Isotropic cell fluctuations

These equations assume isotropic cell fluctuations, whereas Eqs. [84] allow anisotropy in cell dynamics. Note that different thermostats are coupled to the system and barostat variables in Eq. [178]. The total Liouvillian governing the evolution of the phase space variables, [q, p, r, Vy, V, v, q, is given by... 

When studying systems with mixed fluid and solid directions, it is important to keep in mind that each solid direction should be allowed to breathe and fluid directions need to be scaled isotropically or constrained to a constant value. Allowing two fluid directions to fluctuate independently from one another allows the simulation cell to become flat like a pancake, which we certainly would like to avoid. As an example, consider Figure 15, in which a lamellar block copolymer phase is sheared. The convention would be to have the shear direction parallel to x and the shear gradient direction parallel to y. No reason exists for the simulation cell to distort such that Lxz = Lyz = 0 would not be satisfied on average, so one may fix the values of Lxz and Lyz from the beginning. As a result, one solid direction exists plus two fluid directions. We can also constrain Lxx to a constant value, because the shear direction will always be fluid and another fluid direction can fluctuate. This result means that we should allow the simulation cell to fluctuate independently in only the directions of... 

Equations [76] only allow for isotropic fluctuations in the volume. However, it is sometimes useful to allow the lengths and angles of the simulation box all to vary separately. Fully flexible cell simulations of this type, first carried out by Parrinello and Rahman, can also be formulated in rms of a non-Hamiltonian dynamical scheme. In such a scheme, the matrix h representing the cell, which contains the cell vectors in its columns, is incorporated as a dynamical variable. That is, nine extra variables are added to the phase space along with an additional nine from its corresponding momentum Pg matrix. In terms of the box matrix, the partition function A(N, P, T) is given by... 

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