Entropy production along a stochastic trajectory of a small system

Joseph X. Zhou

Biological Physics group

MPI-PKS Jam session, June 6, 2008

In the time of modern physics about BEC, HT super conductor, cold atom, laser

Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more.

- Arnold Sommerfeld

Why talking about Thermodynamics ?

Thermodynamics laws need to be generalized

Do molecular machines work differently from those in macro-world

A complete thermodynamics laws include non-equilibrium

When we move to small world…

Is thermodynamics an outdated subject like a dinosaur ?

Thermodynamics development history

1820 – 1850, First law and second law of thermodynamics

1900, Equilibrium state,

1930 – 1960, Non-equilibrium, Linear response, Onsager, Green-Kubo

> 1993, Non-equilibrium, Fluctuation Theorem, Jarzynski Equality

Thermodynamic systems characterized by length scales

Energy conserves - why machines have different efficiency

Entropy production is different, and it decides the energy efficiency

All thermal machines are below Carnot cycle efficiency

Car – 25 %,

Power station – 46%

Molecular machine’s efficiency

Low Reynolds-number world is very sticky

All kinetic energy are dissipated

Molecular machine’s efficiency should be low

Molecular machine’s efficiency is quite high

Kinesin – 60%

Ion pump on membrane – 70%

J. Liphardt et. al., Science 296, 1832, 2002

Why molecular machine so efficient?

Find your friend in a crowded dance floor

- By random push from others

- By pushing through the crowds

- By taking random push when it is right direction

Stochastic thermodynamics

First law on a trajectory

Longevin equation for overdamped system

On a single trajectory

So fluctuation dominate in small system, W, q no long definite number, P(W) and P(q) instead will reflect system characteristics.

dqdVdw

)())(,()( ttxFtx

))(,())(,())(,( txftxVtxF x

fdxdV

dw

)()(

dxfV

xwt

0

)()()]([ tdxFxq

0)]([

U. Seifert, J. Stat. Phy., 128, 2007

Fluctuation Theorem

The corresponding Fokker-Planck eq.

Define entropy on the trajectory

Relationship to non-equilibrium ensemble entropy

From this definition, we can derive an integral fluctuation theorem

With concept of time-reverse trajectories, we can derive a more general relation:

),(

)),(),(),((),(

xj

xpDxpxFxp

x

xxx

)(x

),(ln)( xps

)(),(ln),()( sdxxpxpS

1)exp( totals

1)exp(

00

1 xp

xps tm

0 totals

Fluctuation Theorem and Jarzynski equality

• If we define entropy production rate , another form of fluctuation theorem can be written:

• It implies that molecule machines may absorb heat to do work sometimes

• Jarzynski equality can also be derived from generalized integral fluctuation theorem:

• Compare with Equilibrium relationship:

)(

)(lnlimP

P

t

kBtTt

p

Tk

G

Tk

W

BB

expexp

Tk

G

p

p

B

exp*2

*1

Prospects of non-equilibrium thermodynamics

For most biological molecular machines, JE doesn’t apply

Need to generalize JE to arbitrary transitions between nonequilibrium states

Quantum version of Fluctuation Theorem is also on its way

C. Bustamante, Physics Today, 2005

Summary

Nonequilibrium thermodynamics of small systems (biomolecular machines) is still a cutting-edge research field

The second thermodynamics law is generalized to the fluctuation theorem. Negative entropy is possible in the single trajectory.

Molecular machines take advantage of thermal noise in the micro-world to do work, which is the reason why they usually have a higher energy efficiency than their macro-counterparts.

Nonequilibrium thermodynamics still need to develop further to find the relationship of arbitrary transitions between Nonequilibrium states

Acknowledgement:

Thanks Prof. Udo Serfeit for the discussion and his numerous papers on Nonequ.