Optimal Thermodynamic Control and the Riemannian Geometry of Ising magnets Gavin Crooks Lawrence Berkeley National Lab Funding: Citizens Like You! MURI threeplusone.com PRE 92, 060102(R) (2015) NSF, DOE 2 / 27
The 2nd Law of Thermodynamics Clausius inequality (1865) Entropy S total 0 Entropy increases as time progresses Cycles of time R.Penrose (2010) 3 / 27 Once or twice I have been provoked and asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold. It was also negative. Yet I was asking something which is about the scientific equivalent of Have you read a work of Shakespeare's? -- C. P. Snow
Thermodynamic Equilibrium No change in Entropy. No Arrow of time. Future, past and present are indistinguishable 4 / 27
Entropy and Disorder 1 natural unit of entropy equivalent to 1 kt of thermal energy T : Temperature (ambient 300 Kelvin) k : Boltzmann s constant 1 kt = 25 mev = 2.5 kj/mol = 4 zeptojoules average kinetic energy = 1.5 kt 5 / 27
Laser Trap Trap Bead Unfolding of RNA hairpins. (circa 2000) Trap Bead RNA Hairpin Entropy sometimes goes down! Actuator Bead force Pizeoelectic Actuator unfolding length folding Actuator Bead total entropy change probability 0 S total = 1 T W work temperature unfolding F free energy change 6 / 27
The (improved) 2nd Law of Thermodynamics Clausius inequality (1865) Jarzynski identity (1997) h S total i 0 he S total i =1 equality only for reversible process equality far-from-equilibrium Trap Bead Trap Bead S total = 1 T W F Laser Trap Actuator Bead RNA Hairpin Actuator Bead 7 / 27 Pizeoelectic Actuator
Fluctuation Theorems: Dissipation (entropy increase) breaks time-reversal symmetry Work Free Energy Change Inverse Temperature Phase Space Reverse Trajectory Forward Trajectory Time 8 / 27
What are the fundamental operational principles of nano-scale machines? proton gradient mechanical energy chemical energy ATP synthase Free Energy ATP = 20 kt = 0.5 ev 9 / 27
Coupled Systems & the Thermodynamics of prediction Recent Projects Geometry of thermodynamic control Measurement of nonequilibrium free energy steady state PRL (2012) PRL (2012) PRE (2012) PRE (2015) steady state approximation PRL (2012) Nonequilibrium simulation PNAS (2011) PRX (2013) JPC (2014) Dynamics of bacterial cell growth PNAS (2014) PRL (2014) 10 / 27
Optimal thermodynamic control of molecular scale systems Laser Trap Trap Bead Trap Bead RNA Hairpin length Protocol Actuator Bead Pizeoelectic Actuator Actuator Bead time Which finite-time experimental protocols minimize dissipation? 11 / 27
Exact minimum dissipation protocols Control trap position: Schmiedl & Seifert PRL (2007) 12 / 27
Geometry of thermodynamic control Finite time thermodynamics with linear response friction tensor Riemannian metric, minimum dissipation paths are geodesics nonequilibrium excess power P ex (t 0 )= imposed by protocol Λ " d T dt # linear response friction tensor t 0 (t 0 ) apple d dt t 0 Prof. David Sivak (Simon Fraser U.) F. Weinhold (1975), Peter Salamon and Steven Berry (1983), Sivak & Crooks PRL (2012) 13 / 27
Combine linear response and thermodynamic geometry free energy p(x )=e F ( ) E(x, ) inverse temperature controllable parameters ( ) ij = Z 1 0 dt h X j (0) X i (t)i positive semi-definite symmetric matrix i.e. thermodynamic metric tenser nonequilibrium excess power P ex (t 0 )= imposed by protocol Λ " d T dt correlations of conjugate variables # linear response friction tensor t 0 (t 0 ) Sivak & Crooks PRL (2012) apple d dt t 0 14 / 27 10
Geometry of thermodynamic control Linear response friction tensor yields a Riemannian metric Metric tensor measures friction in control space Optimal (minimum dissipation) protocols: are geodesics in control space independent of protocol duration constant excess power dissipation inversely proportional to protocol duration minimize time for fixed dissipation minimize error for free energy calculations Rotskoff & Crooks (2015) Sivak & Crooks (2012) Peter Salamon and Steven Berry (1983) 15 / 27
Thermodynamic Geometry of a Harmonic Trap Finite time thermodynamics with linear response friction tensor Riemannian metric, minimum dissipation paths are geodesics nonequilibrium excess power P ex (t 0 )= imposed by protocol Λ " d T dt # linear response friction tensor t 0 (t 0 ) apple d dt t 0 David Sivak Michael DeWeese 16 / 27 Sivak & Crooks, Phys. Rev. Lett., 2012 Zulkowski, Sivak, Crooks & DeWeese Phys. Rev. E 2012 Patrick Zulkowski
Thermodynamic Geometry of a Harmonic Trap stiff 0.5 2 k z loose 0 0.25 Β 1 hot 1 cold 1 x 3 Zulkowski, Sivak, Crooks & DeWeese Phys. Rev. E 2012 Hyperbolic geometry 17 / 27
The Ising Model H( )= J X <ij> i j h X i i 1.5 optimal protocol 1.0 magnetic field: h 0.5 0.0-0.5 critical point Grant Rotskoff -1.0-1.5 1 2 3 4 5 reduced temperature: t = 1/ J 18 / 27
( ) ij = Z 1 0 dt h X j (0) X i (t)i 19 / 27
Energy-Energy 20 / 22 ( )ij = Z 1 0 dt h Xj (0) Xi (t)i
Energy-Energy 21 / 22
Magnetization-Magnetization 22 / 22
Energy - Magnetization 23 / 22
Fast Marching for Finding Geodesics on a Mesh. Dijkstra Fast Marching Method 24 / 27
Minimum Dissipation Protocol (Geodesics) 1.5 1.0 optimal protocol magnetic field: h 0.5 0.0-0.5 critical point -1.0 25 / 27-1.5 1 2 3 4 5 reduced temperature: t = 1/ J
Minimum Dissipation Protocol (Geodesics) threeplusone.com PRE 26 / 2292, 060102(R) (2015)
Frontiers of thermodynamic control Trap Bead Trap Bead 2 Laser Trap Actuator Bead Pizeoelectic Actuator Experiments RNA Hairpin Actuator Bead 1 0 2 1 0 2 1 0 2 1 0-1 0 1 2 b UHl á 2L -1 0 1 2 b UHl á 1.5L -1 0 1 2 b UHl á 1L -1 0 1 2 b UHl á 0L Optimal Control Minimum dissipation computation Domain scale simulation 27 / 27