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Optimal protocols and optimal transport in stochastic termodynamics

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].

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Optimal protocols and optimal transport in stochastic termodynamics

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  1. 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] Erik Aurell, KTH & Aalto University

  2. Nonequilbriumphysics of small systems J. Liphardt et. al., Science 296, 1832, 2002 Contributions byJarzynski, Bustamante, Cohen, Crooks, Evans, Gawedzki, Kurchan, Lebowitz,Moriss, Peliti, Ritort, Rondoni, Seifert, Spohn,and many others Erik Aurell, KTH & Aalto University

  3. Fluctuation relations “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

  4. Optimal protocols If you admit for single small systems (the example will follow) 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 variance of JE as an estimator Xu Zhou, 2008 Nature blogs Erik Aurell, KTH & Aalto University

  5. 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 PRL95 (2005) Erik Aurell, KTH & Aalto University

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

  7. Optimal control b* depends both on forward and backward processes An ”instantaneous equilibrium” ansatz for the control Erik Aurell, KTH & Aalto University

  8. Burgers equation Erik Aurell, KTH & Aalto University

  9. 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 Erik Aurell, KTH & Aalto University

  10. 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 ...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 (with average over initial or final state) is minimal released heat by a small system Erik Aurell, KTH & Aalto University

  11. The quadratic penalty term means Monge-Ampere-Kantorovich optimal transport Expected generated heat between initial and final states has one entropy change term, and one ”Burgers term” (released heat): 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. Erik Aurell, KTH & Aalto University

  12. The examples of Schmiedl & Seifert T. Schmiedl & U. Seifert ”Optimal Finite-time processes in stochastic thermodynamics”, Phys Rev Lett98 (2007): 108301 Initial state in equilibrium. Final state is not fixed: final control is. Optimizing over r and q in ”Burgers formula” for the work gives (Seifert’s ”protocol jump formula”) Erik Aurell, KTH & Aalto University

  13. More complicated optimal transport to optimize protocols in stochastics Estimating free energy differencesusingJarzynski’sequation has statistical fluctuations – whichcanbeminimized in the sameway as for heat and workabove J. Liphardt et. al., Science 296, 1832, 2002 …with some auxiliary field Erik Aurell, KTH & Aalto University

  14. Conclusions and open problems We can solve the problems of optimal protocols in the nonequilibrium physics of smallsystems 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? Erik Aurell, KTH & Aalto University

  15. Thanks to Carlos Meija-Monasteiro Paolo Muratore-Ginanneschi Ralf Eichhorn Stefano Bo Erik Aurell, KTH & Aalto University

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