Random vector on a sphere: Difference between revisions
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The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]] | The ability to generate a randomly orientated vector is very useful in [[Monte Carlo]] simulations of anisotropic [[models]] | ||
or molecular systems. | or molecular systems. | ||
==Marsaglia algorithm== | ==Marsaglia algorithm== | ||
This is the algorithm proposed by George Marsaglia ( | This is the algorithm proposed by George Marsaglia <ref name="Marsaglia">[http://dx.doi.org/10.1214/aoms/1177692644 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics '''43''' pp. 645-646 (1972)]</ref> | ||
*Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a uniform distribution on (-1,1) such that | *Independently generate V<sub>1</sub> and V<sub>2</sub>, taken from a uniform distribution on (-1,1) such that | ||
:<math>S=(V_1^2+V_2^2) < 1</math> | :<math>S=(V_1^2+V_2^2) < 1</math> | ||
*The random vector is then ( | *The random vector is then (Eq. 4 in <ref name="Marsaglia"> </ref> ): | ||
:<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math> | :<math>\left(2V_1 \sqrt{1-S},~ 2V_2 \sqrt{1-S},~ 1-2S\right)</math> | ||
==Fortran 90 implementation== | ==Fortran 90 implementation== | ||
This Fortran 90 implementation is adapted from Ref. | This Fortran 90 implementation is adapted from Ref. <ref>Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)</ref> The function '''ran()''' calls a [[Random_numbers | randon number generator]]: | ||
<small><pre> | <small><pre> | ||
! The following is taken from Allen & Tildesley, p. 349 | ! The following is taken from Allen & Tildesley, p. 349 | ||
| Line 41: | Line 40: | ||
==References== | ==References== | ||
<references/> | |||
'''Related reading''' | |||
* M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989) | |||
{{Source}} | |||
[[category: random numbers]] | [[category: random numbers]] | ||
[[category: computer simulation techniques]] | [[category: computer simulation techniques]] | ||
Revision as of 11:54, 21 October 2010
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 [1]
- Independently generate V1 and V2, taken from a uniform distribution on (-1,1) such that
- The random vector is then (Eq. 4 in [1] ):
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
- ↑ 1.0 1.1 George Marsaglia "Choosing a Point from the Surface of a Sphere", The Annals of Mathematical Statistics 43 pp. 645-646 (1972)
- ↑ Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications" p. 410 Academic Press (1996)
Related reading
- M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", p. 349 Clarendon Press (1989)
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