# Re: A question on Jarkynski equation

From: Jun Zhang (coolrainbow_at_yahoo.cn)
Date: Tue Jun 14 2011 - 23:03:53 CDT

Hello Jerome:

that we can calculate the probability that the work done is
negative is:

P( W<-D ) <= exp( -beta*D ) where D>0 and beta =
1/(k*T)

Of course, this inequality is very weak, since assuming D =
1 kcal/mol and T = 300K gives P(W<-1) < 0.99998 which
seems a  trivial result. I think for soft spring and
slow pulling, the random kick from solvent can make the work
negative in which case is not too "rare". but for stiff
spring and fast pulling, maybe the work being negative can
occur only in miracle, which accounts for the high-weight
low-sampling rare event. So, I think that my question is in
fact due to a sampling error, am I right?

Cheers up!

Jun Zhang
Nankai University
coolrainbow_at_yahoo.cn

> > Re: namd-l: A question on Jarkynski equation
> >  "Jun Zhang" <coolrainbow_at_yahoo.cn>
> >  "NAMD MAILLIST" <namd-l_at_ks.uiuc.edu>
> > 2011 6 14 ,4:43
> > Hi,
> >
> > 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:
> > http://arxiv.org/abs/cond-mat/0401311
> >
> > Cheers,
> > Jerome
> >
> >
> > On 14 June 2011 05:01, Jun Zhang <coolrainbow_at_yahoo.cn>
> > 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
> > > coolrainbow_at_yahoo.cn
> > >
> > >
> >
>

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