Consider a classical system of N particles. The state of the
system is uniquely specified by the 3N spatial co-ordinates
, 
, 
and the 3N momentum co-ordinates
, 
, 
. The 6N dimensional vector 
consisting of the positions and momenta for all
particles is a point in the phase space of the system. The
dynamics of the system evolving in time is described by the
trajectory of the phase point through phase space. The trajectory
is given by Hamilton's equations:
Linear transport coefficients, (e.g. 
and 
)
can be calculated from computer simulations of molecules obeying eq.
(
) by means of
the Green-Kubo relations. These have proved to be of enormous value in
experimental and theoretical applications. If we wish to extend the theory to
non-linear steady states, then we are faced with two problems; one being the
inclusion of higher order terms in the perturbation due to the external field,
and the other due to thermostatting. In a dissipative process, such as shear
flow, the temperature would rise in the system if the heat generated is not
removed by a heat sink. This effect is second order in the small field limit
(
, with L being the linear transport coefficient relating 
to J), and can be ignored in the linear theory, but needs to be included in
non-linear theories.
The most obvious way of modelling the non-equilibrium steady state is to
include the interactions of the system with the outside world. This method is
impractical owing to the high complexity of these interactions, and the large
surface effects inherent in modelling small systems (computer simulations
typically involve 
to 
molecules). It turns out that the heat
sink can be modelled by means of a friction-like term, which acts as a
thermostat, and that provided the dissipation is not too large, the Transient
Time Correlation Functions we will derive are independent of what model
thermostat is used [Evans and Morriss
[1984], Evans
and Holian
[1985]]
.
The thermostatted equations of motion are
where 
describes the force on molecule i due to all the other
molecules. The momenta in this equation are peculiar ,
i.e. measured with respect to the motion of the centre of mass so that
.
The first model thermostat is produced by requiring that the kinetic energy be
a constant of motion:
This is known as the Gaussian thermostat, after Gauss's principle of least constraint [Hoover et al. [1982] , Evans [1983]] . This feedback mechanism does not constrain the actual value of the kinetic energy required, it only constrains the time derivative to be zero. This implies that the propagator (to be defined in the next section) will commute with the operation of differentiation with respect to temperature, allowing equilibrium fluctuation expressions for the second order, or derived thermodynamic quantities, such as specific heats. Such a thermostat is called differential. The equilibrium distribution in this case is given by
where 
, 
, 
.
This distribution is called the isokinetic distribution.
The other common thermostatting method was first proposed by Nosé
[1984a,1984b]. The original formulation by Nosé involved a cumbersome
external reservoir and a non-linear time transformation. Hoover
[1985] made significant simplifications to the method, which has
since become known as the Nosé-Hoover thermostat. The basic idea is to
extend phase space by adding the variable 
, whose equation of motion
is given by
Thus 
acts to keep the kinetic energy fluctuating about the
target value 
, with the timescale of the fluctuations being proportional
to Q. The parameter Q is arbitrary, but the Q=0 case, which corresponds
to the kinetic energy being rigidly constrained, has infinitely stiff
equations of motion. In practice, its value is determined from numerical
experiments. The equilibrium distribution generated by the Nosé-Hoover
equations of motion is canonical:
where 
.
In contrast to the former thermostat, this is an integral thermostat, with the
value of 
depending on all past states of the system.
