1. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University1 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]

2. KTH/CSC September 28, 2010Erik Aurell, KTH & Aalto University2 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

3. KTH/CSC September 28, 2010Erik Aurell, KTH & Aalto University3 “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” Fluctuation relations

4. KTH/CSC September 28, 2010Erik Aurell, KTH & Aalto University4 Optimal protocols If you admit for single small systems (the example will follow) then you can optimize expected dissipated work or released heat Xu Zhou, 2008 Nature blogs Related to efficiency of the small system e.g. molecular machines such as kinesin or ion pumps Another motivation is the variance of JE as an estimator

5. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University5 The stochastic thermodynamics model (Langevin Equation) (no control before initial time) (no control after final time) (Stratonovich sense) Sekimoto Progr. Theor. Phys.180 (1998); Seifert PRL 95 (2005)

6. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University6 Released heat with initial & final states re-writing δQ with the Itô convention gives in expectation: Density evolution, forward Fokker-Planck Optimal control, Bellman equation

7. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University7 Optimal control b * depends both on forward and backward processes An ”instantaneous equilibrium” ansatz for the control

8. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University8 Burgers equation

9. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University9 Burgers is free motion if no shocks solved by Hopf-Cole transformation if there are and by Monge-Ampere equation if only initial and final mass distributions are known

10. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University10 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 (with average over initial or final state) is minimal released heat by a small system...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

11. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University11 Expected generated heat between initial and final states has one entropy change term, and one ”Burgers term” (released heat): The quadratic penalty term means Monge- Amp e re-Kantorovich optimal transport 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.

12. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University12 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. The examples of Schmiedl & Seifert Optimizing over r and q in ”Burgers formula” for the work gives (Seifert’s ”protocol jump formula”)

13. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University13 More complicated optimal transport to optimize protocols in stochastics 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

14. KTH/CSC September 28, 2010Erik Aurell, KTH & Aalto University14 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 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?

15. KTH/CSC March 15, 2011Erik Aurell, KTH & Aalto University15 Thanks to Carlos Meija-Monasteiro Paolo Muratore-Ginanneschi Ralf Eichhorn Stefano Bo