From the classical virial theorem
we arrived at the equipartition theorem:
where 
are the N Cartesian momenta of the
N particles in a system.
This says that the microscopic function of the N momenta
that corresponds to the temperature, a macroscopic observable
of the system, is given by
The ensemble average of K can be related directly to the temperature
is known as an estimator (a term taken over
from the Monte Carlo literature) for the temperature. An estimator is
some function of the phase space coordinates, i.e., a function of
microscopic states,
whose ensemble average gives rise to a physical observable.
An estimator for the pressure can be derived as well, starting from the basic thermodynamic relation:
with
The volume dependence of the partition function is contained in the
limits of integration, since the range of integration for the
coordinates is determined by the size of the physical container.
For example, if the system is confined within a cubic box
of volume 
, with L the length of a side, then
the range of each q integration will be from 0 to L.
If a change of variables is made to 
, then
the range of each s integration will be from 0 to 1.
The coordinates 
are known as scaled coordinates.
For containers of a more general shape, a more general transformation is
In order to preserve the phase space volume element, however, we need to ensure that the transformation is a canonical one. Thus, the corresponding momentum transformation is
With this coordinate/momentum transformation, the phase space volume element transforms as
Thus, the volume element remains the same as required. With this transformation, the Hamiltonian becomes
and the canonical partition function becomes
Thus, the pressure can now be calculated by explicit differentiation with respect to the volume, V:
Thus, the pressure estimator is
and the pressure is given by
For periodic systems, such as solids and currently used models of liquids,
an absolute Cartesian coordinate 
is ill-defined. Thus, the
virial part of the pressure estimator 
must be
rewritten in a form appropriate for periodic systems.
This can be done by recognizing that the force 
is obtained
as a sum of contributions 
, which is the force on particle i
due to particle j. Then, the classical virial becomes
where 
is now a relative coordinate. must be computed
consistent with periodic boundary conditions, i.e., the relative coordinate
is defined with respect to the closest periodic image of particle j
with respect to particle i. This gives rise to surface contributions,
which lead to a nonzero pressure, as expected.