diff --git a/doc/src/pair_dpd.rst b/doc/src/pair_dpd.rst index 5bc8cedaed..aef085ca6e 100644 --- a/doc/src/pair_dpd.rst +++ b/doc/src/pair_dpd.rst @@ -56,8 +56,10 @@ field. This pairwise thermostat can be used in conjunction with any :doc:`pair style `, and in leiu of per-particle thermostats like :doc:`fix langevin ` or ensemble thermostats like Nose Hoover as implemented by :doc:`fix nvt `. To use -*dpd/tstat* as a thermostat for another pair style, use the :doc:`pair_style hybrid/overlay ` 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 ` 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 ` command as in @@ -146,8 +149,8 @@ I,J pairs must be specified explicitly. These pair styles do not support the :doc:`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 ` table option is not relevant for these pair styles. diff --git a/doc/src/pair_dpd_fdt.rst b/doc/src/pair_dpd_fdt.rst index 25fe2581db..133e7ab52c 100644 --- a/doc/src/pair_dpd_fdt.rst +++ b/doc/src/pair_dpd_fdt.rst @@ -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 ` command as in the examples above, -or in the data file or restart files read by the -:doc:`read_data ` or :doc:`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 ` command as in the +examples above, or in the data file or restart files read by the +:doc:`read_data ` or :doc:`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 `). 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 ` -command as in the examples above, or in the data file or restart files -read by the :doc:`read_data ` or :doc:`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 +` command as in the examples above, or in the data file or +restart files read by the :doc:`read_data ` or +:doc:`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) `. -In the cases when these pair styles are combined with the +Shardlow splitting integration scheme described by :ref:`(Lisal) +`. In the cases when these pair styles are combined with the :doc:`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 ` 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 +` page for more info. Pair styles *dpd/fdt* and *dpd/fdt/energy* require use of the :doc:`comm_modify vel yes ` option so that velocities are