SLIDE 1
Patricia Edwins
National Science Foundation UCSD funds: Kurt Shuler, Misha Galperin, Katja Lindenberg
- - PowerPoint PPT Presentation
- - PowerPoint PPT Presentation
Hiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii National Science Foundation UCSD funds: Kurt Shuler, Misha Galperin, Katja Lindenberg Patricia Edwins Chemical Austria C. Dellago Belgium Flemish C. Van den Broeck
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SLIDE 3
- Chemical
- Physical
- Bio-physical
Austria C. Dellago Belgium Flemish C. Van den Broeck Belgium Walloon P. Gaspard Czech Republic
- E. Hulicius
Finland
- J. Pekola
Germany
- U. Seifert
Ireland
- J. Gleeson
Norway A. Hansen Spain JMR Parrondo Sweden Hongqi Xu Switzerland MO Hongler
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Tuesday night 6:30 El Torito (Mexican Restaurant) UCSD campus Marriott Residence Inn
200 m
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T E S T T E S T
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Christian Van den Broeck Universiteit Hasselt christian.vandenbroeck@uhasselt.be
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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.
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W Qh ≤ ηc ηc =1 − Tc Th
Th Qh Tc Qc
Equality Sign: Reversible Process
W = Qh − Qc ΔS = Qh Th + Qc Tc ≥ 0 ΔS = dQ T
rev
∫
≥ 0
Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärme selbst ableiten lassen Réflexions sur la puissance motrice du feu et sur les machines propres a developper cette puissance.
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η = W Qh ≤ 1 − Tc Th
Exceed Carnot at Small Scale? Erase 1 bit: ΔS= kB ln2 Yes? No! Tc=Th
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η = W Qh ≤ 1 − Tc Th
Below Carnot at Small Scale?
Under Steady State Conditions?
- J. Parrondo P. Espagnol, Am J Phys 64, 1125 (1996)
- C. Van den Broeck, R. Kawai, and P. Meurs,
PRL.93, 090601 (2004)
Irreversible Heat Flux
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Reversible: diS/dt=0 hence η=ηc. If J1= J2 =0 for X1 and X2 nonzero. Only possible if determinant of matrix L is zero: L11L22=(L12)2
Carnot efficiency
Architectural constraint of strong coupling J2/J1= L21/L11 .
Carnot efficiency?
- C. Van den Broeck, Adv Chem Phys
135, 189 (2007)
Thermal X1=ΔT/T2 Mechanical X2=F/T Heat flux J1 Rotation Speed J2
L11L22=(L12)2
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FL Curzon B Ahlborn
- Am. J. Phys. 43, 22 (1975)
η = ηC /2 + ηC
2 /8 + 6ηC 3 /96
Efficiency at maximum power
η = W Qh =
approximation endoreversible
1 − Tc Th
Exact in linear approx. for : L11L22=(L12)2
- C. Van den Broeck, Phys Rev Lett 95,
190602 (2005)
1−ηC = Tc Th
Thermal X1=ΔT/T2 Heat flux J1 Mechanical X2=F/T Motion J2
η = W Q = ˙ W ˙ Q = − FJ2 J1 = − ΔT T X2J2 X1J1 = −ηC X2J2 X1J1 =
coupling strong
−ηC X2L21 X1L11 =
power max
1 2 ηC ?
power max
X2(L21X1 + L22X2) max for X2 X1 = − L21 2L22
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suppose efficiency η = ˙ W ˙ Q = η(T
1
T0 ) is unchanged upon inserting heat bath η(T
1
T0 ) = ˙ W '+ ˙ W '' ˙ Q = η(T' T0 ) ˙ Q + η(T
1
T') ˙ Q ' ˙ Q = η(T' T0 ) ˙ Q + η(T
1
T')( ˙ Q −η(T' T0 ) ˙ Q ) ˙ Q t = T
1
T0 x = T
1
T' η(t) = η(t /x) + η(x)(1−η(t /x)) ∀x T0 /T
1 < x <1
⇒ η(t) =1− tα α =1 Carnot α =1/2 Curzon Ahlborn
Concatination property implies:
SLIDE 15
η = ηC /2 + ηC
2 /8 + 3ηC 3 /96
Carnot Cycle for Brownian particle
- T. Schmiedl U. Seifert, EPL 81, 20003 (2008)
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η = ηC /2 + ηC
2 /8 + 7ηC 3 /96
Thermal Engine via Kramer’s Escape
Z.C. Tu, J Phys 41, 312003 (2008)
classical particle
ε1,T1 ε2,T2
W = a e−(V −ε )/T
V
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ε1,T1 ε2,T2
W→ = a e
−x1 x1 = (V −ε1)/T
W← = a e
−x2 x2 = (V −ε2)/T
P = ˙ W = a(e
−x2 −e −x1)(ε1 −ε2) = aT2(e −x2 −e −x1) x2 −(1−ηc)x1
[ ]
˙ Q = a(e
−x2 −e −x1)(V −ε2) = a(e −x2 −e −x1)T2x2
η = ˙ W ˙ Q =1−(1−ηc) x1 x2 ∂P ∂x2 = 0 ⇒ (e
−x2 −e −x1 ) = e −x2 x2 −(1−ηc)x1
[ ]
∂P ∂x1 = 0 ⇒ (e
−x2 −e −x1 )(1−ηc) = e −x1 x2 −(1−ηc)x1
[ ]
x1 =1− 1 ηc ln(1−ηc) x2 = 1−1−ηc ηc ln(1−ηc) η = ηc
( )
2
ηc −(1−ηc)ln(1−ηc) ≈ ηc 2 + ηc
( )
2
8 + 3 ηc
( )
3
96
W = a e−(V −ε )/T
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η = ηC /2 + ηC
2 /8 + (7 + a)ηC 3 /96
- M. Esposito K. Lindenberg C. Van den Broeck
EPL 85, 60010 (2009)
Thermo-electric quantum dot fermions
Win = af Wout = a(1− f ) f = 1 e(ε−µ)/T +1
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- M. Esposito K. Lindenberg C. Van den Broeck
PRL 102,130602 (2009)
Maser
bosons
Wabs = Γn Wemis = Γ(1+ n) n = 1 ehν /T −1
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Stochastic Thermodynamics: General Proof of 1/8
- M. Esposito K. Lindenberg C. Van den Broeck PRL 102,130602 (2009)
µ1,T1 µ2,T2 εi,Ni εj,Ni
η = ηC /2 + ηC
2 /8 + ...
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Conclusion Time-reversibility in linear regime Symmetry Onsager matrix Prigogine minimum entropy production Efficiency at max power η=ηc/2 Time-reversibility in nonlinear regime Efficiency at max power η=ηc/2+ηc
2/8
(strong coupling, spatial symmetry)
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Strong Coupling J1~ J2 Thermal Chemical Entropic Machines
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