Introduction to Nonequilibrium Thermodynamics: From Onsager to ...

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, Prague 6th 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 the first 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 thermodynamics Basic notes

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Description of macroscopic systems

Small fluctuations can be neglected √ ∆E N 1 ' '√ N N

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(1)




(1)

Equilibrium: state of a system, where we cannot observe any change of measurable quantities

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(1)

Equilibrium: state of a system, where we cannot observe any change of measurable quantities Structure of Thermodynamics:

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

Equilibrium thermodynamics Laws of thermodynamics

First law (Claussius 1850, Helmholtz 1847): Energy is conserved. dU = δQ − δW Second law (Carnot 1824, Claussius 1854, Kelvin): Heat cannot be fully transformed into work. dS ≥

(2)

δQ T

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

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(3)

Equilibrium thermodynamics Laws of thermodynamics

First law (Claussius 1850, Helmholtz 1847): Energy is conserved. dU = δQ − δW Second law (Carnot 1824, Claussius 1854, Kelvin): Heat cannot be fully transformed into work. dS ≥

(2)

δQ T

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

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(3)

Nonequilibrium thermodynamics

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For quasistatic reversible process we have dU X ∂SR dSR = + Yi dXi Yi = T ∂Xi i

From the second law we know that ∆SR =

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Q T

(4)


For quasistatic reversible process we have dU X ∂SR dSR = + Yi dXi Yi = T ∂Xi

(4)

i

Q T


For irreversible process we get an extra entropy ∆S = where ∆Si > 0

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Q T

+ ∆Si



(4)

i

Q T



Entropy production rate: dS X ∂S ∂Xi = dt ∂Xi ∂t i

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Q T

+ ∆Si

(5)



(4)

i

Q T



Entropy production rate: dS X ∂S ∂Xi = dt ∂Xi ∂t

Q T

+ ∆Si

(5)

i

aim of Nonequilibrium TD: to compute entropy production rate

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Nonequilibrium thermodynamics Linear thermodynamics

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There exists no unified theory of nonequilibrium thermodynamics.

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There exists no unified theory of nonequilibrium thermodynamics. Near equilibrium exists a linear theory that is universal.

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There exists no unified theory of nonequilibrium thermodynamics. 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

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There exists no unified theory of nonequilibrium thermodynamics. 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 = S a (Xia ) + S b (Xib ) + . . . Entropy production rate for a subsystem a: dS a X a ˙ a X a a Yi Ji = Yi Xi = σa = dt i

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i

(6)

Nonequilibrium thermodynamics Current and Affinity

Jia is generalized current, at equilibrium Jia = 0

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a b Γab i := Yi − Yi is affinity

Affinity - deviation from equilibrium TD force

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A system brought from equilibrium reacts by creating a current X Ji = Lij Γj (7) j

Lij nonequilibrium response coefficients

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A system brought from equilibrium reacts by creating a current X Ji = Lij Γj (7) j

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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Nonequilibrium thermodynamics Onsager relations

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We can rewrite entropy production as X X σ= Ji Γi = Lij Γi Γj i

From the second law:

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dS dt

(8)

ij

≥ 0, which implies det L ≥ 0, Lii ≥ 0



dS dt

(8)

ij

From the second law: ≥ 0, which implies det L ≥ 0, Lii ≥ 0 In case of two currents we get L11 L22 − L21 L12 ≥ 0

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dS dt

(8)

ij

From the second law: ≥ 0, which implies det L ≥ 0, Lii ≥ 0 In case of two currents we get L11 L22 − L21 L12 ≥ 0 Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. Lij = Lji For two currents: L212 ≤ L11 L22

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(8)

ij

dS dt

From the second law: ≥ 0, which implies det L ≥ 0, Lii ≥ 0 In case of two currents we get L11 L22 − L21 L12 ≥ 0 Onsager relations (L. Onsager, Nobel prize 1968): The matrix L is symmetric, i.e. Lij = Lji For two currents: L212 ≤ L11 L22 It says more than second law of TD: if L = LS + LA , then X X X σ= Lij Γi Γj = LSij + LAij Γi Γj = LSij Γi Γj ≥ 0. ij

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ij

ij

(9)

Application: Brownian motors Microsystems

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

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



Volume scales as L3 - inertial forces, weight,... Surface scales as L2 - friction, heat transfer,... friction 1 inertia ∼ L - for small systems become friction forces important For microsystems is the thermalization time very small - instant thermalization

Macromotor: based on inertia and temperature difference

Micromotor: based on random fluctuations

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Brownian motors Transport in living cells

In living cells we can observe a few types of transport mechanisms One is transport of kinesin protein with cargo on the actin filament 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 motors Ratchets

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

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In equilibrium: No. (fluctuations are neglected)

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Out of equilibrium: Yes!

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In order to get some useful work we use spatial and temporal asymmetry (ratchet effect)

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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 ratchet

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Brownian motors Flashing ratchet

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

Examples of potentials: asymmetric sawtooth, V (x) = sin(x) + 14 sin 2x + π4

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

When the potential is on - particles tend to get to minimums localization: p(x) ' exp(−βV (x))

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−x 2 2Dt




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

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 on average

We can observe a particle flow

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−x 2 2Dt


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Brownian motors Rocking ratchet

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Brownian motors Rocking ratchet

We use again the asymmetric potential, but instead of switching on and off, we tilt the potential a little bit: V (x, t) = V0 (x) + c1 x sin(c2 t)

Again, due to asymmetry of the potential is the current produced with zero average force.

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(10)

Brownian motors Chemical ratchet

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

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For that we use a chemical reaction ATP ADP + P

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(11)



For that we use a chemical reaction ATP ADP + P

(11)

Chemical ratchet is kind of flashing ratchet, where the energy to switch of the potential is from the reaction of ATP

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Brownian motors Efficiency of a Chemical ratchet

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Efficiency is defined as a ratio between the performed work and consumed energy ˙ W W η=− (12) =− ˙ Q Q

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Efficiency is defined as a ratio between the performed work and consumed energy ˙ W W η=− (12) =− ˙ Q Q We define the chemical force, which is nothing else than difference between chemical potentials, ∆µ = µL − µR . The ˙ = r ∆µ, where r is consumed energy per unit time is Q chemical reaction rate.

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Efficiency is defined as a ratio between the performed work and consumed energy ˙ W W η=− (12) =− ˙ Q Q We define the chemical force, which is nothing else than difference between chemical potentials, ∆µ = µL − µR . The ˙ = r ∆µ, where r is consumed energy per unit time is Q chemical reaction rate. Similarly we obtain the performed work per unit time, which is ˙ = fext v , where fext is a external force and v is the velocity of W particles. The efficiency is then fext v η=− (13) r ∆µ

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

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v

= L11 fext + L12 ∆µ

r

= L21 fext + L22 ∆µ


Near to equilibrium we can consider a linear thermodynamics, which means that currents are linear functions of forces v

= L11 fext + L12 ∆µ

r

= L21 fext + L22 ∆µ

The efficiency for linear regime has the form η=− where a = fext /∆µ.

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L11 a2 + L12 a L21 a + L22

(14)


Near to equilibrium we can consider a linear thermodynamics, which means that currents are linear functions of forces v

= L11 fext + L12 ∆µ

r

= L21 fext + L22 ∆µ

The efficiency for linear regime has the form η=−

L11 a2 + L12 a L21 a + L22

(14)

where a = fext /∆µ.

The maximal efficiency is given by the relation maximal value is in terms of Λ = ηmax =

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L212

∂η ∂a

= 0 and the

L11 L22 :

1−

√

1 − Λ2 Λ

(15)


The maximal efficiency we get for L212 = L11 L22 which means maximal permissible coupling of currents from second law of thermodynamics, the efficiency is therefore η = 1! In comparison to macromotors, where the efficiency is limited by η ≤ 1 − TThc , here is no restriction to maximal efficiency and micromotors have usually much higher efficiency than macromotors.

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Recent developments of nonequilibrium TD Fluctuation theorem

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The second law of TD tells us, that entropy production is always non-negative

The second law is nevertheless a statistical statement which holds only in thermodynamical limit

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

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

The quantification gives us Fluctuation theorem (Evans, Cohen, Morris, 1993) ¯ t = A) P(Σ (16) ¯ t = −A) = exp(At) P(Σ ¯ t is time-averaged irreversible entropy production. where Σ

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

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

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

A corollary of FT is Second law inequality that says

¯ t ≥ 0 ∀t, Σ

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so ensemble average of entropy production is always positive 21 / 23

Recent developments of nonequilibrium TD Jarzynski equality

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In thermodynamics can be for quasistatic process derived an inequality between free energy and work ∆F ≤ W

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(18)



(18)

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

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(18)


the relation is called Jarzynski equality (Jarzynski, 1997) −∆F −W exp = exp kB t kB t

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(19)



(18)


the relation is called Jarzynski equality (Jarzynski, 1997) −∆F −W exp = exp kB t kB t

(19)

The line indicated all possible realizations of an external process that takes 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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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 examples close 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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