Multidimensional Extensions of Multicanonical Algorithm

In the isobaric-isothermal ensemble [119-122], the probability distribution Pnpt(E, V T, V) for potential energy E and volume V at temperature T and pressure V is given by 

the density of states n(E,V) is given as a function of both E and V, and H is the enthalpy (without the kinetic energy contributions)  

This weight factor produces an isobaric-isothermal ensemble at constant temperature (T) and constant pressure (V), and this ensemble yields bell-shaped distributions in both E and V. 

To perform the isobaric-isothermal MC simulation [122], we perform Metropolis sampling on the scaled coordinates r, = L 1qi (qi are the real coordinates) and the volume V (here, the particles are placed in a cubic box of size L = /V). The trial moves from state x with the scaled coordinates r with volume V to state x with the scaled coordinate r and volume V are generated by uniform random numbers. The enthalpy is accordingly changed from Ti(E(r, V), V) to 7i E r, V), V) by these trial moves. The trial moves will be accepted with the probability 

As for the MD method in this ensemble, we just present the Nose-Andersen algorithm [119-121]. The equations of motion in (4.11)-(4.14) are now generalized as follows  

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