Random vector on a sphere

The ability to generate a randomly orientated vector is very useful in Monte Carlo simulations of anisotropic models or molecular systems.

Marsaglia algorithm

This is the algorithm proposed by George Marsaglia (Ref. 1):

• Independently generate V₁ and V₂, taken from a unifor distribution on (-1,1) such that

$S=(V_{1}^{2}+V_{2}^{2})<1$

• The random vector is then (Ref. 1 Eq. 4):

$\left(2V_{1}{\sqrt {1-S}},~2V_{2}{\sqrt {1-S}},~1-2S\right)$

Fortran 90 implementation

This Fortran 90 implementation is adapted from Ref. 2. The function ran() calls a randon number generator:

!    The following is taken from Allen & Tildesley, p. 349
!    Generate a random vector towards a point in the unit sphere
!    Daniel Duque 2004

subroutine random_vector(vctr)

  implicit none

  real, dimension(3) :: vctr

  real:: ran1,ran2,ransq,ranh
  real:: ran

  do
     ran1=1.0-2.0*ran()
     ran2=1.0-2.0*ran()
     ransq=ran1**2+ran2**2
     if(ransq.le.1.0) exit
  enddo

  ranh=2.0*sqrt(1.0-ransq)

  vctr(1)=ran1*ranh
  vctr(2)=ran2*ranh
  vctr(3)=(1.0-2.0*ransq)

end subroutine random_vector

References

1. George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics 43 pp. 645-646 (1972)
2. Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)
3. M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989)