optimal transport
2 RS
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.
KTH/CSC
Erik Aurell, KTH & Aalto University 12March 15, 2011
T. Schmiedl & U. Seifert ”Optimal Finite-time processes in stochastic thermodynamics”, Phys Rev Lett 98 (2007): 108301Initial state in equilibrium. Final state is not fixed: final control is.
The examples of Schmiedl & Seifert
2
21)( ii xxU
2
2)(~ f
cf xxU
.),(2
21 ConstxtxR iii
.),(2
2 ConstqxtxR fr
ff
Optimizing over r and q in ”Burgers formula” for the work gives
2
tc
tcq (Seifert’s ”protocol jump formula”)
KTH/CSC
Erik Aurell, KTH & Aalto University 13March 15, 2011
FW ee
More complicated optimal transport tooptimize protocols in stochastics
We 2
0log2222
61 mxxxxt
0])[(1 mm xxt
J. Liphardt et. al., Science 296, 1832, 2002
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
…with some auxiliary
field
KTH/CSC
Erik Aurell, KTH & Aalto University 14September 28, 2010
Conclusions and open problems
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 BurgersField. 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?