It will now be shown how the basic principles of non-Hamiltonian statistical mechanics can be employed to design of MD equations of motion for generating the canonical and isothermal-isobaric ensembles. In fact, the above principles lead to a procedure for designing general-purpose MD algorithms [28].
The Nosé-Hoover chain (NHC) dynamics method [29] is a non-Hamiltonian MD
scheme
for generating the canonical ensemble. In this method, the ordinary phase space
is extended to include a set of M thermostat variables 
and their conjugate momenta 
, which act as a heat bath
coupled to the system. The equations of motion take the form
where the thermostat forces are,
The parameters, 
, given by 
, 
, determine the
time scale of the thermostat motion via the single time scale 
, which
should be chosen corresponding to a characteristic time scale of the system, e.g., a
vibrational period. In Eqs. (5.1)
M thermostats which successively thermostat each other, are coupled to the
particles thereby controlling/modulating the kinetic energy fluctuations of both
particle and thermostat degrees of freedom.
Let us apply principles 1 and 2 above to the NHC equations. First, the compressibility and determine the invariant measure are computed. The compressibility can be seen to be
Since 
, it is clear that 
,
and the invariant measure according to Eq. (4.9) is
In order to determine the microcanonical partition function, the conserved quantities of Eqs. (5.1) are needed. One of the conserved quantities is the total energy of the extended system,
In addition to the energy, there are d additional conservation laws if there
are no external forces (i.e., when 
). These conservation
laws take the form
as can be seen by direct differentiation. Given the conservation laws, the microcanonical partition function can now be constructed according to Eq. (4.13). Suppose d=3 and M=2,
In order to show that Eq. (5.7) generates a canonical distribution in
the system Hamiltonian, 
, the integrals over the thermostat
variables need to be performed. The energy 
-function can be
used to perform the integral over 
, which requires that
.
One of the remaining -functions can be used to integrate
over 
, which will leave only ratios of components of 
in the other two -functions. Finally, by changing variables to
center-of-mass and normal mode momenta, 
, the expression
can be simplified to yield
where 
are the normal mode and total masses, 
,
and the proportionality constant depends on E and 
.
Equation (5.8) is the correct canonical
partition function for the Hamiltonian, . Hence, the NHC equations
generate the canonical distribution function (within constants).
Following the same
procedure, it can be shown that the Nosé-Hoover thermostat method
(corresponding to M=1 in Eqs. (5.1)) does not produce the
correct canonical distribution when [28].
It is also interesting to note that
if more than one thermostat chain is coupled to the system,
then the momentum conservation
laws are no longer present, and the proof simplifies. An extreme example,
which has proved useful in a number of applications, is the coupling of a
separate thermostat to each degree of freedom in the system. This scheme
leads to very rapid equilibration of a system and plays an important
role both in path integral molecular dynamics and in simulations involving biological
macromolecules [30, 31].
Next, the isothermal-isobaric or NPT ensemble is considered. In this ensemble,
the volume V of the system must fluctuate such that the average internal pressure
of the system, 
,
is equal to an external applied pressure, 
. Thus, in
designing a MD algorithm for the NPT ensemble, one should both incorporate
the volume as a dynamical variable [24], and employ
a thermostat to ensure that both instantaneous temperature and pressure
fluctuations are generated properly. A useful
non-Hamiltonian scheme for the NPT ensemble [32]
is defined by the following equations of motion,
Here, 
is a momentum conjugate to the logarithm of the volume, W
is its associated mass parameter, and 
,
is the external applied
pressure and 
is the instantaneous internal pressure of the system given by
Thus, the variable acts as a ``barostat'' which drives the system to
the steady state 
.
For simplicity, Eqs. (5.9) are written with a single thermostat
variable coupling both to the particles and to the barostat. Thus,
the case 
will be considered. However,
it is clear that when , a thermostat
chain should be used. In fact, the optimal algorithm employs separate thermostat
chains on the particles and on the barostat.
The compressibility associated with Eqs. (5.9) is
from which it can be seen that 
, and the invariant
measure is
If the forces do not sum to zero, then the only conservation law that is always present is the energy,
Using Eq. (4.13), the microcanonical partition function corresponding to Eqs. (5.9) can be constructed,
Integrating over 
using the -function and then over 
and yields,
the correct partition function for the NPT ensemble.
These examples show how the principles of classical non-Hamiltonian statistical
mechanics can be applied in the design of MD algorithms that generate
different statistical ensembles. Note, non-Hamiltonian dynamics are different
from Hamiltonian dynamics. These non-Hamiltonian systems reduce to approximate
Hamiltonian dynamics when the extended system coupling
parameters (here, Q and W) are large. An important outstanding question
concerns the possibility of employing non-Hamiltonian dynamical schemes to
generate a grand canonical or 
ensemble sampling method.
