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National Science Foundation UCSD funds: Kurt Shuler, Misha Galperin, Katja Lindenberg Patricia Edwins

• Chemical Austria C. Dellago Belgium Flemish C. Van den Broeck Belgium Walloon P. Gaspard Czech Republic E. Hulicius Finland J. Pekola Germany U. Seifert • Physical Ireland J. Gleeson Norway A. Hansen Spain JMR Parrondo Sweden Hongqi Xu Switzerland MO Hongler • Bio-physical

Tuesday night 6:30 El Torito (Mexican Restaurant) UCSD campus Marriott Residence 200 m Inn

T T E E S S T T

Christian Van den Broeck Universiteit Hasselt christian.vandenbroeck@uhasselt.be

NOT TO PERPETUATE A NAME, WHICH MUST ENDURE WHILE THE PEACEFUL ARTS FLOURISH, BUT TO SHOW THAT MANKIND HAVE LEARNT TO HONOR TH0SE WHO BEST DESERVE THEIR GRATITUDE,å THE KING HIS MINISTERS, AND MANY OF THE NOBLES AND COMMONERS OF THE REALM, RAISED THIS MONUMENT TO JAMES WATT, WHO, DIRECTING THIE FORCE OF AN ORIGINAL GENIUS, EARLY EXERCISED IN PHILOSOPHIC RESEARCH, TO THE IMPROVEMENT OF THE STEAMENGINE, ENLARGED THE RESOURCES OF HlS COUNTRY, INCREASED THE POWER OF MAN, AND ROSE TO AN EMINENT PLACE AMONG TRE MOST ILLUSTRIOUS FOLLOWERS OF SCIENCE AND THE REAL BENEFACTORS OF THE WORLD. BORN AT GREENOCE, IUDCCXXXVI. DIED AT BEATRFIELD, IN STAFFORDSHIRE, MDCCCXIX.

T h T c W ≤ η c Q h Q c Q h η c = 1 − T c T h W = Q h − Q c Réflexions sur la puissance motrice du feu et sur les machines propres a developper cette puissance. Δ S = Q h + Q c ≥ 0 T h T c Equality Sign: Reversible Process dQ ∫ Δ S = ≥ 0 Ueber die bewegende Kraft der Wärme und die Gesetze, T rev welche sich daraus für die Wärme selbst ableiten lassen

Exceed Carnot at Small Scale? η = W ≤ 1 − T c Q h T h Yes? No! T c =T h Erase 1 bit: Δ S= k B ln2

η = W ≤ 1 − T c Q h T h Below Carnot at Irreversible Heat Flux Small Scale? Under Steady State J. Parrondo P. Espagnol, Am J Phys 64, 1125 (1996) C. Van den Broeck, R. Kawai, and P. Meurs, Conditions? PRL.93, 090601 (2004)

Thermal X 1 = Δ T/T 2 Mechanical X 2 =F/T Heat flux J 1 Rotation Speed J 2 Carnot efficiency? C. Van den Broeck, Adv Chem Phys 135 , 189 (2007) Reversible: d i S/dt=0 hence η = η c . If J 1 = J 2 =0 for X 1 and X 2 nonzero. Only possible if determinant of matrix L is zero: L 11 L 22 =(L 12 ) 2 Carnot efficiency Architectural constraint of strong coupling J 2 /J 1 = L 21 /L 11 . L 11 L 22 =(L 12 ) 2

FL Curzon B Ahlborn 1 − η C = T c Am. J. Phys. 43, 22 (1975) T h Efficiency at maximum power η = W T c endoreversible 1 − = Q h T h approximation Exact in linear approx. for : L 11 L 22 =(L 12 ) 2 2 /8 + 6 η C 3 /96 η = η C /2 + η C C. Van den Broeck, Phys Rev Lett 95, 190602 (2005) ˙ η = W W Q = − FJ 2 = − Δ T T X 2 J 2 = − η C X 2 J 2 Q = ˙ Thermal X 1 = Δ T/T 2 J 1 X 1 J 1 X 1 J 1 strong Heat flux J 1 X 2 L 21 1 max − η C = 2 η C = X 1 L 11 coupling power Mechanical X 2 =F/T X 2 ( L 21 X 1 + L 22 X 2 ) max for X 2 = − L 21 max Motion J 2 ? X 1 2 L 22 power

˙ W Q = η ( T 1 suppose efficiency η = ) ˙ Concatination T 0 is unchanged upon inserting heat bath property η ( T ' Q + η ( T ) ˙ T ') ˙ 1 Q ' W ' + ˙ ˙ η ( T W '' T 0 1 ) = = ˙ ˙ T 0 Q Q η ( T ' Q + η ( T Q − η ( T ' ) ˙ T ')( ˙ ) ˙ 1 Q ) T 0 T 0 = ˙ Q t = T x = T 1 1 T 0 T ' η ( t ) = η ( t / x ) + η ( x )(1 − η ( t / x )) ∀ x T 0 / T 1 < x < 1 implies: ⇒ η ( t ) = 1 − t α α = 1 Carnot α = 1/2 Curzon Ahlborn

Carnot Cycle for Brownian particle 2 /8 + 3 η C 3 /96 η = η C /2 + η C T. Schmiedl U. Seifert, EPL 81, 20003 (2008)

Thermal Engine via Kramer’s Escape 2 /8 + 7 η C 3 /96 η = η C /2 + η C Z.C. Tu, J Phys 41, 312003 (2008) classical particle W = a e − ( V − ε )/ T V ε 1 ,T 1 ε 2 ,T 2

− x 1 x 1 = ( V − ε 1 )/ T W → = a e − x 2 x 2 = ( V − ε 2 )/ T W ← = a e P = ˙ − x 2 − e − x 2 − e − x 1 )( ε 1 − ε 2 ) = aT 2 ( e − x 1 ) x 2 − (1 − η c ) x 1 [ ] W = a ( e W = a e − ( V − ε )/ T ˙ − x 2 − e − x 2 − e − x 1 )( V − ε 2 ) = a ( e − x 1 ) T 2 x 2 Q = a ( e ˙ W = 1 − (1 − η c ) x 1 η = ˙ Q x 2 ∂ P − x 1 ) = e − x 2 x 2 − (1 − η c ) x 1 − x 2 − e [ ] = 0 ⇒ ( e ∂ x 2 ∂ P − x 1 )(1 − η c ) = e − x 1 x 2 − (1 − η c ) x 1 − x 2 − e [ ] = 0 ⇒ ( e ∂ x 1 x 1 = 1 − 1 ln(1 − η c ) x 2 = 1 − 1 − η c ln(1 − η c ) η c η c ε 1 ,T 1 ε 2 ,T 2 2 2 3 ( ) ( ) ( ) + 3 η c η c + η c ≈ η c η = η c − (1 − η c )ln(1 − η c ) 2 8 96

Thermo-electric quantum dot M. Esposito K. Lindenberg C. Van den Broeck EPL 85, 60010 (2009) 2 /8 + (7 + a ) η C 3 /96 η = η C /2 + η C fermions 1 W in = af W out = a (1 − f ) f = e ( ε − µ )/ T + 1

Maser M. Esposito K. Lindenberg C. Van den Broeck PRL 102,130602 (2009) 1 bosons W abs = Γ n W emis = Γ (1 + n ) n = e h ν / T − 1

Stochastic Thermodynamics: General Proof of 1/8 M. Esposito K. Lindenberg C. Van den Broeck PRL 102,130602 (2009) 2 /8 + ... η = η C /2 + η C ε i ,N i µ 1 ,T 1 µ 2 ,T 2 ε j ,N i

Conclusion Time-reversibility in linear regime Symmetry Onsager matrix Prigogine minimum entropy production Efficiency at max power η = η c /2 Time-reversibility in nonlinear regime 2 /8 Efficiency at max power η = η c /2+ η c (strong coupling, spatial symmetry)

Strong Coupling J 1 ~ J 2 Thermal Chemical Entropic Machines

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