Equilibrium Information from
Nonequilibrium Measurements
in an Experimental Test of
Jarzynski’s Equality
J. Liphardt et al., Science, 296, 5574, 1832-1835 (2002)
Simon-Shlomo Poil 9.december 2005
Single Molecule Biophysics
Equilibrium / Nonequilibrium Thermodynamic
• Irreversible processes drive a Nonequilibrium system
towards an Equilibrium system
• Equilibrium systems have one uniform temperature, and
state functions of energy and entropy
- e.g. U = U(T,V,N) and S = S(T,V,N)
•In Nonequilibrium systems the energy and entropy need to
be described using energy and entropy densities.
Jarzynski’s equality
normal
or
C. Jarzynski 1997
N
= # of work trajectories
z
= order parameter
Experiment
p5abc
• Reversible slow switching rate 2
to 5 pN/s
•Irreversible fast switching rates
34 and 52 pN/s
Compare three different
estimates
• Average work
•Fluctuation-dissipation
•Jarzynski’s estimate
Force-extension curves
Reversible
Irreversible
green,
34pN/s
Red,
52 pN/s
Jarzynski’s equality as a sum
If the distribution is Gaussian, (near equilibrium state)
To use Jarzynski’s equality experimentally;
fluctuations should be below KbT
Convergence of Jarzynski
Conclusion
•
and
are equal in near-equilibrium and
unequal in far-equilibrium.
• Jarzynski gives the same
at the two different
irreversible switching rates (within 0.3 KbT )
• The difference between reversible and irreversible work
is less than 0.6 KbT
•Jarzynski overestimate
largest
Reversible :
Jarzynski :
where the work dissipation is