Time asymmetric driving and entropy production Juan MR Parrondo - PowerPoint PPT Presentation

Time asymmetric driving and entropy production Juan MR Parrondo (GISC and Universidad Complutense de Madrid) Micromachines Reversible transport (1998) Hidden pumps (M. Esposito, 2014) Carnot cycles (Leo Granger and Johannes Hoppenau, 2015) Kyoto, July 28th 2015

Micromachines Autonomous Driven Non-feedback Feedback Symmetric Non-symmetric

Reversible transport JMRP, PRE (1998). Reversible ratchets as Brownian particles in an adiabatically changing periodic potential. An overdamped Brownian particle in a driven periodic potential: x ( t ) = − V � ( x ; λ ( t )) + ξ ( t ) ˙ V (0; λ ) = V ( L ; λ ) λ ( t ) , 0 ≤ t ≤ τ Quasistatic limit: 1 ˙ Z ( λ ( t )) e − β V ( x ; λ ( t )) λ ( t ) → 0 ⇒ ρ ( x, t ) = Zero current, BUT...

Reversible transport Integrated current: Total work: Z τ Z τ J = δ J ( t ) W = δ W ( t ) 0 0 � L � x � � · ˙ � � � · ˙ � � W ( t ) = � kT � λ ln Z − ( � ( t )) � ( t ) dt dx � � + ( x ; � ( t )) � λ � � ( x � ; � ( t )) � J ( t ) = � ( t ) dt dx 0 0 ρ ± ( x ; λ ) = e ± β V ( x ; λ ) Exact differential Z ± Not an exact differential

Reversible transport Z τ J = δ J ( t ) 0 FIG. 3. Net fraction of particles crossing x 0 to the right, Efficiency: 0.06 0.3 0.04 0.2 η η 0.02 0.1 0.00 0.0 0 2 4 6 8 0 2 4 6 8 F α 4 1 Irreversible Irreversible Reversible ratchet ratchet ratchet 3 2

Reversible transport Z τ Line integral (it only depends on the J = δ J ( t ) path in the parameter space) 0 d λ ( t ) � L � x � � · ˙ � dx � � + ( x ; � ( t )) � λ � � ( x � ; � ( t )) � J ( t ) = � ( t ) dt dx 0 0 V max (t) time In a flashing ratchet with an asymmetric potential J=0 Time asymmetry is not enough to induce reversible transport

Adiabatic pumps (Astumian, PRL 2003) An auxiliary state a modulated E a = ∞ barriers E 2 E 1 E a = ∞ 1 2 a E a p 1 p 2 1 2 1 a 2 a barrier state E a 1 2 1 2 a a 1 2 a

Hidden pumps (Esposito & JMRP . PRE 2015) λ t A pump biases a transition, i.e., creates an effective force. The idea: design a protocol such that, at some coarse-grain level (hidden pump), the dynamics of the network is identical to that of an autonomous Markovian system. R L T he original motivation: To compare R L L chemistry R /demon R chemical motors and Maxwell demons R L ∆ E R L (Horowitz, Sagawa, JMRP . PRL 2013). L spatial diffusion

Hidden pumps λ t time X p i ( t + ∆ t ) = p i ( t ) + ( w j ! i − w i ! j ) p j ( t ) ∆ t j 6 = i Markovianity at the coarse grained level: A cyclic protocol with period ∆ t ⌧ 1 /w i → j time In each cycle a small amount of probability is transferred. λ t makes a Low entropy production: cycle in every ∆ t Protocol must be slow compared to the kinetics of the hidden states: ∆ t � (hidden rates) − 1

Hidden pumps An auxiliary state a modulated E a = ∞ barriers E a E 2 E 1 E a = ∞ 1 2 a p 1 p 2 1 2 1 a 2 a barrier state p 1 e − β ( E a − E 1 ) E a We transfer an amount of 1 2 1 2 a a probability from 1 to 2 : p 1 e − β ( E a − E 1 ) ' p 1 w 1 → 2 ∆ t 1 2 a time Poissonian Irreversible λ t rate makes a leak! cycle in every ∆ t

Hidden pumps g) a) E (0) (a) (b) a b) E a f) E a e) E (2) a d) c) E (1) a a barrier states E 2 E 1 E (0) b 2 1 E b E (1) E b b Effective force b E (2) induced by the b pump e − β ( E (1) − E 1 ) = w 1 → 2 ∆ t barrier states a w 1 → 2 = e − β ( E 2 − E 1 − F eff 21 ) 21 ≡ E (2) F e ff − E (1) a b w 2 → 1

Hidden pumps Entropy production of the λ t coarse-grained description: S (cg) tot = J 12 F (e ff ) � T ˙ δ ˙ F ij ≥ 0 − 12 all links Real entropy production: � T ˙ δ ˙ S tot = − F ij ≥ 0 all links but 12 It is even possible to have zero entropy S (cg) T ˙ tot ≥ T ˙ S tot production with finite current!

Hidden variables Entropy production: p ( x, y ; x � , y � ; { λ t } ) ln p ( x, y ; x � , y � ; { λ t } ) � ~ = time reversal S tot = k y ; { ˜ p (˜ x � , ˜ y � ; ˜ x, ˜ λ t } ) ( x,y ) ( x � ,y � ) Coarse grained entropy production: Marginal probability distribution p ( x, x � ; { λ t } ) ln p ( x, x � ; { λ t } ) S (cg) � X p ( x, x 0 ; { λ t } ) = p ( x, y ; x 0 , y 0 ; { λ t } ) tot = k x ; { ˜ p (˜ x � , ˜ λ t } ) y,y 0 x,x � One can prove: S (cg) tot ≤ S tot Two y = y ˜ (overdamped systems + no magnetic fields) assumptions! No hidden driving.

Hidden pumps Entropy production of the λ t coarse-grained description: S (cg) tot = J 12 F (e ff ) � T ˙ δ ˙ F ij ≥ 0 − 12 all links Real entropy production: � T ˙ δ ˙ S tot = − F ij ≥ 0 all links but 12 It is even possible to have zero entropy S (cg) T ˙ tot ≥ T ˙ S tot production with finite current!

Hidden pumps F e ff 21 The pump induces the production of A. (b) 1 2 0.6 d ) cg S tot 12 a ) b ) c ) e ) f ) g ) 0.5 B 10 A E b 8 0.4 = µ A − µ B > 0); 6 E a 0.3 4 1 a , 2 b 400 1 b , 2 a 400 2 0.2 num S tot 0 0.000 0.002 0.004 0.006 0.008 0.010 0.1 S tot 0.0 0 2 4 6 8 10 w reac F e ff 21 1 2 (c) Finite current with zero entropy F ext 6 3 production 5 4

Carnot cycles ∆ S h B < 0 Isothermal Adiabatic Isothermal Adiabatic A compression compression expansion expansion Temperature T h T h Temperature Stiffness ∆ S T c T c ∆ S c B > 0 Entropy (1) (2) (3) (4) Time 0 τ 700 1.25 700 B C D 600 F κ ( × 10 − 3 µ m 2 ) 1.00 600 T part (K) T part (K) 500 0.75 500 400 0.50 400 300 0.25 300 200 0 200 0 5 10 15 20 25 0 5 10 15 20 25 -0.8 -0.6 -0.4 -0.2 0.0 0.2 κ (pN /µ m) κ (pN /µ m) ∆ S/k • Martínez, Roldán, Dinis, Petrov, JMRP , Rica. Brownian Carnot engine. arXiv:1412.1282v3 (2015).

Carnot cycles Hoppenau, Granger, Dinis, JMRP ∆ S h B < 0 Temperature v u bath T h ∆ S T c ∆ S c B > 0 Entropy x L f piston v (3) L 0 0 v (2) 0 x 0 v (1) 0 t T f 0 1. � Hoppenau, J., Niemann, M. & Engel, A. Carnot process with a single particle. Phys Rev E 87, 062127 (2013).

Carnot cycles Hoppenau, Granger, Dinis, JMRP ∆ S h B < 0 Temperature T h ρ ( ∆ S c B , ∆ S h B ) = const ∆ S h B ∆ S h ∆ S c B i ∆ S c T c S ∗ B ∆ S c B > 0 Entropy h ∆ S h B i − S ∗ Carnot line

Conclusions Driven systems apparently perform much better than autonomous systems. Time asymmetry is not enough to have reversible transport. Most likely reversible trajectories have finite power. • JMRP . Reversible ratchets as Brownian particles in an adiabatically changing periodic potential . Phys Rev E 57, 7297 (1998). • JMRP , Blanco, Cao, Brito. Efficiency of Brownian motors . Europhys Lett 43, 248 (1998). • Horowitz, Sagawa, JMRP . Imitating Chemical Motors with Optimal Information Motors. Phys. Rev. Lett. 111, 010602 (2013). • Esposito, JMRP . Stochastic thermodynamics of hidden pumps . Phys Rev E 91, 052114 (2015). • Martínez, Roldán, Dinis, Petrov, JMRP , Rica. Brownian Carnot engine. arXiv:1412.1282v3 (2015). • Hoppenau, Granger, Dinis, JMRP . Efficiency of finite-time small Carnot engines: optimal designs and fluctuations. In preparation.