Massimiliano Esposito - uni-augsburg.de · Efficiencies and fluctuations in small...
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Efficiencies and fluctuations Efficiencies and fluctuations in small out-of-equilibrium devicesin small out-of-equilibrium devices
Université Libre de Bruxelles
Massimiliano Esposito
Center for Nonlinear Phenomena and Complex Systems
Dresden, March 2011
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Large fluctuations Small systems
Far from equilibrium(large surf/vol ratio)
Quantum effects(low temperatures)
Average behavior contains limited information
Higher moments or full probability distribution is needed
Linear response theories fail
Nonlinear response and nonperturbative methods
need to be developed
Traditional stochastic descriptions fail
Stochastic descriptions taking coherent effects
into account are needed
Challenges when dealing with small systemsChallenges when dealing with small systems
Stochastic thermodynamicsStochastic thermodynamics
Esposito, Harbola, Mukamel Rev. Mod. Phys. 81 1665 (2009)
Esposito, Lindenberg, Van den BroeckNJP 12 013013 (2010)
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OutlineOutline
I) Introduction to stochastic thermodynamics
A) Thermodynamics at the level of averages
B) The trajectory level and fluctuation theorems
II) Efficiency of small devices at maximum power A) Steady state devices
B) Externally driven devices
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I) Introduction to stochastic thermodynamics:I) Introduction to stochastic thermodynamics:
A) Thermodynamics at the level of averagesA) Thermodynamics at the level of averages
Esposito, Harbola, Mukamel, PRE 76 031132 (2007)
Esposito, Van den Broeck, PRE 82 011143 & 011144 (2010)
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Stochastic master equation
Reservoir at equilibrium“Local detailed balance”
Rate matrix
Very slow external driving:
Stochastic dynamics:Stochastic dynamics:
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First lawFirst law
System energy:
Work: Heat:
Heat from reservoir :
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System entropy:
Second lawSecond law
Entropy flow: “Local detailed balance”
Measures the breaking of detailed balance
Entropy production:
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the stationary distribution satisfies detailed balance (equilibrium)
If all the reservoirs have the same thermodynamic properties
If the reservoirs have different thermodynamic properties
the stationary distribution breaks detailed balance
Equilibrium and nonequilibrium steady state:Equilibrium and nonequilibrium steady state:
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For a slow transformation (reversible transformation):
Externally driven system (single reservoir):Externally driven system (single reservoir):
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Thermodynamics emerges at the level of averages from an underlying stochastic dynamics
What did we gain?
Time scales
Finite time thermodynamics
Fluctuations
Fluctuation theoremsEfficiency at maximum power
Optimal work extraction
Trajectory level thermodynamics
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I) Introduction to stochastic thermodynamics:I) Introduction to stochastic thermodynamics:
B) The trajectory level and Fluctuation TheoremsB) The trajectory level and Fluctuation Theorems
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First Law:
Second Law:
Thermodynamics at the trajectory level:Thermodynamics at the trajectory level:
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Fluctuation theorem:
Entropy production:
Symmetry relations betweennonlinear coefficients
Fluctuation theoremsFluctuation theorems
Crooks J.Stat.Phys. 90 1481 (1998) & PRE 60 2721 (1999)
Jarzynski PRL 78 2690 (1997) Andrieux, Gaspard JSTAT P01011 (2006)Esposito, Harbola, Mukamel PRB 75 155316 (2007)
Current FT:Work FT:
Andrieux, Gaspard JSM P02006
General formulation: Esposito, Van den Broeck, PRL 104 090601 (2010)
Seifert PRL 95 040602 (2005)
Review: Esposito, Harbola, Mukamel, Rev. Mod. Phys. 81 1665 (2009)
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II) Efficiency at maximum powerII) Efficiency at maximum power
A) Steady state devicesA) Steady state devices
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Efficiency of heat-matter transportEfficiency of heat-matter transport
Heat:
Power:
Steady state:
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Efficiency:
Tight coupling:
Fluxes:
Forces:
Entropy production:
constantCurrent independent
Collapse of forces
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Van den Broeck, PRL 95, 190602 (2005)
Esposito, Lindenberg, Van den Broeck, PRL 102, 130602 (2009)
I. I. Novikov & P. Chambadal (1957) ; F. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975).
Linear Nonlinear(in presence of a
left-right symmetry)
Efficiency at maximum power:
Efficiency at equilibrium: Carnot efficiency
Seifert, PRL 106, 020601 (2011)
zero power!But
Not too far from equilibrium:
Far from equilibrium: Everything is possible
tight coupling
Phenomenologically:
tight coupling
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Photoelectric nanocellPhotoelectric nanocellRutten, Esposito, Cleuren, PRB 80, 235122 (2009)
Tight coupling No tight coupling
Maximum power vs
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II) Efficiency at maximum powerII) Efficiency at maximum power
B) Externally driven devicesB) Externally driven devices
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Curzon-Ahlborn in the symmetric dissipation:
Esposito, Kawai, Lindenberg, Van den Broeck, PRL 105 150603 (2010)
Expansion in :
Finite-time Carnot cycleFinite-time Carnot cycleWeak
Dissipationregime
Maximize power vs and :
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is the dot occupation probability
Optimal work extractionOptimal work extraction(driven quantum dot)(driven quantum dot)
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Esposito, Kawai, Lindenberg, Van den Broeck ,EPL 89, 20003 (2010).
The optimal protocol contains an initial and final jump!
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Efficiency of the finite-time Carnot cycleEfficiency of the finite-time Carnot cycle
Esposito, Kawai, Lindenberg, Van den Broeck, Phys. Rev. E 81, 041106 (2010).
In the low dissipation regimeefficiency at maximum power:
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ConclusionsConclusions
● It can be used to study finite time thermodynamics
- Efficiencies: Tight coupling and Curzon-Ahlborn efficiency
- Optimal work extraction: jumps in the protocol
● It can be used to study fluctuations around average behaviors
- Fluctuation theorems, ...
● Further directions: quantum effects, feedbacks and information...
● Stochastic thermodynamics is a natural and powerful tool to study small systems:
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Collaborators:Collaborators:
Christian Van Den Broeck, University of Hasselt, Belgium.
Katja Lindenberg, University of California San Diego, USA.
Shaul Mukamel, University of California Irvine, USA.
Bart Cleuren & Bob Cleuren, University of Hasselt, Belgium.
Ryoichi Kawai, University of Alabama at Birmingham, USA.