On optimal protocols in stochastic thermodynamics Anomalous - PowerPoint PPT Presentation

KTH/CSC On optimal protocols in stochastic thermodynamics Anomalous transport: From Billiards to Nanosystems Sperlonga, September 20-24 2010 joint work (in progress) with Paolo Muratore-Ginanneschi, U of Helsinki Carlos Mejía-Monasterio, U of Helsinki & Madrid September 23, 2010 Erik Aurell, KTH & Aalto University 1

Thermodynamics of small systems KTH/CSC Xu Zhou, 2008 Nature blogs J. Liphardt et. al., Science 296, 1832, 2002 Contributions by Jarzynski, Crooks, Bustamante, Ritort, Peliti, Imparato, Seifert, Rondoni, Vulpiani, Puglisi, Kurchan, Gawedzki, Lebowitz, and many others September 23, 2010 Erik Aurell, KTH & Aalto University 2

KTH/CSC “The free energy landscape between − β − β ∆ = two equilibrium states is well related W F e e to the irreversible work required to drive the system from one state to the other” September 23, 2010 Erik Aurell, KTH & Aalto University 3

Why optimization & optimal protocols? KTH/CSC Jarzynski estimator unbiased for exponentiated Free energy diff. [ ] [ ] S S ∑ ∑ ( i ) ( i ) < − β > = − β = − β ∆ − β − β W W F < > ≡ W W E e E e e 1 e e 1 S S S S = = i 1 i 1 But has a statistic error, scaling with sample number S [ ]   S 1 ∑ W i − β − β ∆ − β − β ∆ − = − F 2 2 W 2 F E ( e e ) ( E e e ) S (protocol dep.)   S   = i 1 NB: Jarzynski estimator is biased for the Free energy difference S [ ] [ ] ∑ < > ≡ ( i ) < > = ≥ ∆ W 1 W E W E W F S S S = i 1 September 23, 2010 Erik Aurell, KTH & Aalto University 4

Other motivation(s) for optimization KTH/CSC If you admit for single small systems (the example will follow) ⇒ δ = + δ < δ >=< > + < δ > W dU Q W dU Q δ = ∆ + W F W and if the initial and final states are known diss < >=< δ > + ∆ W diss Q T S then the expected dissipated work is Minimizing expected dissipated work is maximizing efficiency of the small system Molecular machines have quite high efficiency : Kinesin – 60% Xu Zhou, 2008 Nature blogs Ion pump on membrane – 70% September 23, 2010 Erik Aurell, KTH & Aalto University 5

Which optimization tasks? What to optimize? Which constraints? KTH/CSC What to optimize: some expectation value over the paths [ ] [ ] [ ] [ ] − β 2 W Var W E W E e E Q Which constraints Class I: given initial and final states For instance, equilibrium at the same T but different potentials. Connected to Schrödinger ”Über die Umkehrung des Naturgesetze”, Sitzung der Preuss. Akad. Wissen. Berlin 144 (1931); and to Guerra & Morato (1983); Dai Pra (1991) Class II: given initial state and final control Schmiedl & Seifert, Phys. Rev. Lett. 98 (2007); Gomez-Marin, Schiedl & Seifert, J. Chem. Phys. 129 (2008); Then & Engel, Phys Rev E77 (2008). September 23, 2010 Erik Aurell, KTH & Aalto University 6

What is stochastic thermodynamics? KTH/CSC  1 ξ = − ∂ ξ λ + τβ ω  V ( , ) 2 (Langevin Equation) τ ξ t t t t ξ < = ξ V ( ; t t ) U ( ) (no control before initial time) t i t ~ ξ > = ξ V ( ; t t ) U ( ) (no control after final time) t f t    ξ ⋅ −∂ = τ ξ ⋅ ξ − τβ ω  ( V ) ( 2 ) (Stratonovich) ξ t t t t t t    ∫ ∫ = τ ξ ⋅ ξ − τβ ω = λ ⋅ ∂ ξ λ f  f Q ( 2 ) W V ( , ) t t t λ t t t t t i i ~ − = ξ − ξ = ∆ W Q U ( ) U ( ) U Sekimoto Progr. Theor. Phys. 180 f i (1998); Seifert PRL 95 (2005) September 23, 2010 Erik Aurell, KTH & Aalto University 7

Class I: minimizing expected released heat with given initial and final states KTH/CSC ξ ∈ + = µ ξ ∈ + = µ Pr( [ x , x dx ]) ( dx ) Pr( [ x , x dx ]) ( dx ) i i i i i i f f f f f f Re-writing Q with the Ito convention gives, in expectation: [ ] ∫ dt t = −∂ ξ λ b V ( , ) µ µ = < − ∂ ⋅ > f 2 E Q | , ( | b | T b ) ξ t t ξ i f τ t i b (for short) is the stochastic control of the Langevin equation  1 ξ = + τβ ω  b 2 τ t t which gives a forward Fokker-Planck equation for the density 1 1 ∂ + ∂ = ∂ 2 = µ m ( bm ) m m ( x , t ) dx ( dx ) τ t x x βτ i i September 23, 2010 Erik Aurell, KTH & Aalto University 8

Class I: minimizing expected released heat with given initial and final states KTH/CSC Density evolution, forward Fokker-Planck t t f i µ µ f dx ( ) i dx ( ) Optimal control, Bellman equation f i + ∂ ⋅ ⋅ ∂ 2 (| b | T b ) ( b ) S 1 − ∂ = + + ∂ 2 S S t , x t τ τ τβ µ ∫ f dx ( ) = µ S ( t ) m ( x , t ) S ( x , t ; f ) dx f − t dt , y [ ] µ µ = E Q | , S ( t ) i f i Problem: b * depends ∗ = ∂ − ∂ b R S = R T log m both on the forward and the backward proceses September 23, 2010 Erik Aurell, KTH & Aalto University 9

Class I: minimizing expected released heat with given initial and final states KTH/CSC ∂ − ( R S ) 1 ∂ + ∂ = ∂ 2 m [ x m ] m = R T log m t x x τ βτ 2 ∂ − ∂ − 2 ( R S ) 1 | ( R S ) | 1 ∂ + ∂ + ∂ = − − ∂ − 2 2 [ x ] S x ( R S ) t x x x τ τβ τ βτ 2 4 2 1 ∂ + ⋅ ∂ = ψ = − S + v ( v ) v 0 ( R ) t τ 2 ∂ + ∂ ⋅ = m ( vm ) 0 = ∂ ψ v t x Burgers’ equation and mass transport by Burgers’ field September 23, 2010 Erik Aurell, KTH & Aalto University 10

Burgers’ equation with initial and final densities is a problem we know well KTH/CSC x , t a , t f i ∂ m ( x ) x = f det( ) ∂ a m ( a ) i − 2 ( x a ) = ∂ ψ v ( a , t ) ( a ) = ψ − a ( x ) arg max[ ( a ) ] 0 i a 0 o ∆ 2 t − x a ( x ) = = Valid as long as there v ( x , t ) v ( a , t ) f 0 i ∆ t are no shocks in the field Monge-Ampère-Kantorovich Reconstruction of the early Universe Frisch et al Nature (2002), 417 260; Brenier et al MNRAS (2003), 346 501 September 23, 2010 Erik Aurell, KTH & Aalto University 11

« True » equations « Zeldovich » « Burgers » equation of early Universe approximation and mass transport KTH/CSC v v caustic shock x caustic September 23, 2010 Courtesy Sobolevskii & Mohayaee 12

MAK-reconstruction applied to optimal protocols in stochastic thermodynamics KTH/CSC One can now express released heat from initial to final state as ∗ < > = τ ∆ < > + − f 2 Q t v TS TS i 0 i i f ~ ~ < > =< > − < > rel Q U U In relaxation, no work is done: f f rel < >=< > + < > W U Q So using the equivalence ∗ < > = τ ∆ < > + − rel 2 W t v F F We have also i 0 i f i ~ − =< + ∆ − − + ∆ ∂ > F F U ( a v t ) U ( a ) T log( 1 t v ) Since f i 0 a 0 i It should also be possible to optimize over final state f (Class II) September 23, 2010 Erik Aurell, KTH & Aalto University 13

Comparison with Seifert’s examples KTH/CSC T. Schield & U. Seifert ”Optimal Finite-time processes in stochastic thermodynamics”, Phys Rev Lett 98 (2007): 108301 We have not worked this out yet..... (that’s why this is work in progress...) September 23, 2010 Erik Aurell, KTH & Aalto University 14

Some other problems can be solved this way also (but not easily) KTH/CSC Mimimizing the statistical error of the Jarzynski estimator [ ] − β 2 W E e Can be turned into a non-linear transport problem 2 ∂ + ∂ ⋅ ∂ φ = m [( log ) m ] 0 t x x τ 2 ∂ φ − ∂ ⋅ ∂ φ = [( log m ) ] 0 t x x τ Which is a bit more difficult than mass transport by Burgers’ eq. September 23, 2010 Erik Aurell, KTH & Aalto University 15

Conclusions and open problems KTH/CSC Work out the mixed backward-forward equations for other problems in stochastic thermodynamics. Are there other examples that are as solvable as Burgers’ eq.? Compare to Seifert’s state-independent protocols for the harmonic trap (shame on us, we have not done so as yet). 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? September 23, 2010 Erik Aurell, KTH & Aalto University 16

Thanks to Collaborators on the project Paolo Muratore-Ginanneschi Carlos Mejía-Monasteiro For posing and framing the problem Udo Seifert For contributing slides Xu Zhou (through Nature blogs) Andrei Sobolevskii & Roya Mohayaee