Re: A question on Jarkynski equation

From: Jérôme Hénin (
Date: Tue Jun 14 2011 - 03:43:55 CDT


You put your finger on a very interesting question. The Jarzynski
equality is formally exact, hence the estimator will converge towards
zero, at least on paper. As you note, since the reversible work is
zero, all measured work is irreversible: we expect it to be positive
basically all the time. Then how can the average be zero?

Actually, statistical mechanics dictates that a few trajectories will
give negative work values, i.e. negative irreversible work, i.e. a net
decrease in entropy. This is why such situations are sometimes called
'second law violations'. They are not really violations, just a
reminder that the second law applies to macroscopic systems only (even
if we in the molecular modeling community often stretch the notion of
macroscopic a little far). In the case of a solute moving through a
solvent, the 'negative work' case would be a trajectory where random
fluctuations in the solvent happen to push the solute along its path,
instead of slowing it down in a normal frictional behavior.

Such negative work values will be exceedingly rare, but they have a
huge weight in the exponential average of the Jarzynski formula. That
is why a few negative values are enough to make the average zero even
though almost all values are positive. Because these events are so
rare, numerical convergence will be awful, so it is unlikely that you
will manage to get a zero free energy value from a numerical
simulation. To some extent, the same can be said of any application of
the Jarzynski estimator, and is also true of FEP calculations with the
exponential formula: these averages are dominated by rare events,
which results in various degrees of convergence problems.

For more details on 'second law violations', see for example:


On 14 June 2011 05:01, Jun Zhang <> wrote:
> Hello Everyone:
> I want to use Jarkynski's equation combined with SMD to compute the binding free energy of a protein and its ligand (eg. JCP, 120, 5946). However, I was puzzled by some theoretical issues.
> For example, a system composed of water and a ligand. if we move a single ligand in aqueous for some distances, the work done cannot be zero, but the free energy change should be zero since the state of the ligand has not changed.
> It seems somewhat strange, and may be a naive question. But I am really puzzled by it, so I'm looking for help. Thank you in advance!
> Cheers up!
> Jun Zhang
> Nankai University

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