Entropy Produce Molecule Motor
Post on 19-Jan-2015
877 Views
Preview:
DESCRIPTION
Transcript
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Is thermodynamics an outdated subject like a dinosaur ?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.
top related