The microcanonical average of a physical quantity *A* is obtained
as a time average on the trajectory:

(26) |

There are other important alternatives to the NVE ensemble, that we will mention here only briefly. The basic idea is that of integrating other equations in place of Newton's equations, in such a way that sampling is performed in another statistical ensemble. Averages of physical quantities in the new ensemble will be again obtained as time averages, similarly to eq. (3.18).

A scheme for simulations in the isoenthalpic-isobaric ensemble
(NPH) has been developed by Andersen [22].
Here, an additional degree of freedom representing the volume
of the box has been introduced, and all the particle
coordinates are given in units relative to the box.
The volume *V* of the box becomes a dynamical variable, with
a kinetic energy and a potential energy which just *PV*,
where *P* is the external pressure. The enthalpy
*H*=*E*+*PV* is a conserved quantity.

Parrinello and Rahman [23] developed a variant where the shape of the box can vary as well as the volume. This is achieved by introducing 9 new degrees of freedom instead of 1: the components of the three vectors spanning the MD box. Each of them is a new dynamical variable, evolving accordingly to equation of motion derived from an appropriate Lagrangian. This scheme allows to study structural phase transitions as a function of pressure, where for example the system abandons a certain crystal structure in favor of a more compact one.

Another very important ensemble is the canonical ensemble
(NVT). In a method developed by Nosè and Hoover [24],
this is achieved by introducing a time-dependent frictional
term, whose time evolution is driven by the imbalance
between the instantaneous kinetic energy and the average
kinetic energy (3*N*/2) *k*_{B} *T*.