# [gmx-users] Rerefence on Berendsen vs Nose-Hoover

John D. Chodera jchodera at gmail.com
Mon Oct 15 22:50:48 CEST 2007

Hi Jochen,

I think the paper you are referring to is this one, commonly cited on
the AMBER mailing list and by others in support of the idea that
there is little difference between Berendsen and Nose-Hoover:

* T. Morishita. Fluctuation formulas in molecular-dynamics
simulations with the weak coupling heat bath. J. Chem. Phys., 113(8):
2976, 2000.

However, this paper actually concludes the opposite:

"The present study shows that the WC thermostat does not produce the
canonical ensemble but another ensemble."

The paper demonstrates that, in the limit as the coupling time \tau -
> 0, the isokinetic ensemble is produced, while in the limit that
\tau -> \infinity, the microcanonical ensemble is produced.

For expectations involving only coordinates (and not momenta), the
agreement with the canonical ensemble is achieved only if terms on
the order of 1/N are neglected -- which is precisely the rate at
which all ensembles become equivalent, the "thermodynamic limit".  In
other words, if your system is big enough to reach the "thermodynamic
limit" (and who knows how big that is?) then you are just as well off
using *any* ensemble -- NVE, NPT, NPH, NVT -- it doesn't matter anymore.

Allen and Tildesley, for example, states on the difference between
ensembles: "the small relative magnitude of the correction term can
be seen explicitly: it decreases as O(N^-1)." (pg. 45 in my edition)

So only in the thermodynamic limit will there be correspondence
between Berendsen and Nose-Hoover ensembles for properties dependent
on the configuration space distribution.  Whether your system is
large enough to have reached this limit is something that you will
have to establish for yourself.

However, if you are interested in kinetic properties and make use of
the Berendsen weak-coupling algorithm, you will likely want to
investigate whether there is any \tau dependence in your computed
properties or observed behavior, since the width of the kinetic
energy distribution is a strong function of \tau.

Cheers,

- John

--
Dr. John D. Chodera <jchodera at gmail.com>      | Mobile    : 415.867.7384
Postdoctoral researcher, Pande lab            | Lab phone : 650.723.1097
Department of Chemistry, Stanford University  | Lab fax   : 650.724.4021
http://www.dillgroup.ucsf.edu/~jchodera
>
> Date: Mon, 15 Oct 2007 15:37:40 +0200
> From: Jochen Hub <jhub at gwdg.de>
> Subject:
> To: Discussion list for GROMACS users <gmx-users at gromacs.org>
> Message-ID: <47136D24.8070103 at gwdg.de>
> Content-Type: text/plain; charset=ISO-8859-1; format=flowed
>
> Hi,
>
> I think there is a paper around which says that for parcticle purposes
> it hardly matters whether one uses Berendsen or Nose-Hoover T-
> coupling.
> Does anyone remember the reference?
>
> Jochen
>
> --
> ************************************************
> Jochen Hub
> Max Planck Institute for Biophysical Chemistry
> Computational biomolecular dynamics group
> Am Fassberg 11
> D-37077 Goettingen, Germany
> Email: jhub[at]gwdg.de
> ************************************************
>
>