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Optimal Processes within Stochastic Thermodynamics and beyond Udo - PowerPoint PPT Presentation

San Diego meeting, July 2009 Optimal Processes within Stochastic Thermodynamics and beyond Udo Seifert II. Institut f ur Theoretische Physik, Universit at Stuttgart Thanks to Tim Schmiedl (PhD thesis work) 1 Intro: Classical vs


  1. San Diego meeting, July 2009 Optimal Processes within Stochastic Thermodynamics and beyond Udo Seifert II. Institut f¨ ur Theoretische Physik, Universit¨ at Stuttgart Thanks to Tim Schmiedl (PhD thesis work) 1

  2. • Intro: Classical vs Stochastic thermodynamics • Optimization – directed processes – cyclic processes ∗ heat engines ∗ temperature ratchets ∗ biochemical machines: motor proteins • beyond 2

  3. • Thermodynamics of macroscopic systems [19 th cent] ������ ������ ��� ��� ������ ������ ��� ��� W ������ ������ ��� ��� ������ ������ ��� ��� ������ ������ ��� ��� T λ 0 λ t – First law energy balance: W = ∆ E + Q = ∆ E + T ∆ S M – Second law: ∆ S tot ≡ ∆ S + ∆ S M > 0 W > ∆ E − T ∆ S ≡ ∆ F W diss ≡ W − ∆ F > 0 3

  4. • Macroscopic vs mesoscopic vs molecular machines [Bustamante et al , Physics Today, July 2005] 4

  5. • Stochastic thermodynamics for small systems W T , p λ 0 λ t driving: mechanical hydrodynamical (bio)chemical – First law: how to define work, internal energy and exchanged heat? – fluctuations imply distributions: p ( W ; λ ( τ )) ... – entropy: distribution as well? 5

  6. • Nano-world Experiment: Stretching RNA [Liphardt et al, Science 296 1832, 2002.] – distributions of W diss : 6

  7. • Stochastic thermodynamics applies to such systems where – non-equilibrium is caused by mechanical or chemical forces – ambient solution provides a thermal bath of well-defined T – fluctuations are relevant due to small numbers of involved molecules Energy conservation (1 st law) and entropy production • Main idea: (2 nd law) along a single stochastic trajectory • Review: U.S., Eur. Phys. J. B 64 , 423, 2008 • Precursors: – notion “stoch th’dyn” by Nicolis, van den Broeck mid ‘80s ( on ensemble level) – stochastic energetics (1 st law) by Sekimoto late ‘90s – .... 7

  8. • Paradigm for mechanical driving: V ( x, λ ) f ( λ ) ��������������� ��������������� x 4 ��������������� ��������������� x 1 ��������������� ��������������� ������������������������������ ������������������������������ x 6 ��������������� ��������������� ������������������������������ ������������������������������ ��������������� ��������������� x 3 λ ( τ ) ������������������������������ ������������������������������ x 0 ��������������� ��������������� x 5 ������������������������������ ������������������������������ ��������������� ��������������� V ( x, λ ) ������������������������������ ������������������������������ x 2 ��������������� ��������������� ������������������������������ ������������������������������ ��������������� ��������������� ������������������������������ ������������������������������ ��������������� ��������������� ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ ������������������������������ x λ ( τ ) x = µ [ − V ′ ( x, λ ) + f ( λ )] – Langevin dynamics ˙ + ζ � ζζ � = 2 µ k B T � �� � � �� � F ( x,λ ) ( ≡ 1) – external protocol λ ( τ ) • First law [(Sekimoto, 1997)] : dw = du + dq – applied work: dw = ∂ λ V ( x, λ ) dλ + f ( λ ) dx – internal energy: du = dV – dissipated heat: dq = dw − du = F ( x, λ ) dx = Tds m 8

  9. • Experimental illustration: Colloidal particle in V ( x, λ ( τ )) [V. Blickle, T. Speck, L. Helden, U.S., C. Bechinger, PRL 96, 070603, 2006] – work distribution – non-Gaussian distribution ⇒ – Langevin valid beyond lin response [T. Speck and U.S., PRE 70, 066112, 2004] 9

  10. • Stochastic entropy [U.S., PRL 95, 040602, 2005] – Fokker-Planck equation ∂ τ p ( x, τ ) = − ∂ x j ( x, τ ) = − ∂ x ( µF ( x, λ ) − D∂ x ) p ( x, τ ) [ D = µk B T ] – Common non-eq ensemble entropy [ k B ≡ 1] � dx p ( x, τ ) ln p ( x, τ ) S ( τ ) ≡ − – Stochastic entropy for a single trajectory x ( τ ) s ( τ ) ≡ − ln p ( x ( τ ) , τ ) with � s ( τ ) � = S ( τ ) – ∆ s tot ≡ ∆ s m + ∆ s – � exp[ − ∆ s tot ] � = 1 ⇒ � ∆ s tot � ≥ 0 ∗ integral fluctuation theorem for total entropy production ∗ arbitrary initial state, driving, length of trajectory 10

  11. • General integral fluctuation theorem 1 = � exp[ − q [ x ( τ )] + ln p 1 ( x t ) /p 0 ( x 0 )] � for any (normalized ) p 1 ( x t ) � �� � − ∆ s m • Jarzynski relation (1997) W 2 nd law: T , p � W � | λ ( τ ) ≥ ∆ F ≡ F ( λ t )) − F ( λ 0 ) λ 0 λ t – � exp[ − W ] � = exp[ − ∆ F ] or � exp[ − W d ] � = 1 ∗ p 0 ( x 0 ) ≡ exp[ − ( V ( x 0 , λ 0 ) − F ( λ 0 )] ∗ p 1 ( x t ) ≡ exp[ − ( V ( x t , λ t ) − F ( λ t )] 11

  12. • Optimal finite-time processes in stochastic thermodynamics [T. Schmiedl and U.S., PRL 98, 108301, 2007] W T λ i λ f – optimal protocol λ ∗ ( τ ) minimizes � W � for given λ i , λ f and finite t 12

  13. • Ex 1: Moving a laser trap V ( x, λ ) = ( x − λ ( τ )) 2 / 2 V ( x, 0) V ( x, t ) λ f λ lin( τ ) ∆ λ λ ∗ ( τ ) ∆ λ 0 λ f 0 t – λ ∗ ( τ ) requires jumps at beginning and end ∆ λ = λ f / ( t + 2) 1 ≥ W ∗ ( t ) /W lin ( t ) ≥ 0 . 88 – gain 13

  14. V ( x, λ ) = λ ( τ ) x 2 / 2 • Ex 2: Stiffening trap – typical size of the jump λf λi ∆ λ/λ f ( W ∗ − W qs ) /W jp (d) 0 100 0.9 1 (a) 0.7 0.9 10 0.6 0.8 0.55 0.7 5 λ f /λ i = 2 , λ i t = 0 . 1 1 0.6 λ f /λ i = 2 , λ i t = 1 λ i t λ f /λ i = 2 , λ i t = 10 0.5 λ f /λ i = 5 , λ i t = 0 . 1 4 λ f /λ i = 5 , λ i t = 1 0.1 ∆ λ/λ f λ f /λ i = 5 , λ i t = 10 ( W ∗ − W qs ) /W jp 3 λ ∗ /λ i 0.01 10 100 1000 10000 2 λ f /λ i 1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 τ/t – might help to improve – jumps are generic convergence of � exp( − W ) � 14

  15. • Underdamped dynamics: role of inertia [A. Gomez-Marin, T.Schmiedl , U.S., J Chem Phys 129 024114 (2008)] x + γx + V ′ ( x, λ ) = ξ m ¨ ∗ jumps and delta-functions at the boundaries ∗ W ∗ /W lin >> 1 possible 15

  16. • Heat engines at maximal power – Curzon-Ahlborn (1975) – Carnot (1824) Th − Qh = α ( Th − Tm h ) Tm h Th − W − Qh Tm c − W Qc = β ( Tm − Tc ) c Tc – efficiency at maximum power Qc � η ca ≡ 1 − T c /T h Tc – η c ≡ 1 − T c /T h – universality(?) but zero power [cf van den Broeck, PRL 2005] – what about fluctuations? 16

  17. • Brownian heat engine at maximal power [T. Schmiedl and U.S., EPL 81 , 20003, (2008)] V V 1 p a p b 1 T h 0.8 2 0.6 4 η ∗ 0.4 T c α = 1 / 2 α = α CA 0.2 V V α = 1 3 η C 0 0 1 2 3 4 5 p b p a ( T h − T c ) /T c η ∗ = η c with α = 1 / 2 for temp-independent mobility – 2 − αη c Curzon-Ahlborn neither universal nor a bound – 17

  18. b • Optimizing potentials for temperature ratchets [F. Berger, T. Schmiedl, U.S., PRE 79 , 031118, 2009] T c T h T h T c T c T h 2.5 V(x) T(x) 2 2 V ( x ) 1.5 f V(x) T(x) 1 1 0.5 0 0 -0.5 0 0.2 0.4 0.6 0.8 1 0 L / 2 L x 0.5sin(2 π x)+1 T(x)=0.5sin(2 π x)+1 7 2.5 d=1 j(d) d=1 1.4 d=0.1 d=0.1 6 2 d=0.005 d=0.005 5 1.5 1.2 1 4 T(x) V(x) 0.01 0.1 1 1 3 0.8 2 1 0.6 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x 18

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