Equilibrium Information from

Nonequilibrium Measurements

in an Experimental Test ofJarzynski’s Equality

Simon-Shlomo Poil 9.december 2005 Single Molecule Biophysics

J. Liphardt et al., Science, 296, 5574, 1832-1835 (2002)

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

or

normal

C. Jarzynski 1997

N = # of work trajectories

z = order parameter

Experiment• 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

p5abc

Force-extension curves

Reversible

Irreversiblegreen,

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 where the work dissipation is largest

Reversible :

Jarzynski :