Fluctuation Theorem & Jarzynski Equality Zhanchun Tu ( 涂展春 ) [email protected] Department of Physics Tamkang University — new developments in nonequilibrium.

Fluctuation Theorem & Jarzynski Equality

Zhanchun Tu (涂展春 )

[email protected]

Department of Physics

Tamkang University

— new developments in nonequilibrium process

Outline

• I. Introduction

• II. Fluctuation theorem (FT) & Jarzynsk

i equality (JE)

• III. Generalized JE

• IV. Proposed experiments & Summary

I. Introduction

Thermodynamics

• Object (for classic thermodynamics)– Systems: a large number of particles (~1023)

– Short-range interaction between particles (ideal gas, vdW gas, plasma, polymer; gravity system, + or - charged system)

– An isolated system can reach thermal equilibrium through finite-time relaxation

• Four thermodynamic laws

– 0th: it is possible to build a thermometer

– 1st: energy is conserved

– 2nd: not all heat can be converted into work

– 3rd: absolute zero temperature cannot be reached via a finite reversible steps

1st+2nd+const. T:

Statistical mechanics• Function

Time reversibleMacroscopic, reversibleTime reversible

Time irreversible?

Newtonian mechanics

N-particle system

ThermodynamicsStat. Mech.

(Ensemble average)

• Macrostate: thermodynamic EQ state– e.g. PVT, HMT etc.

• Microstate: phase point (q,p)

A macrostate

q

p

Each macrostate corresponds to many microstates!

Non-equilibrium process

• The system is driven out of the equilibrium by the external field

• Two equalities are proved to hold still for the system far away from the equilibrium

• FT: probability of violating the 2nd Law of thermodynamics along a micropath in NEQ process

• JE: extract free energy difference between two EQ states from the NEQ work performed on the system in the process between these two EQ states

Classicsystem

external field

(finite time interval)

II. FT & JE

FT [Adv. Phys. 51 (2002) 1529]

q

p

Bt=t2

q

pA

γ(t)

t=t1

Entropy production function

Phase space contraction factor

FT:

p(s) represents the probability distribution of the entropy production function taking the value s along the micro-path γ

• Preconditions for FT– Microscopic dynamics: time reversible

– Initial distribution is symmetric under the time reversal mapping

– Ergodic consistency:

Time reversal mapping

Γm(0)

Γm(t)

JE [PRL 78 (1997) 2690]

JE:

w: work performed on the system along each micro-pathΔF: free energy difference between two macrostates< >: average for all micro-paths

t=t2

t=t1 Macrostate 1Temperature T

Macrostate 2Temperature T

q

p

q

p

B1

A1

B2

A2

B3

A3

…γ1

γ2

γ3

Examples for JE

• Gas & piston

• Unfolding RNA hairpin

[Nature 437 (2005) 231]

Relation between FT & JE

• FT JE

– Crooks: proof for stochastic, microscopically time reversible dynamics [PRE 60 (1999) 2721]

– Evans: proof for time reversible deterministic dynamics [Mol. Phys. 101 (2003) 1551]

q

p

Evans’ proof: FTJE

• General description

q

pB

t=t2

q

pγ(t)

t=t1 t=t’2

Macrostate 1Temperature T

Macrostate 2Temperature T

EQ EQSwitch external field(parameter from λ1 to λ2) Long time relaxation

Attention: the system always contact with the constant temperature thermal bath

Do work! No work

A

• FT between 2 macrostates– Note: Original FT is valid for the ensemble cont

aining all paths beginning from all microstates at time t1

– Evans: FT holds also for the ensemble only containing all paths connecting the microstates corresponding to macrostates 1 and 2 [Mol. Phys. 101 (2003) 1551]

<>: average for all micro-paths beginning from the microstates corresponding to the macrostate 1

• Initial and final distribution functions

• Phase space contraction factor

Effective dynamics of isothermal system

Number of particles

thermostat multiplier ensuring the kinetic temperature of the system to be fixed at a temperature T. It reflects the heat exchange between the system and the thermal bath.

Crucial condition in the derivation of JE from FT!

• Energy conservation along the micro-path

Work performed on the system along the micro-path

Heat absorbed by the system form the thermal bath along the micro-path

• Entropy production function

• JE

FT, JE and 2nd law

• FT permits the existence of micro-pathes violating the 2nd law

[Crooks FT, 1999]

• JE: macro-work satisfies 2nd law

w1 w2<w>

w

p(w)

ΔFRNA unfolding [Nature 437 (2005) 231]

• Remarks

– JE satisfies 2nd law

– 2nd law cannot be derived from JE: 2nd law

holds in a much wider realm than JE does

– Proof of JE implies microscopic time reversibility

can result in macroscopic time irreversibility

III. Generalized JE (GJE)

cond-mat/0512443

Two gedanken expts. on JE• Expt. 1

[J. Phys. Chem. B 109 (2005) 6805]

vvp

• Expt. 2

JE is violated!

Why?

• Jarzynski and Crooks’ argument: the JE fails because the initial distribution function is not canonical in the second expt.

• Our viewpoint: the initial distribution function is still canonical but a more underlying reason makes the JE fail. In other words, a generalized JE may exist.

Investigate 2 gedanken expts.• Crucial condition (emphasize again)

– time integral of the phase space contraction factor is exactly expressed as the entropy production resulting from the heat absorbed by the system from the thermal bath

The dynamics of two gedanken expts. may not be described as the form of Hamiltonian dynamics with the thermostat multiplier, we should check whether the crucial condition holds or not for 2 gedanken expts.

• Expt. 1

Effective dynamics (ideal gas)

Influence of piston movement

Phase space contraction factor

(Crucial condition still holds)

Thus JE holds!

Switch parameter is V

[Evans’s book]

• Expt. 2

Effective dynamics (ideal gas)

Phase space contraction factor

Crucial condition does not hold JE fails!

Volume expansion has no effect on the momentum of the particles

• Expt. 2 (continued)

However, following the derivation of JE from FT, we obtain

We have known:

Although JE fails, but above general form holds!

Hint: a more general version of JE may exist.

GJE• Time integral of phase space contraction factor

• GJE

Special cases:

Conjecture: most of macroscopic systems satisfy σ=0.

Heuristic viewpoint

• Expt. 1: M→∞

• Expt. 2: M→0

• Intermediate case: 3rd gedanken expt.– The experimental setup is same as the first one. T

he mass of the piston M is finite. At time t1, we remove the pins P1 and P2. The gas will push the piston to the right wall of the container. Once the piston contacts with the wall, it adheres to the wall without bounce. After a long time relaxation, the system arrives at an equilibrium state at time t2.

3rd gedanken expt.• Effective dynamics

• GJE for 3rd gedanken expt.

• Determine g: numerical simulation

Ideal gas: βm=1, 1000<N<10000

0.2<M/m<1000, 1.1<V2/V1<1.9

<>: average on 500 systems with different initial microstates corresponding to the same macrostate.

• Fitting results

Small x (inset):

ν=0.53 (fitting)

• Prediction

Take M/m=4000, N=1000 and V2/V1=1.1

Numerical result: g= 0.8846

This fact implies that our conjecture on the form of g is reasonable!

IV.Proposed expts. & summary

Proposed expts.• Inert gas in a long single-walled CNT

1. The friction between C60 and (10,10) SWNT is very small2. Long (10,10) SWNT can be achieved in recent nanotech3. The inert gas with large radius, for example Ar, cannot go out fr

om the small gap between C60 and (10,10) SWNT. 4. The whole setup is put in vacuum5. Small CNT coordinates the initial position of C60. At some time,

pull out the small CNT suddenly to a new position. Measure the velocity of C60 when it arrives at the new position.

6. Repeat expt. and calculate σ. We expect σ ≠ 0.

• Polymer in CNT

L

1. Initial position L=L1

2. Finial position L=L2

3. L1<L2<R0

• Special macroscopic system

Summary

• FT & JE• Micro. reversibility macro. irreversibility• Three gedanken expts. are analyzed, whic

h implies a generalized JE may exist

• σ=0 for most of macroscopic systems• Expts. for σ≠0 are proposed

Thank you for your attention!

http://biox.itp.ac.cn/~tzc/index.html