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Geodesic optimization in thermodynamic control

This talk explores the use of geometric frameworks and optimization techniques to design optimal computational devices and predict efficient biomolecules in thermodynamics through the study of geodesics and Riemannian metrics.

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Geodesic optimization in thermodynamic control

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  1. Geodesic optimization inthermodynamic control 1110110101… 1000110101… 1000110101… 0010010110… Mike DeWeese with: Patrick Zulkowski, Dibyendu Mandal, Katie Klymko, Neha Wadia, Ryan Zarcone, David Sivak, & Gavin Crooks UC Berkeley Manoa Mini-Symposium on Physics of Adaptive Computation January 7, 2019

  2. Some long-term goals: • New thermodynamic relations • Predict signatures of efficient biomolecules • Design optimal computational devices

  3. Talk Outline • Geometric framework • Optimal bit erasure • New generalization of geometric framework from Slow Perturbation Theory 0 1

  4. W W W

  5. Geometry in thermodynamics • Long history, e.g.: • “Thermodynamic length” • Assumed endoreversibility (system in equilibrium, if not same as environment) • Macroscopic description Weinhold, J Chem Phys 63, 2479 (1975) Ruppeiner, PRA 20, 1608 (1979) Schlögl, Z Phys B 59, 449 (1985) Salamon, Nulton, & Ihrig, J Chem Phys 80, 436 (1984) Salamon & Berry, PRL 51, 1127 (1983) Brody & Rivier, PRE 51, 1006 (1995) …

  6. Geometry in thermodynamics • Microscopic rather than macroscopic: • Extend to nonequilibrium systems (beyond endoreversibility): • Derived in the linear response regime • Since extended (Mandal & Jarzynski, others) Crooks, PRL 99, 100602 (2007) Burbea & Rao, J. Multivariate Anal. 12, 575 (1982) Feng & Crooks, PRE 79, 012104 (2009) Sivak & Crooks, PRL 108, 190602 (2012)

  7. Linear resp. optimal protocols = geodesics Conjugate forces External control parameters Riemannian Metric Sivak & Crooks, PRL (2012)

  8. Some properties of optimal protocols (in linear resp. regime): • Follow geodesics • Require constant excess power • Independent of total duration except for global rescaling • Can be calculated based on equilibrium aspects of system Sivak & Crooks, PRL (2012)

  9. Optimal protocols are geodesics (Even simple geometries can yield complex optimal protocols in terms of the control parameters)

  10. Once you know the metric, how do you find geodesics? • No systematic way that’s guaranteed to work, but there are tools: • Christoffel symbols • Geodesic equation • Scaler curvature (Ricci scaler; useful in 2-D) • Killing vector fields (isometries/symmetries) • Try various coordinate transformations to simplify eqs., and recognize familiar spaces

  11. 1st model: Particle in optical tweezers (or driven torsion pendulum, or nanomechanicaloscil.) Inertial Langevin dynamics, 3 control params: Low β Small k Large k High β Time Time Large yo Small yo Zulkowski, Sivak, Crooks & DeWeese PRE (2012)

  12. 1st model: Particle in optical tweezers Inertial Langevin dynamics: with Gaussian white noise: Zulkowski, Sivak, Crooks & DeWeese PRE (2012)

  13. 1st model: Particle in optical tweezers • Decoupling of y0 from (β, k) • : constant negative curvature • hyperbolic geometry for (β, k)submanifold Metric tensor: y0 β k y0 β k Zulkowski, Sivak, Crooks & DeWeese PRE (2012)

  14. 1st model: Particle in optical tweezers Change of variables: hyperbolic geometry 1/ Zulkowski, Sivak, Crooks & DeWeese PRE (2012)

  15. 1st model: Particle in optical tweezers Optimal paths are complex! 1/ Zulkowski, Sivak, Crooks & DeWeese PRE (2012)

  16. Optimal protocol can be much more efficient than nearby paths 1/

  17. Notes and open questions: • Can optimize 3 paramsat once • (k,y0 :Aurell 2012; Seifert 2007, 2008) • Treating β as a control parameter; measuring dissipation in units of entropy • What do the 2 conserved quantities mean? • Are there more? (can be up to 6 Killing fields) • What is full range of applicability for this approach? (> lin. resp. & deriv. truncation?) • What does scalar curvature (R) represent? (seems different than R of George Ruppeiner, but • P.S. Krishnaprasad calc. suggests deep connection…)

  18. Optimal bit erasure: a simple, but exactly solvable, model

  19. Optimal bit erasure: a simple, but exactly solvable, model Overdamped, Brownian particle in double square well; Fokker-Planck for geometric approach 2 control params: 0 1 VB VL

  20. Optimal bit erasure Scalar curvature vanishes everywhere for this 2-D model: R = 0 Flat manifold! 0 1 Zulkowski & DeWeese PRE (2014)

  21. Optimal bit erasure 0 1 Zulkowski & DeWeese PRE (2014)

  22. Optimal bit erasure Linear Response: Exact: 0 1 Zulkowski & DeWeese PRE (2014)

  23. Optimal bit erasure Landauer Finite t cost Aurell et al., J Stat Phys (2012) Esposito et al., Europhys Let (2010) Diana, Bagci & Esposito, PRE (2013) 0 1 (K is twice the square of theHellinger Distance) Zulkowski & DeWeese PRE (2014)

  24. Beyond erasure PL(0) K 0 1 PL(tf) Erasure Zulkowski & DeWeese PRE (2014)

  25. New generalization from Slow Perturbation Theory • Time Dependent Fokker-Planck Equation Explicit terms to all orders Foker-Planck operator Metric Tensor 1st correction to equil. dist. Wadia, Zarcone, DeWeese, Mandal, in prep.

  26. Thanks! Collaborators:David Sivak, Gavin Crooks, Vivienne Ming, Paul King Postdocs:Dibyendu Mandal, Alfonso Apicella, Mohammad Dastjerdi, Louis Kang Grad Students:Patrick Zulkowski, Neha Wadia, Ryan Zarcone, Katie Klymko, Chris Rodgers, Joel Zylberberg, Eric Dodds, Nicole Carlson, Jesse Livezey, Peter Battaglino, JaschaSohl-Dickstein, Charles Frye, BadrAlbanna, Sharif Corinaldi, Mat Leonard, ChayutThanapirom, Vanessa Carels, Sarah Marzen, Joseph Thurakal, Charles Frye, Mayur Mudigonda, PoonehMohammadiara, Pratik Sachdeva, Michael Fang, James Arnemann Undergrads & Postbacks: Jason Murphy, Teddy Yerxa, David Pfau, Marvin Thielk, Trevor Dolinajec, Ambika Rustagi, Vuong Vu, Sarah Kochik, Lily Lin, Yassi Sabahi, Christian Fernandez, Pooja Upadhyaya, Camille Malatt, Daniel Resnick, Jacob Simon, Daniel Cho, Tanling Nancy Hsu, Ramon Martinez, Winward Choy, Carol Pham, Vivek Ayer, Jason Zhang, Josh Chong, Amish Shah, Rachel Lim, Charles Choi, Tommy Li, Danielle Jobe, Yuliy Tsank, S. Zayd Enam, Steven Munn, Shayaan Abdullah, Doron Reuven, Ali Al Setr, Ahyeon Hwang, TeppAhn, Christopher Bryant, Julian Bacon, Emil Barkovich, Anna Yap, Julia Wong, Pauldeep Singh, Angie Kim, Celeste Zamora, Marvin Alcantara, Jenna Pinkham, Nathan Olsy, Bastian Pardede, Ned Udomkesmalee, Peter Dotti, Salman Kahn, Ziyan Feng, Bryon Pavlacka, Steven Munn, Sharon Lin, Ngoc Huyen Tran Nguyen, Adam Bruce, Trevor Grandpre, Kevin Gutowski, Nikhil Sthalekar, Jason Belling, Kyle Chism, Alison Kim, Bernal Jimenez, Christian Merino, Greg Martin, GurubalaKotta, Andrew Berger, Brandon Lazar, Qianjin Wu, Josh Goldman Funding: Lab: NSF, McKnight, McDonnell, Hellman, Rennie, DOD/ARO Student: NSF, NARSAD, Fullbright, NSERC, Thailand Govnt., HHMI, NDSEG, Google This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office Under Contract No.W911NF-13-1-0390.

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