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Louis Hoffenberg
2023-12-14 11:47:08 -05:00
parent 8d252d75cf
commit c78e2a5dd3
1 changed files with 23 additions and 10 deletions
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33
doc/src/fix_hmc.rst
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@ -33,10 +33,13 @@ Examples
Description
"""""""""""
This fix performs the the Hybrid/Hamiltonian Monte Carlo (HMC) algorithm in line with the following order of steps:
This fix performs the the Hybrid/Hamiltonian Monte Carlo (HMC)algorithm
in line with the following order of steps:
The new particle configuration (positions and velocities) is calculated by invoking the velocity-Verlet time integration algorithm.
Before these configuration changes are performed, the proposed change in the Hamiltonian,
The new particle configuration (positions and velocities) is calculated
by invoking the velocity-Verlet time integration algorithm.
Before these configuration changes are performed, the proposed change
in the Hamiltonian,
:math:`\Delta{H}`
is calculated following the equation:
@ -45,29 +48,39 @@ is calculated following the equation:
\Delta{H} = H(q′,p′) - H(q,p)
This new proposed configuration is then accepted/rejected according to the Metropolis criterion with probability:
This new proposed configuration is then accepted/rejected according to
the Metropolis criterion with probability:
.. math::
p^{acc} = min(1,e^{\frac{-\Delta{H}}{k_B T}})
Upon acceptance, the new proposed particle configuration positions and velocities are updated.
Upon rejection, the old particle configuration is kept, and particle momenta (and therefore velocities) are randomly resampled from a normal distribution:
Upon acceptance, the new proposed particle configuration positions and
velocities are updated. Upon rejection, the old particle configuration
is kept, and particle momenta (and therefore velocities) are randomly
resampled from a normal distribution:
.. math::
p_{x,y,z} = \textbf{N}(0,1) \sqrt{\frac{k_B T}{2 m^2}}
The algorithm then continues, proposing a new configuration of particles and velocities N integration steps later.
The algorithm then continues, proposing a new configuration of particles
and velocities N integration steps later.
----------
The keyword/value options are used in the following ways.
The *mom* keyword sets the linear momentum of the ensemble of particles. If mom = yes, the linear momentum of the ensemble of velocities is zeroed. If mom = no, the linear momentum of the ensemble of velocities is not zeroed.
The *ra* keyword decides whether velocities are resampled upon acceptance. If ra = yes, velocities are resampled upon acceptance. If ra = no, velocities are not resampled upon acceptance.
The *mom* keyword sets the linear momentum of the ensemble of particles.
If mom = yes, the linear momentum of the ensemble of velocities is
zeroed. If mom = no, the linear momentum of the ensemble of velocities
is not zeroed.
The *ra* keyword decides whether velocities are resampled upon acceptance.
If ra = yes, velocities are resampled upon acceptance. If ra = no,
velocities are not resampled upon acceptance.
----------
Restrictions
""""""""""""