Re: A question on Jarkynski equation

From: Jrme Hnin (
Date: Tue Jun 14 2011 - 09:09:13 CDT

Yi, thanks for your remark. This emphasizes one point that I should
have mentioned: the "rare event" problem is highly dependent on
pulling speed. At slow speeds in a low-viscosity solvent like water,
friction is small or comparable to Brownian forces, so-called 'second
law violations' become fairly frequent and the exponential estimator
can converge. Similarly, in a FEP calculation between very similar
states, convergence is much improved (hence the use of staged


On 14 June 2011 15:11, Wang Yi <> wrote:
> Hi Jun,
> Dr. Hénin has pointed out the general concerns regarding SMD and JE. But for
> processes like moving in water, you might not have to worry too much about
> that. What I mean is:
> The work done for dragging a ligand in a water box is mostly against the
> "friction" from solvent. Since water molecules move in all directions, the
> "frictions" the ligand is experiencing changes all the time. Thus, you will
> notice the SMD force fluctuating roughly around zero (albeit with large
> swings). Then after integrating the force curve, the positive work and
> negative work will mostly cancel each other. And after applying JE, although
> there exists the issue of "heavey-weight rare events", the final value would
> be pretty small (compared to thermo energy). That's especially true in a
> unbinding process simulation, where the unbinding part has a much larger
> magnitude than the "free moving in water" part.
> That is based on my understanding.
> Best,
> ___________________________
> Yi (Yves) Wang
> Duke University
> 在 2011-6-14,上午4:43, Jérôme Hénin 写道:
> 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:
> Cheers,
> Jerome
> 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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