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fix multiple typesetting issues and make consistent

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This commit is contained in:
Axel Kohlmeyer
2022-09-14 23:26:01 -04:00
parent 69034167b8
commit aae96b9cb0
2 changed files with 70 additions and 71 deletions
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49
doc/src/pair_dpd.rst
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@ -56,8 +56,10 @@ field. This pairwise thermostat can be used in conjunction with any
:doc:`pair style <pair_style>`, and in leiu of per-particle thermostats
like :doc:`fix langevin <fix_langevin>` or ensemble thermostats like
Nose Hoover as implemented by :doc:`fix nvt <fix_nh>`. To use
*dpd/tstat* as a thermostat for another pair style, use the :doc:`pair_style hybrid/overlay <pair_hybrid>` command to compute both the desired
pair interaction and the thermostat for each pair of particles.
*dpd/tstat* as a thermostat for another pair style, use the
:doc:`pair_style hybrid/overlay <pair_hybrid>` command to compute both
the desired pair interaction and the thermostat for each pair of
particles.
For style *dpd*, the force on atom I due to atom J is given as a sum
of 3 terms
@ -68,29 +70,30 @@ of 3 terms
F^C = & A w(r) \\
F^D = & - \gamma w^2(r) (\hat{r_{ij}} \bullet \vec{v_{ij}}) \\
F^R = & \sigma w(r) \alpha (\Delta t)^{-1/2} \\
w(r) = & 1 - r/r_c
w(r) = & 1 - \frac{r}{r_c}
where :math:`F^C` is a conservative force, :math:`F^D` is a dissipative
force, and :math:`F^R` is a random force. :math:`r_{ij}` is a unit
vector in the direction :math:`r_i - r_j`, :math:`v_{ij}` is the vector
difference in velocities of the two atoms :math:`= \vec{v}_i -
\vec{v}_j`, :math:`\alpha` is a Gaussian random number with zero mean and
unit variance, dt is the timestep size, and w(r) is a weighting factor
that varies between 0 and 1. :math:`r_c` is the cutoff. :math:`\sigma`
is set equal to :math:`\sqrt{2 k_B T \gamma}`, where :math:`k_B` is the
Boltzmann constant and T is the temperature parameter in the pair_style
command.
force, and :math:`F^R` is a random force. :math:`\hat{r_{ij}}` is a
unit vector in the direction :math:`r_i - r_j`, :math:`\vec{v_{ij}}` is
the vector difference in velocities of the two atoms :math:`\vec{v}_i -
\vec{v}_j`, :math:`\alpha` is a Gaussian random number with zero mean
and unit variance, *dt* is the timestep size, and :math:`w(r)` is a
weighting factor that varies between 0 and 1. :math:`r_c` is the
pairwise cutoff. :math:`\sigma` is set equal to :math:`\sqrt{2 k_B T
\gamma}`, where :math:`k_B` is the Boltzmann constant and *T* is the
temperature parameter in the pair_style command.
For style *dpd/tstat*, the force on atom I due to atom J is the same
as the above equation, except that the conservative Fc term is
dropped. Also, during the run, T is set each timestep to a ramped
value from Tstart to Tstop.
For style *dpd/tstat*, the force on atom I due to atom J is the same as
the above equation, except that the conservative :math:`F^C` term is
dropped. Also, during the run, *T* is set each timestep to a ramped
value from *Tstart* to *Tstop*.
For style *dpd*, the pairwise energy associated with style *dpd* is
only due to the conservative force term Fc, and is shifted to be zero
at the cutoff distance Rc. The pairwise virial is calculated using
all 3 terms. For style *dpd/tstat* there is no pairwise energy, but
the last two terms of the formula make a contribution to the virial.
For style *dpd*, the pairwise energy associated with style *dpd* is only
due to the conservative force term :math:`F^C`, and is shifted to be
zero at the cutoff distance :math:`r_c`. The pairwise virial is
calculated using all 3 terms. For style *dpd/tstat* there is no
pairwise energy, but the last two terms of the formula make a
contribution to the virial.
For style *dpd*, the following coefficients must be defined for each
pair of atoms types via the :doc:`pair_coeff <pair_coeff>` command as in
@ -146,8 +149,8 @@ I,J pairs must be specified explicitly.
These pair styles do not support the :doc:`pair_modify <pair_modify>`
shift option for the energy of the pair interaction. Note that as
discussed above, the energy due to the conservative Fc term is already
shifted to be 0.0 at the cutoff distance Rc.
discussed above, the energy due to the conservative :math:`F^C` term is already
shifted to be 0.0 at the cutoff distance :math:`r_c`.
The :doc:`pair_modify <pair_modify>` table option is not relevant
for these pair styles.

92
doc/src/pair_dpd_fdt.rst
View File

@ -58,32 +58,27 @@ given as a sum of 3 terms
F^C = & A w(r) \\
F^D = & - \gamma w^2(r) (\hat{r_{ij}} \bullet \vec{v_{ij}}) \\
F^R = & \sigma w(r) \alpha (\Delta t)^{-1/2} \\
w(r) = & 1 - r/r_c
w(r) = & 1 - \frac{r}{r_c}
where :math:`F^C` is a conservative force, :math:`F^D` is a dissipative
force, and :math:`F^R` is a random force. :math:`r_{ij}` is a unit
vector in the direction :math:`r_i - r_j`, :math:`V_{ij} is the vector
difference in velocities of the two atoms :math:`= \vec{v}_i -
\vec{v}_j, :math:`\alpha` is a Gaussian random number with zero mean and
unit variance, dt is the timestep size, and w(r) is a weighting factor
that varies between 0 and 1. Rc is the cutoff. The weighting factor,
:math:`\omega_{ij}`, varies between 0 and 1, and is chosen to have the
following functional form:
force, and :math:`F^R` is a random force. :math:`\hat{r_{ij}}` is a
unit vector in the direction :math:`r_i - r_j`, :math:`\vec{v_{ij}}` is
the vector difference in velocities of the two atoms, :math:`\vec{v}_i -
\vec{v}_j`, :math:`\alpha` is a Gaussian random number with zero mean
and unit variance, *dt* is the timestep size, and :math:`w(r)` is a
weighting factor that varies between 0 and 1, :math:`r_c` is the
pairwise cutoff. Note that alternative definitions of the weighting
function exist, but would have to be implemented as a separate pair
style command.
.. math::
\omega_{ij} = 1 - \frac{r_{ij}}{r_{c}}
Note that alternative definitions of the weighting function exist, but
would have to be implemented as a separate pair style command.
For style *dpd/fdt*, the fluctuation-dissipation theorem defines :math:`\gamma`
to be set equal to :math:`\sigma^2/(2 T)`, where T is the set point
temperature specified as a pair style parameter in the above examples.
The following coefficients must be defined for each pair of atoms types
via the :doc:`pair_coeff <pair_coeff>` command as in the examples above,
or in the data file or restart files read by the
:doc:`read_data <read_data>` or :doc:`read_restart <read_restart>` commands:
For style *dpd/fdt*, the fluctuation-dissipation theorem defines
:math:`\gamma` to be set equal to :math:`\sigma^2/(2 T)`, where *T* is the
set point temperature specified as a pair style parameter in the above
examples. The following coefficients must be defined for each pair of
atoms types via the :doc:`pair_coeff <pair_coeff>` command as in the
examples above, or in the data file or restart files read by the
:doc:`read_data <read_data>` or :doc:`read_restart <read_restart>`
commands:
* A (force units)
* :math:`\sigma` (force\*time\^(1/2) units)
@ -94,9 +89,9 @@ cutoff is used.
Style *dpd/fdt/energy* is used to perform DPD simulations under
isoenergetic and isoenthalpic conditions. The fluctuation-dissipation
theorem defines :math:`\gamma` to be set equal to :math:`sigma^2/(2
\theta)`, where :math:theta` is the average internal temperature for the
pair. The particle internal temperature is related to the particle
theorem defines :math:`\gamma` to be set equal to :math:`\sigma^2/(2
\theta)`, where :math:`\theta` is the average internal temperature for
the pair. The particle internal temperature is related to the particle
internal energy through a mesoparticle equation of state (see :doc:`fix
eos <fix>`). The differential internal conductive and mechanical
energies are computed within style *dpd/fdt/energy* as:
@ -116,15 +111,15 @@ where
\sigma^{2}_{ij} = & 2\gamma_{ij}k_{B}\Theta_{ij} \\
\Theta_{ij}^{-1} = & \frac{1}{2}(\frac{1}{\theta_{i}}+\frac{1}{\theta_{j}})
:math:`\zeta_ij^q` is a second Gaussian random number with zero mean and unit
variance that is used to compute the internal conductive energy. The
fluctuation-dissipation theorem defines :math:`alpha^2` to be set
equal to :math:2k_B\kappa`, where :math:`\kappa` is the mesoparticle thermal
conductivity parameter. The following coefficients must be defined for
each pair of atoms types via the :doc:`pair_coeff <pair_coeff>`
command as in the examples above, or in the data file or restart files
read by the :doc:`read_data <read_data>` or :doc:`read_restart <read_restart>`
commands:
:math:`\zeta_ij^q` is a second Gaussian random number with zero mean and
unit variance that is used to compute the internal conductive
energy. The fluctuation-dissipation theorem defines :math:`alpha^2` to
be set equal to :math:`2k_B\kappa`, where :math:`\kappa` is the
mesoparticle thermal conductivity parameter. The following coefficients
must be defined for each pair of atoms types via the :doc:`pair_coeff
<pair_coeff>` command as in the examples above, or in the data file or
restart files read by the :doc:`read_data <read_data>` or
:doc:`read_restart <read_restart>` commands:
* A (force units)
* :math:`\sigma` (force\*time\^(1/2) units)
@ -135,23 +130,23 @@ The last coefficient is optional. If not specified, the global DPD
cutoff is used.
The pairwise energy associated with styles *dpd/fdt* and
*dpd/fdt/energy* is only due to the conservative force term Fc, and is
shifted to be zero at the cutoff distance Rc. The pairwise virial is
calculated using only the conservative term.
*dpd/fdt/energy* is only due to the conservative force term :math:`F^C`,
and is shifted to be zero at the cutoff distance :math:`r_c`. The
pairwise virial is calculated using only the conservative term.
The forces computed through the *dpd/fdt* and *dpd/fdt/energy* styles
can be integrated with the velocity-Verlet integration scheme or the
Shardlow splitting integration scheme described by :ref:`(Lisal) <Lisal3>`.
In the cases when these pair styles are combined with the
Shardlow splitting integration scheme described by :ref:`(Lisal)
<Lisal3>`. In the cases when these pair styles are combined with the
:doc:`fix shardlow <fix_shardlow>`, these pair styles differ from the
other dpd styles in that the dissipative and random forces are split
from the force calculation and are not computed within the pair style.
Thus, only the conservative force is computed by the pair style,
while the stochastic integration of the dissipative and random forces
are handled through the Shardlow splitting algorithm approach. The
Shardlow splitting algorithm is advantageous, especially when
performing DPD under isoenergetic conditions, as it allows
significantly larger timesteps to be taken.
Thus, only the conservative force is computed by the pair style, while
the stochastic integration of the dissipative and random forces are
handled through the Shardlow splitting algorithm approach. The Shardlow
splitting algorithm is advantageous, especially when performing DPD
under isoenergetic conditions, as it allows significantly larger
timesteps to be taken.
----------
@ -162,8 +157,9 @@ significantly larger timesteps to be taken.
Restrictions
""""""""""""
These commands are part of the DPD-REACT package. They are only
enabled if LAMMPS was built with that package. See the :doc:`Build package <Build_package>` page for more info.
These commands are part of the DPD-REACT package. They are only enabled
if LAMMPS was built with that package. See the :doc:`Build package
<Build_package>` page for more info.
Pair styles *dpd/fdt* and *dpd/fdt/energy* require use of the
:doc:`comm_modify vel yes <comm_modify>` option so that velocities are