# Tdamp definition

Hello All,

I am trying to get a deeper understanding of the Nose-Hoover
thermostat. In my reading I believe I found a difference between the
thermostat's period as defined by Nose and the Tdamp parameter
implemented by LAMMPS. Should those be exactly the same? From
fix_nh.cpp it appears that

Q = f*k*T/t_freq^2 = f*k*T*t_period^2

=> t_period = ( Q/(f*k*T) )^1/2

where "f" is the number of degrees of freedom and I believe t_period
is the same as Tdamp. The best explanation I've been able to find in
a journal was by Nose (Molecular Physics, 1984, 52, 2, 255-268). In
equation 2.30 he states the thermostat period is

t0 = 2*pi*( Q*<s>^2 / (2*f*k*T) )^1/2

The differences are the 2*pi in front, which implies t_period =
2*pi/t_freq, and the 2 in the denominator, which I don't yet
understand. The latter period is about four times bigger and appears
to be closer to the one I observe in my simulations. Perhaps Tdamp
and t0 are not supposed to be the same and it's quite possible that
I'm misunderstanding the definitions. Any insight would be greatly
appreciated.

Best,
Corey Musolff

I'll let Aidan comment. We could be off by 2 pi.
I think of the "period" of the temperature oscillations
as only a rough estimate and only something you
need to worry about if you try to use the thermostat
to instantly equilibrate from a T1 to a (very) different
T2 and it takes a while to settle down to T2. Which
isn't a very good way to use a thermostat.

Steve

Since the thermostat does not produce perfect periodic dynamics (that
would be bad), the period can
not be defined precisely. Instead, it makes sense to define a
characteristic timescale tau. For the LAMMPS thermostat,
the timescale is given by:

tau^2 = Q/NkT

The apparent periodic observed in an oscillatory transient should be of
the same magnitude as tau, but the precise ration will depend on initial
conditions, anharmonicity, etc. The takehome message is that LAMMPS' Tdamp
is a good estimator of the effective timscale.

A similar analysis exists for the barostat. Unfortunately, int hat case,
the relationship between tau and the characteristic timescale seems to be
more complicated, perhaps depending on the relevant elastic modulus.

For a good understanding of the equations of motion that LAMMPS uses, I
recommend the papers cited on the doc page, particularly the most recent
one by Mark Tuckerman.

Aidan