# [gmx-developers] Two questions about Nose-Hoover chains

Shirts, Michael (mrs5pt) mrs5pt at eservices.virginia.edu
Sun Aug 12 23:18:50 CEST 2012

```Hi, all-

Guess this is my baliwick-

> (1) Do you happen to know of any good review-type articles or tutorials on
> the concept of Nose-Hoover chains.  Unfortunately, I am not so strong in
> either statistical mechanics or Lagrangian mechanics, so when I read (for
> example) the original Nose-Hoover chain papers by Martyna, I get lost very
> quickly.

Not right off.  I got my understanding reading the original paper.  Frenkel
and Smit has some additional explanations.

> I understand that the basic Nose-Hoover method relies on the introduction of
> a heat bath and an associated friction parameter; this friction parameter
> modifies the equation of motion (i.e., Newton's equation) by adding an
> additional term which contains the momentum of the friction parameter xi and
> a mass parameter Q.

It's not really a friction parameter per se.  Think of it as a coupling term
that allows kinetic energy to pass between a bath and the system of
interest. There's another equivalent way to think of it as changing the
length of each timestep, but that's a little confusing.

> I also understand that the basic Nose-Hoover method can be non-ergodic and
> to remedy this, Nose-Hoover chains are introduced.  Are Nose-Hoover chains
> thermostats "chained" to the original Nose-Hoover thermostat?  Can you
> recommend any overview reading to help me get the basic conceptual idea?

Each new link in the chain is a new bath -- so each bath is itself sharing
getting energy from another bath.  As you add chains, the system becomes
chaotic, moving the system closer to ergodicity.  The original paper +
Frenkel and Smit are the sources I used.

> (2) Why are (multiple) Nose-Hoover chains not implemented in the leap frog
> (md) integrator?  This is NOT a criticism; I am just curious if there is any
> conceptual idea behind the inability to use chains with the leap frog
> algorithm.

It's possible in either integrator.  The simple answer is too complicated
bookkeeping.  You have to write out the equations in a somewhat different
way that just got too hard to code together.  I'm working on ways of
simplifying the bookkeeping in 5.0 so that all integration methods are
available (where possible) in both leapfrog and verlet, and are put together
in a more intuitive, extensible way.

Note that the ergodicity in NH chains occurs only in the limit of inifinite
chains.  Likely in 5.0, NH chains will be replaced with NH-Langevin, where
the NH bath is thermostatted using Langevin dynamics.  This is provably
ergodic and is simpler.

Best,
~~~~~~~~~~~~
Michael Shirts
Assistant Professor
Department of Chemical Engineering
University of Virginia
michael.shirts at virginia.edu
(434)-243-1821

```