264 Isobaric ensembles
I 0 (x) =1
π∫π0dθe±xcosθ.b. Calculate the equation of state by determining the one-dimensional “pres-
sure”P. Do you obtain an ideal-gas equation of state? Why or why not?
You might find the following properties of modified Bessel functions use-
ful:
dIν(x)
dx=
1
2
[Iν+1(x) +Iν− 1 (x)]Iν(x) =I−ν(x).c. Write down integral expressions for the position and length distribution
functions in the isothermal-isobaric ensemble.5.10. Write a program to integrate the isotropicNPTequations of motion (5.9.5)
for the one-dimensional periodic potential in eqn. (5.9.8) using the integrator
in eqn. (5.12.4). The program should be able to generate the distributions in
Fig. 5.3.5.11. How should the algorithm in Section 4.13 for calculating the radialdistribu-
tion function be modified for the isotropicNPTensemble?∗5.12. Generalize the ROLL algorithm of Section 5.13 to the case of anisotropic
cell fluctuations based on eqns. (5.10.2) and the integrator defined by eqns.
(5.12.10) and (5.12.11).5.13. a. Using the constraint condition on the box matrixhαβ= 0 forα > β,
show using Lagrange undetermined multipliers, that overall cell rotations
in eqns. (5.10.2) can be eliminated simply by working with an upper
triangular box matrix.b. Using the constraint condition thatpg−pTg = 0, show using Lagrange
undetermined multipliers, that overall cell rotations in eqns. (5.10.2) can
be eliminated by explicity symmetrization of the pressure tensorPαβ(int).
Why is this scheme easier to implement within the ROLL algorithm of
Section 5.13?