Introduction to Nonequilibrium Thermodynamics: From ...sophia.dtp.fmph.uniba.sk/~tatry/text12/Korbel.pdf · Introduction to Nonequilibrium Thermodynamics: From Onsager to Micromotors

Introduction to Nonequilibrium Thermodynamics:From Onsager to Micromotors

Based on the lecture “Nonequilibrium phenomena in micro and nanosystems” taught at Freie Universität Berlin

Jan Korbel

Faculty of Nuclear Sciences and Physical Engineering, CTU, Prague6th Student Colloquium and School on Mathematical Physics,Stará Lesná

25. 8. 2012

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Outline

� History & Motivation� Introduction to nonequilibrium thermodynamics� Application: Brownian motors� Recent developments in nonequilibrium TD

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History & Motivation

� Theory of nonequilibrium thermodynamics originates from thefirst half of 20. century

� It was mainly developed by Onsager, Rayleigh...� Aim: to extend a formalism of equilibrium processes to

dissipative or fast processes� Many processes observed in real system exhibit behavior of

irreversible processes� Applications: biophysics, nanosystems,...

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Equilibrium thermodynamicsBasic notes

� Description of macroscopic systems� Small fluctuations can be neglected

∆E< E >

'√

NN' 1√

N(1)

� Equilibrium: state of a system, where we cannot observe anychange of measurable quantities

� Structure of Thermodynamics:

� General laws� System-specific response coefficients: cp, cv , βT , . . .

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∆E< E >

'√

NN' 1√

N(1)




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∆E< E >

'√

NN' 1√

N(1)




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∆E< E >

'√

NN' 1√

N(1)


� Structure of Thermodynamics:� General laws� System-specific response coefficients: cp, cv , βT , . . .

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Equilibrium thermodynamicsLaws of thermodynamics

� First law (Claussius 1850, Helmholtz 1847): Energy isconserved.

dU = δQ − δW (2)

� Second law (Carnot 1824, Claussius 1854, Kelvin): Heatcannot be fully transformed into work.

dS ≥ δQT

(3)

� Third law: We cannot bring the system into the absolute zerotemperature in a finite number of steps.

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Equilibrium thermodynamicsLaws of thermodynamics

� First law (Claussius 1850, Helmholtz 1847): Energy isconserved.

dU = δQ − δW (2)

� Second law (Carnot 1824, Claussius 1854, Kelvin): Heatcannot be fully transformed into work.

dS ≥ δQT

(3)

� Third law: We cannot bring the system into the absolute zerotemperature in a finite number of steps.

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Nonequilibrium thermodynamics

� For quasistatic reversible process we have

dSR =dUT

+∑

i

YidXi

(Yi =

∂SR

∂Xi

)(4)

� From the second law we know that ∆SR = QT

� For irreversible process we get an extra entropy ∆S = QT + ∆Si

where ∆Si > 0� Entropy production rate:

dSdt

=∑

i

∂S∂Xi

∂Xi

∂t(5)

� aim of Nonequilibrium TD: to compute entropy production rate

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dSR =dUT

+∑

i

YidXi

(Yi =

∂SR

∂Xi

)(4)




dSdt

=∑

i

∂S∂Xi

∂Xi

∂t(5)


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dSR =dUT

+∑

i

YidXi

(Yi =

∂SR

∂Xi

)(4)



where ∆Si > 0

� Entropy production rate:

dSdt

=∑

i

∂S∂Xi

∂Xi

∂t(5)


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dSR =dUT

+∑

i

YidXi

(Yi =

∂SR

∂Xi

)(4)




dSdt

=∑

i

∂S∂Xi

∂Xi

∂t(5)


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dSR =dUT

+∑

i

YidXi

(Yi =

∂SR

∂Xi

)(4)




dSdt

=∑

i

∂S∂Xi

∂Xi

∂t(5)

� aim of Nonequilibrium TD: to compute entropy production rate6 / 23

Nonequilibrium thermodynamicsLinear thermodynamics

� There exists no unified theory of nonequilibriumthermodynamics.

� Near equilibrium exists a linear theory that is universal.� Let us consider a system which we divide into small subsystems.

We assume that every system is in local equilibirium

� Total entropy is: S = Sa(X ai ) + Sb(X b

i ) + . . .� Entropy production rate for a subsystem a:

σa =dSa

dt=∑

i

Y ai X a

i =∑

i

Y ai Ja

i (6)

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σa =dSa

dt=∑

i

Y ai X a

i =∑

i

Y ai Ja

i (6)

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� Near equilibrium exists a linear theory that is universal.

� Let us consider a system which we divide into small subsystems.We assume that every system is in local equilibirium



σa =dSa

dt=∑

i

Y ai X a

i =∑

i

Y ai Ja

i (6)

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σa =dSa

dt=∑

i

Y ai X a

i =∑

i

Y ai Ja

i (6)

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σa =dSa

dt=∑

i

Y ai X a

i =∑

i

Y ai Ja

i (6)

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Nonequilibrium thermodynamicsCurrent and Affinity

� Jai is generalized current, at equilibrium Ja

i = 0

� Γabi := Y a

i − Y bi is affinity

� Affinity - deviation from equilibrium TD force� A system brought from equilibrium reacts by creating a current

Ji =∑

j

LijΓj (7)

� Lij nonequilibrium response coefficients� Generally are Lij functions of Γ’s, but near equilibrium are assumed to

be constants - Ji ’s are linear functions of Γ’s

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i = 0� Γab

i := Y ai − Y b

i is affinity� Affinity - deviation from equilibrium TD force

� A system brought from equilibrium reacts by creating a current

Ji =∑

j

LijΓj (7)



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i = 0� Γab

i := Y ai − Y b

i is affinity� Affinity - deviation from equilibrium TD force� A system brought from equilibrium reacts by creating a current

Ji =∑

j

LijΓj (7)

� Lij nonequilibrium response coefficients

� Generally are Lij functions of Γ’s, but near equilibrium are assumed tobe constants - Ji ’s are linear functions of Γ’s

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i = 0� Γab

i := Y ai − Y b

i is affinity� Affinity - deviation from equilibrium TD force� A system brought from equilibrium reacts by creating a current

Ji =∑

j

LijΓj (7)



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Nonequilibrium thermodynamicsOnsager relations

� We can rewrite entropy production as

σ =∑

i

JiΓi =∑

ij

LijΓiΓj (8)

� From the second law: dSdt ≥ 0, which implies det L ≥ 0, Lii ≥ 0

� In case of two currents we get L11L22 − L21L12 ≥ 0Onsager relations (L. Onsager, Nobel prize 1968):The matrix L is symmetric, i.e. Lij = Lji

� For two currents: L212 ≤ L11L22

� It says more than second law of TD: if L = LS + LA, then

σ =∑

ij

LijΓiΓj =∑

ij

(LS

ij + LAij

)ΓiΓj =

∑ij

LSij ΓiΓj ≥ 0. (9)

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σ =∑

i

JiΓi =∑

ij

LijΓiΓj (8)





σ =∑

ij

LijΓiΓj =∑

ij

(LS

ij + LAij

)ΓiΓj =

∑ij


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σ =∑

i

JiΓi =∑

ij

LijΓiΓj (8)


� In case of two currents we get L11L22 − L21L12 ≥ 0

Onsager relations (L. Onsager, Nobel prize 1968):The matrix L is symmetric, i.e. Lij = Lji



σ =∑

ij

LijΓiΓj =∑

ij

(LS

ij + LAij

)ΓiΓj =

∑ij


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σ =∑

i

JiΓi =∑

ij

LijΓiΓj (8)





σ =∑

ij

LijΓiΓj =∑

ij

(LS

ij + LAij

)ΓiΓj =

∑ij


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σ =∑

i

JiΓi =∑

ij

LijΓiΓj (8)





σ =∑

ij

LijΓiΓj =∑

ij

(LS

ij + LAij

)ΓiΓj =

∑ij


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Application: Brownian motorsMicrosystems

� In nonequilibrium TD fluctuations cannot be neglected� Laws are the same, but importance of quantities is different

� Volume scales as L3 - inertial forces, weight,...� Surface scales as L2 - friction, heat transfer,...� friction

inertia ∼1L - for small systems become friction forces important

� For microsystems is the thermalization time very small - instantthermalization

� Macromotor: based on inertia and temperature difference� Micromotor: based on random fluctuations

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� Volume scales as L3 - inertial forces, weight,...� Surface scales as L2 - friction, heat transfer,...

� frictioninertia ∼

1L - for small systems become friction forces important



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Brownian motorsTransport in living cells

� In living cells we can observe a few types of transportmechanisms

� One is transport of kinesin protein with cargo on the actinfilament

� We can see a directed “walking” of kinesin on the filament� The mechanism is based on nonequilibrium fluctuations -

Brownian motors

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Brownian motorsRatchets

Question: Can exist an engine that exploits random fluctuations inorder to produce some work?

� In equilibrium: No. (fluctuations are neglected)� Out of equilibrium: Yes!� In order to get some useful work we use spatial and temporal

asymmetry (ratchet effect)

� Flashing (on-off) ratchet� Rocking ratchet� Correlation ratchet (based on the disruption of

fluctuation-dissipation theorem)� Chemical ratchet

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Question: Can exist an engine that exploits random fluctuations inorder to produce some work?� In equilibrium: No. (fluctuations are neglected)

� Out of equilibrium: Yes!� In order to get some useful work we use spatial and temporal




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Question: Can exist an engine that exploits random fluctuations inorder to produce some work?� In equilibrium: No. (fluctuations are neglected)� Out of equilibrium: Yes!

� In order to get some useful work we use spatial and temporalasymmetry (ratchet effect)



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Question: Can exist an engine that exploits random fluctuations inorder to produce some work?� In equilibrium: No. (fluctuations are neglected)� Out of equilibrium: Yes!� In order to get some useful work we use spatial and temporal




12 / 23


Question: Can exist an engine that exploits random fluctuations inorder to produce some work?� In equilibrium: No. (fluctuations are neglected)� Out of equilibrium: Yes!� In order to get some useful work we use spatial and temporal

asymmetry (ratchet effect)Types of ratchets� Flashing (on-off) ratchet� Rocking ratchet� Correlation ratchet (based on the disruption of

fluctuation-dissipation theorem)� Chemical ratchet12 / 23

Brownian motorsFlashing ratchet

� The transport is based on switching on and off of an periodic,asymmetric potential

� Examples of potentials: asymmetric sawtooth,V (x) = sin(x) + 1

4 sin(2x + π

4

)� When the potential is off - diffusion: p(x , t) ' exp

(−x2

2Dt

)� When the potential is on - particles tend to get to minimums -

localization: p(x) ' exp(−βV (x))

� Because the potential is periodic, no force is present onaverage

� We can observe a particle flow

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4 sin(2x + π

4

)

� When the potential is off - diffusion: p(x , t) ' exp(−x2

2Dt





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4 sin(2x + π

4


(−x2

2Dt





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4 sin(2x + π

4


(−x2

2Dt





13 / 23


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Brownian motorsRocking ratchet

� We use again the asymmetric potential, but instead of switchingon and off, we tilt the potential a little bit:

V (x , t) = V0(x) + c1x sin(c2t) (10)

� Again, due to asymmetry of the potential is the currentproduced with zero average force.

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Brownian motorsRocking ratchet

� We use again the asymmetric potential, but instead of switchingon and off, we tilt the potential a little bit:

V (x , t) = V0(x) + c1x sin(c2t) (10)

� Again, due to asymmetry of the potential is the currentproduced with zero average force.

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Brownian motorsChemical ratchet

� Motivated by biological background, another possibility how toforce the particle to diffuse, is to give it some additional energy,so it can get from the minimum of the potential,

� For that we use a chemical reaction

ATP ADP + P (11)

� Chemical ratchet is kind of flashing ratchet, where the energy toswitch of the potential is from the reaction of ATP

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ATP ADP + P (11)


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ATP ADP + P (11)


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ATP ADP + P (11)


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Brownian motorsEfficiency of a Chemical ratchet

� Efficiency is defined as a ratio between the performed work andconsumed energy

η = −WQ

= −WQ

(12)

� We define the chemical force, which is nothing else thandifference between chemical potentials, ∆µ = µL − µR. Theconsumed energy per unit time is Q = r∆µ, where r ischemical reaction rate.

� Similarly we obtain the performed work per unit time, which isW = fextv , where fext is a external force and v is the velocity ofparticles. The efficiency is then

η = − fextvr∆µ

(13)

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η = −WQ

= −WQ

(12)




(13)

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η = −WQ

= −WQ

(12)




(13)

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η = −WQ

= −WQ

(12)




(13)

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� Near to equilibrium we can consider a linear thermodynamics, whichmeans that currents are linear functions of forces

v = L11fext + L12∆µ

r = L21fext + L22∆µ

� The efficiency for linear regime has the form

η = −L11a2 + L12aL21a + L22

(14)

where a = fext/∆µ.� The maximal efficiency is given by the relation ∂η

∂a = 0 and the

maximal value is in terms of Λ =L2

12L11L22

:

ηmax =1−√

1− Λ2

Λ(15)

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η = −L11a2 + L12aL21a + L22

(14)

where a = fext/∆µ.

� The maximal efficiency is given by the relation ∂η∂a = 0 and the


12L11L22

:

ηmax =1−√

1− Λ2

Λ(15)

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η = −L11a2 + L12aL21a + L22

(14)

where a = fext/∆µ.� The maximal efficiency is given by the relation ∂η

∂a = 0 and the


12L11L22

:

ηmax =1−√

1− Λ2

Λ(15)

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� The maximal efficiency we get for L212 = L11L22 which means

maximal permissible coupling of currents from second law ofthermodynamics, the efficiency is therefore η = 1!

� In comparison to macromotors, where the efficiency is limitedby η ≤ 1− Tc

Th, here is no restriction to maximal efficiency and

micromotors have usually much higher efficiency thanmacromotors.

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Recent developments of nonequilibrium TDFluctuation theorem

� The second law of TD tells us, that entropy production is alwaysnon-negative

� The second law is nevertheless a statistical statement whichholds only in thermodynamical limit

� For small systems driven out of equilibrium we can expectsome entropy fluctuations that can be may also negative

� The quantification gives us Fluctuation theorem (Evans, Cohen,Morris, 1993)

P(Σt = A)

P(Σt = −A)= exp(At) (16)

where Σt is time-averaged irreversible entropy production.

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P(Σt = A)

P(Σt = −A)= exp(At) (16)


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P(Σt = A)

P(Σt = −A)= exp(At) (16)


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P(Σt = A)

P(Σt = −A)= exp(At) (16)


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� With an increasing time or size of the system, negativefluctuations are exponentially supresed. But for small scalesand time intervals can negative fluctuations be observed (andalready have been measured)

� The importance of the theorem is in the fact that FT is valid forall systems arbitrarly far from equilibrium

� A corollary of FT is Second law inequality that says⟨Σt⟩≥ 0 ∀t , (17)

so ensemble average of entropy production is always positive

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Recent developments of nonequilibrium TDJarzynski equality

� In thermodynamics can be for quasistatic process derived aninequality between free energy and work

∆F ≤W (18)

� It is possible to derive a generalization of this inequality for arbitraryprocesses (not only “slow”) from the fluctuation theorem

� the relation is called Jarzynski equality (Jarzynski, 1997)

exp(−∆FkBt

)= exp

(−WkBt

)(19)

� The line indicated all possible realizations of an external process thattakes the system from equilibrium state A to equilibrium state B.States in between these points do not have to be equilibrium states.

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∆F ≤W (18)



exp(−∆FkBt

)= exp

(−WkBt

)(19)


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∆F ≤W (18)



exp(−∆FkBt

)= exp

(−WkBt

)(19)


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∆F ≤W (18)



exp(−∆FkBt

)= exp

(−WkBt

)(19)


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∆F ≤W (18)



exp(−∆FkBt

)= exp

(−WkBt

)(19)


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Robert Zwanzig.Nonequilibrium Statistical Mechanics.Oxford University Press, USA, March 2001.

P. Hänggi, F. Marchesoni, and F. Nori.Brownian motors.Annalen der Physik, 14(1-3):51–70, 2005.

Andrea Parmeggiani, Frank Jülicher, Armand Ajdari, and Jacques Prost.Energy transduction of isothermal ratchets: Generic aspects and specific examplesclose to and far from equilibrium.Phys. Rev. E, 60:2127–2140, Aug 1999.

Denis J. Evans, E. G. D. Cohen, and G. P. Morriss.Probability of second law violations in shearing steady states.Phys. Rev. Lett., 71:2401–2404, Oct 1993.

Harvard Biovisions.Molecular machinery of life: Online video.http://www.youtube.com/watch?v=FJ4N0iSeR8U, February 2011.

Thank you for attention!

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Introduction to Nonequilibrium Thermodynamics: From ...sophia.dtp.fmph.uniba.sk/~tatry/text12/Korbel.pdf · Introduction to Nonequilibrium Thermodynamics: From Onsager to Micromotors