Introduction to Stochastic Thermodynamics: Introduction to Stochastic Thermodynamics: Application to Thermo- and Photo-electricity Application to Thermo- and Photo-electricity

in small devicesin small devices

Université Libre de Bruxelles

Massimiliano Esposito

Center for Nonlinear Phenomena and Complex Systems

Hasselt, October 2011

Collaborators: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.

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)

OutlineOutline

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

I) Introduction to stochastic thermodynamics:I) Introduction to stochastic thermodynamics:

A) From stochastic dynamics to nonequilibrium A) From stochastic dynamics to nonequilibrium thermodynamics (at the level of averages)thermodynamics (at the level of averages)

Esposito, Harbola, Mukamel, PRE 76 031132 (2007)

Esposito, Van den Broeck, PRE 82 011143 & 011144 (2010)

Markovian stochastic master equation

Each reservoir is always at equilibrium

“Local Detailed Balance”

Time-dependent rate matrix

Very slow external driving:

Stochastic dynamics of closed systemsStochastic dynamics of closed systems(i.e. energy but no particle exchange):

External driving force

ReservoirsSystem states

Steady state: No external driving:

1)

2)

First law of thermodynamicsFirst law of thermodynamics

System energy:

Work: Heat:

Heat from reservoir :

System entropy:

Second law of thermodynamicsSecond law of thermodynamics

Entropy flow: “Local detailed balance”

Measures the breaking of detailed balance

Entropy production:

the stationary distribution satisfies detailed balance (i.e. equilibrium)

If all the reservoirs have the same thermodynamic properties

If the reservoirs have different thermodynamic properties

the stationary distribution breaks detailed balance

Autonomous system (i.e. no external driving):Autonomous system (i.e. no external driving):

Equilibrium Steady State

Nonequilibrium Steady State

For a slow transformation (reversible transformation):

Nonautonomous system (i.e. externally driven)Nonautonomous system (i.e. externally driven)in contact with a single reservoir:in contact with a single reservoir:

For a fast transformation (irreversible transformation):

Clausius inequality

Integrated heat becomes a state function

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

I) Introduction to stochastic thermodynamics:I) Introduction to stochastic thermodynamics:

B) The trajectory level and Fluctuation TheoremsB) The trajectory level and Fluctuation Theorems

First Law:

Second Law:

Thermodynamics at the trajectory level:Thermodynamics at the trajectory level:

represents a singletrajectory (i.e. an history)

=0 if detail balance is satisfied

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)

II) Efficiency at maximum powerII) Efficiency at maximum power

A) Steady state (or autonomous) devicesA) Steady state (or autonomous) devices

(i.e. no external time-dependent driving)

Open systems: energy and matter echangeOpen systems: energy and matter echange

Heat:

Local Detailed Balance

Entropy flow

Energy current Matter current

different reservoirs

At steady stateAt steady state

Power:

Energy and matter conservation at steady state:

Entropy conservation at steady state

Fluxes: Forces:

Entropy production:

Entropy and matter conservation at steady state

Efficiency:

Work extracted from the system

Heat extracted from the hot reservoirThermal machines:

Efficiency of thermal machinesEfficiency of thermal machines

Carnot efficiency

where

since

only at equilibrium:

Therefore and

Generically (but not always!) this will require:

Constant (independent of the thermodynamic forces)

Current Independent

efficiency

Collapse of forces

Tight couplingTight coupling

is reached at equilibrium and leads to finite value of

But we still have that because at equilibrium

Linear response:

Maximum power vs leads to

Efficiency at maximum power in the linear regimeEfficiency at maximum power in the linear regime

Strong couplingcase

Strong couplingcase

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)

Seifert, PRL 106, 020601 (2011)

Not too far from equilibrium:

Far from equilibrium:

Phenomenologically:

tight coupling

Efficiency at maximum power beyond linear regimeEfficiency at maximum power beyond linear regime

Study of model systems Everything is possible

Application: Thermoelectric single level quantum dotApplication: Thermoelectric single level quantum dotM.E., Lindenberg, Van den Broeck, EPL 85, 60010 (2009)

Fermi distribution

1: empty electronic level2: filled electronic level

Two states:

Maximum power:

Change of variable:

Details of the calculationDetails of the calculation

One can prove perturbatively:

Application: Photoelectric nanocellApplication: Photoelectric nanocellRutten, Esposito, Cleuren, PRB 80, 235122 (2009)

Fermi distribution

Bose distribution

Three states(two electronic levels l & r)

0 : l and r emptyr : r filled and l empty l : r empty and l filled

Steady state currents:

Power:

Efficiency:

Forces:

Entropy production:

Maximum power vs

Tight coupling

No tight coupling

II) Efficiency at maximum powerII) Efficiency at maximum power

B) Externally driven (or nonautonomous) devicesB) Externally driven (or nonautonomous) devices

Finite time Carnot cycleFinite time Carnot cycle

for reversible cycle

Carnot efficiencyEfficiency:

Power:

Maximize power vs and :

Extracted power:

Assumptions:

- cycle with a proper reversible limit - weak dissipation

- instantaneous adiabatic step

Curzon-Ahlborn in the symmetric dissipation:

M.E., Kawai, Lindenberg, Van den Broeck, PRL 105 150603 (2010)

Expansion in :

is the dot occupation probability

Application: Driven quantum dotApplication: Driven quantum dot

M.E., Kawai, Lindenberg, Van den Broeck ,EPL 89, 20003 (2010).

The optimal protocol contains an initial and final jump!

Optimal work extractionOptimal work extraction

Efficiency of the finite-time Carnot cycleEfficiency of the finite-time Carnot cycle

M.E., Kawai, Lindenberg, Van den Broeck, Phys. Rev. E 81, 041106 (2010).

In the low dissipation regimeefficiency at maximum power:

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: