Multiple Time Steps Algorithms for the Isothermal-Isobaric Ensemble

The integrators developed in the previous section generates dynamics in the microcanonical ensemble where total energy, number of particles and volume are conserved. The derivation based on the Liouvillean and the corresponding propagator, however lends itself to a straightforward generalization to non microcanonical ensembles. Simulations of this kind are based on the concept of extended system and generate trajectories that sample the phase space according to a target distribution function. The extended system method is reviewed in many excellent textbooks and papers [12,86,87,88,89,90,91,25] to which we refer for a complete and detailed description. Here it suffices to say that the technique relies on the clever definition of a modified or extended Lagrangian which includes extra degrees of freedom related to the intensive properties (e.g. pressure or temperature) one wishes to sample with a well defined distribution function. The dynamics of the extended system is generated in the microcanonical ensemble with the true $n$ degrees of freedom and, additionally, the extra degrees of freedom related to the macroscopic thermodynamic variables. With an appropriate choice, the equations of motion of the extended system will produce trajectories in the extended phase space generating the desired equilibrium distribution function upon integration over the extra (extended) variables. There are several extended system techniques corresponding to various ensembles, e.g. constant pressure in the NPH ensemble simulation with isotropic [92] and anisotropic [93] stress, constant temperature simulation [94] in the NVT ensemble and isothermal-isobaric simulation [95] in the NPT ensemble. As we shall see, the dynamic of the real system generated by the extended system method is never Hamiltonian. Hence, symplecticness is no longer an inherent property of the equations of motion. Nonetheless, the Liouvillean formalism developed in the preceding section, turns out to be very useful for the derivation of multiple time step reversible integrators for a general isothermal-isobaric ensemble with anisotropic stress, or N PT^3.1. This extended system is the most general among all non microcanonical simulations: The NPT, NPH the NVT and even NVE ensemble may be derived from this Lagrangian by imposing special constraints and/or choosing appropriate parameters [25,27]