Introduction to Stochastic Thermodynamics: Application to Thermo- and Photo-electricity in small devices

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1 Université Libre de Bruxelles Center for Nonlinear Phenomena and Complex Systems Introduction to Stochastic Thermodynamics: Application to Thermo- and Photo-electricity in small devices Massimiliano Esposito Hasselt, October 2011

2 Collaborators: Bart Cleuren & Bob Cleuren, University of Hasselt, Belgium. Ryoichi Kawai, University of Alabama at Birmingham, USA. Katja Lindenberg, University of California San Diego, USA. Shaul Mukamel, University of California Irvine, USA. Christian Van Den Broeck, University of Hasselt, Belgium.

3 Challenges when dealing with small systems Small systems Quantum effects Large fluctuations (low temperatures) Far from equilibrium (large surf/vol ratio) Average behavior contains limited information Higher moments or full probability distribution is needed Traditional stochastic descriptions fail Linear response theories fail Nonlinear response and nonperturbative methods need to be developed Stochastic thermodynamics Stochastic descriptions taking coherent effects into account are needed Esposito, Harbola, Mukamel Rev. Mod. Phys (2009) Esposito, Lindenberg, Van den Broeck NJP (2010)

4 Outline I) Introduction to stochastic thermodynamics A) From stochastic dynamics to nonequilibrium thermodynamics (at the level of averages) B) The trajectory level and fluctuation theorems II) Efficiency at maximum power A) Steady state (or autonomous) devices B) Externally driven (or nonautonomous) devices

5 I) Introduction to stochastic thermodynamics: A) From stochastic dynamics to nonequilibrium thermodynamics (at the level of averages) Esposito, Harbola, Mukamel, PRE (2007) Esposito, Van den Broeck, PRE & (2010)

6 Stochastic dynamics of closed systems (i.e. energy but no particle exchange): Time-dependent rate matrix 1) Markovian stochastic master equation External driving force System states 2) Reservoirs Each reservoir is always at equilibrium Local Detailed Balance No external driving: Very slow external driving: Steady state:

7 First law of thermodynamics System energy: Work: Heat from reservoir Heat: :

8 Second law of thermodynamics System entropy: Entropy flow: Local detailed balance Entropy production: Measures the breaking of detailed balance

9 Autonomous system (i.e. no external driving): If all the reservoirs have the same thermodynamic properties the stationary distribution satisfies detailed balance (i.e. equilibrium) Equilibrium Steady State If the reservoirs have different thermodynamic properties the stationary distribution breaks detailed balance Nonequilibrium Steady State

10 Nonautonomous system (i.e. externally driven) in contact with a single reservoir: For a fast transformation (irreversible transformation): Clausius inequality For a slow transformation (reversible transformation): Integrated heat becomes a state function

11 Thermodynamics emerges at the level of averages from an underlying stochastic dynamics What did we gain? Time scales Fluctuations Finite time thermodynamics Trajectory level thermodynamics Efficiency at maximum power Optimal work extraction Fluctuation theorems

12 I) Introduction to stochastic thermodynamics: B) The trajectory level and Fluctuation Theorems

13 Thermodynamics at the trajectory level: represents a single trajectory (i.e. an history) First Law: Second Law: =0 if detail balance is satisfied

Fluctuation theorems Entropy production: Seifert PRL 95 040602

nonlinear coefficients Andrieux, Gaspard JSM P02006 Current

Mukamel PRB 75 155316 (2007) General formulation: Esposito,

14 Fluctuation theorems Entropy production: Seifert PRL (2005) Fluctuation theorem: Work FT: Jarzynski PRL (1997) Crooks J.Stat.Phys (1998) & PRE (1999) Symmetry relations between nonlinear coefficients Andrieux, Gaspard JSM P02006 Current FT: Andrieux, Gaspard JSTAT P01011 (2006) Esposito, Harbola, Mukamel PRB (2007) General formulation: Esposito, Van den Broeck, PRL (2010) Review: Esposito, Harbola, Mukamel, Rev. Mod. Phys (2009)

15 II) Efficiency at maximum power A) Steady state (or autonomous) devices (i.e. no external time-dependent driving)

16 Open systems: energy and matter echange different reservoirs Local Detailed Balance Entropy flow Heat: Energy current Matter current

17 At steady state Energy and matter conservation at steady state: Power: Entropy conservation at steady state Entropy and matter conservation at steady state Entropy production: Fluxes: Forces:

18 Efficiency of thermal machines Work extracted from the system Thermal machines: Heat extracted from the hot reservoir where Efficiency: Carnot efficiency since only at equilibrium: Generically (but not always!) this will require: Therefore and

Independent efficiency is reached at equilibrium But

19 Tight coupling Constant (independent of the thermodynamic forces) Collapse of forces Current Independent efficiency is reached at equilibrium But we still have that and leads to finite value of because at equilibrium

20 Efficiency at maximum power in the linear regime Linear response: Strong coupling case Maximum power vs leads to Strong coupling case

21 Efficiency at maximum power beyond linear regime Phenomenologically: I. I. Novikov & P. Chambadal (1957) ; F. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975). Not too far from equilibrium: tight coupling Linear Van den Broeck, PRL 95, (2005) Nonlinear (in presence of a left-right symmetry) Esposito, Lindenberg, Van den Broeck, PRL 102, (2009) Far from equilibrium: Study of model systems Everything is possible Seifert, PRL 106, (2011)

22 Application: Thermoelectric single level quantum dot M.E., Lindenberg, Van den Broeck, EPL 85, (2009) Two states: Fermi distribution 1: empty electronic level 2: filled electronic level

23 Details of the calculation Change of variable: Maximum power:

24 One can prove perturbatively:

25 Application: Photoelectric nanocell Rutten, Esposito, Cleuren, PRB 80, (2009) 0 : l and r empty Three states r : r filled and l empty (two electronic levels l & r) l : r empty and l filled Fermi distribution Bose distribution

26 Steady state currents: Entropy production: Forces: Power: Efficiency:

27 Maximum power vs Tight coupling No tight coupling

28 II) Efficiency at maximum power B) Externally driven (or nonautonomous) devices

29 Finite time Carnot cycle Efficiency: Carnot efficiency for reversible cycle Power:

30 Assumptions: - cycle with a proper reversible limit - weak dissipation - instantaneous adiabatic step Extracted power: Maximize power vs and :

31 M.E., Kawai, Lindenberg, Van den Broeck, PRL (2010) Curzon-Ahlborn in the symmetric dissipation: Expansion in :

33 Application: Driven quantum dot is the dot occupation probability

34 Optimal work extraction M.E., Kawai, Lindenberg, Van den Broeck, EPL 89, (2010). The optimal protocol contains an initial and final jump!

35 Efficiency of the finite-time Carnot cycle M.E., Kawai, Lindenberg, Van den Broeck, Phys. Rev. E 81, (2010). In the low dissipation regime efficiency at maximum power:

36 Conclusions Stochastic thermodynamics is a natural and powerful tool to study small systems: 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...

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