refer to temperature more consistently with :math:T instead of *T*

This commit is contained in:
Axel Kohlmeyer
2022-08-23 05:26:07 -04:00
parent 0b22ad5189
commit 60e5ce0dc2
9 changed files with 48 additions and 44 deletions

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@ -41,20 +41,21 @@ The pressure is computed by the formula
P = \frac{N k_B T}{V} + \frac{1}{V d}\sum_{i=1}^{N'} \vec r_i \cdot \vec f_i
where *N* is the number of atoms in the system (see discussion of DOF
below), :math:`k_B` is the Boltzmann constant, *T* is the temperature, *d*
is the dimensionality of the system (2 for 2d, 3 for 3d), and *V* is the
system volume (or area in 2d). The second term is the virial, equal to
:math:`-dU/dV`, computed for all pairwise as well as 2-body, 3-body, 4-body,
many-body, and long-range interactions, where :math:`\vec r_i` and
:math:`\vec f_i` are the position and force vector of atom *i*, and the
dot indicates the dot product (scalar product). This is computed in parallel
for each sub-domain and then summed over all parallel processes. Thus
:math:`N'` necessarily includes atoms from neighboring sub-domains (so-called
ghost atoms) and the position and force vectors of ghost atoms are thus
included in the summation. Only when running in serial and without
periodic boundary conditions is :math:`N' = N` the number of atoms in the
system. :doc:`Fixes <fix>` that impose constraints (e.g., the :doc:`fix
shake <fix_shake>` command) may also contribute to the virial term.
below), :math:`k_B` is the Boltzmann constant, :math:`T` is the
temperature, *d* is the dimensionality of the system (2 for 2d, 3 for
3d), and *V* is the system volume (or area in 2d). The second term is
the virial, equal to :math:`-dU/dV`, computed for all pairwise as well
as 2-body, 3-body, 4-body, many-body, and long-range interactions, where
:math:`\vec r_i` and :math:`\vec f_i` are the position and force vector
of atom *i*, and the dot indicates the dot product (scalar product).
This is computed in parallel for each sub-domain and then summed over
all parallel processes. Thus :math:`N'` necessarily includes atoms from
neighboring sub-domains (so-called ghost atoms) and the position and
force vectors of ghost atoms are thus included in the summation. Only
when running in serial and without periodic boundary conditions is
:math:`N' = N` the number of atoms in the system. :doc:`Fixes <fix>`
that impose constraints (e.g., the :doc:`fix shake <fix_shake>` command)
may also contribute to the virial term.
A symmetric pressure tensor, stored as a 6-element vector, is also
calculated by this compute. The six components of the vector are

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@ -52,7 +52,7 @@ concentration of species *j* in particle *i*, :math:`u_j` is the
internal energy of species j, :math:`\Delta H_{f,j} is the heat of
formation of species *j*, N is the number of molecules represented
by the coarse-grained particle, :math:`k_B` is the Boltzmann constant,
and *T* is the temperature of the system. Additionally, it is
and :math:`T` is the temperature of the system. Additionally, it is
possible to modify the concentration-dependent particle internal
energy relation by adding an energy correction, temperature-dependent
correction, and/or a molecule-dependent correction. An energy

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@ -261,7 +261,7 @@ pressure of the fictitious gas reservoir by:
\mu^{id} = & k T \ln{\rho \Lambda^3} \\
= & k T \ln{\frac{\phi P \Lambda^3}{k_B T}}
where :math:`k_B` is the Boltzmann constant, *T* is the user-specified
where :math:`k_B` is the Boltzmann constant, :math:`T` is the user-specified
temperature, :math:`\rho` is the number density, *P* is the pressure,
and :math:`\phi` is the fugacity coefficient. The constant
:math:`\Lambda` is required for dimensional consistency. For all unit
@ -320,7 +320,7 @@ this will ensure roughly the same behavior whether or not the
*full_energy* option is used.
Inserted atoms and molecules are assigned random velocities based on
the specified temperature *T*. Because the relative velocity of all
the specified temperature :math:`T`. Because the relative velocity of all
atoms in the molecule is zero, this may result in inserted molecules
that are systematically too cold. In addition, the intramolecular
potential energy of the inserted molecule may cause the kinetic energy

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@ -81,11 +81,11 @@ the particle's velocity. The proportionality constant for each atom is
computed as :math:`\frac{m}{\mathrm{damp}}`, where *m* is the mass of the
particle and damp is the damping factor specified by the user.
:math:`F_r` is a force due to solvent atoms at a temperature *T*
:math:`F_r` is a force due to solvent atoms at a temperature :math:`T`
randomly bumping into the particle. As derived from the
fluctuation/dissipation theorem, its magnitude as shown above is
proportional to :math:`\sqrt{\frac{k_B T m}{dt~\mathrm{damp}}}`, where
:math:`k_B` is the Boltzmann constant, *T* is the desired temperature,
:math:`k_B` is the Boltzmann constant, :math:`T` is the desired temperature,
*m* is the mass of the particle, *dt* is the timestep size, and damp is
the damping factor. Random numbers are used to randomize the direction
and magnitude of this force as described in :ref:`(Dunweg) <Dunweg1>`,

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@ -88,20 +88,23 @@ target temperature Tt obtained from the following equation:
T_t - T = \frac{\left(\frac{1}{2}\left(P + P_0\right)\left(V_0 - V\right) + E_0 - E\right)}{N_{dof} k_B } = \Delta
where *T* and :math:`T_t` are the instantaneous and target temperatures,
*P* and :math:`P_0` are the instantaneous and reference pressures or axial stresses,
depending on whether hydrostatic or uniaxial compression is being
performed, *V* and :math:`V_0` are the instantaneous and reference volumes,
*E* and :math:`E_0` are the instantaneous and reference internal energy (potential
plus kinetic), :math:`N_{dof}` is the number of degrees of freedom used in the
definition of temperature, and :math:`k_B` is the Boltzmann constant. :math:`\Delta` is the
negative deviation of the instantaneous temperature from the target temperature.
When the system reaches a stable equilibrium, the value of :math:`\Delta` should
fluctuate about zero.
where :math:`T` and :math:`T_t` are the instantaneous and target
temperatures, *P* and :math:`P_0` are the instantaneous and reference
pressures or axial stresses, depending on whether hydrostatic or
uniaxial compression is being performed, *V* and :math:`V_0` are the
instantaneous and reference volumes, *E* and :math:`E_0` are the
instantaneous and reference internal energy (potential plus kinetic),
:math:`N_{dof}` is the number of degrees of freedom used in the
definition of temperature, and :math:`k_B` is the Boltzmann
constant. :math:`\Delta` is the negative deviation of the instantaneous
temperature from the target temperature. When the system reaches a
stable equilibrium, the value of :math:`\Delta` should fluctuate about
zero.
The values of :math:`E_0`, :math:`V_0`, and :math:`P_0` are the instantaneous values at the start of
the simulation. These can be overridden using the fix_modify keywords *e0*,
*v0*, and *p0* described below.
The values of :math:`E_0`, :math:`V_0`, and :math:`P_0` are the
instantaneous values at the start of the simulation. These can be
overridden using the fix_modify keywords *e0*, *v0*, and *p0* described
below.
----------

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@ -73,11 +73,11 @@ the particle's velocity. The proportionality constant for each atom is
computed as :math:`\frac{m}{\mathrm{damp}}`, where *m* is the mass of
the particle and damp is the damping factor specified by the user.
:math:`F_r` is a force due to solvent atoms at a temperature *T*
:math:`F_r` is a force due to solvent atoms at a temperature :math:`T`
randomly bumping into the particle. As derived from the
fluctuation/dissipation theorem, its magnitude as shown above is
proportional to :math:`\sqrt{\frac{k_B T m}{dt~\mathrm{damp}}}`, where
:math:`k_B` is the Boltzmann constant, *T* is the desired temperature,
:math:`k_B` is the Boltzmann constant, :math:`T` is the desired temperature,
*m* is the mass of the particle, *dt* is the timestep size, and damp is
the damping factor. Random numbers are used to randomize the direction
and magnitude of this force as described in :ref:`(Dunweg) <Dunweg5>`,

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@ -57,12 +57,12 @@ multiple times to adjust :math:`\gamma` for several atom types.
self-consistent.
In a Brownian dynamics context, :math:`\gamma = \frac{k_B T}{D}`, where
:math:`k_B =` Boltzmann's constant, *T* = temperature, and *D* = particle
diffusion coefficient. *D* can be written as :math:`\frac{k_B T}{3 \pi
\eta d}`, where :math:`\eta =` dynamic viscosity of the frictional fluid
and d = diameter of particle. This means :math:`\gamma = 3 \pi \eta d`,
and thus is proportional to the viscosity of the fluid and the particle
diameter.
:math:`k_B =` Boltzmann's constant, :math:`T` = temperature, and *D* =
particle diffusion coefficient. *D* can be written as :math:`\frac{k_B
T}{3 \pi \eta d}`, where :math:`\eta =` dynamic viscosity of the
frictional fluid and d = diameter of particle. This means :math:`\gamma
= 3 \pi \eta d`, and thus is proportional to the viscosity of the fluid
and the particle diameter.
In the current implementation, rather than have the user specify a
viscosity, :math:`\gamma` is specified directly in force/velocity units.

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@ -102,9 +102,9 @@ The excess chemical potential mu_ex is defined as:
\mu_{ex} = -kT \ln(<\exp(-(U_{N+1}-U_{N})/{k_B T})>)
where :math:`k_B` is the Boltzmann constant, *T* is the user-specified temperature,
U_N and U_{N+1} is the potential energy of the system with N and N+1
particles.
where :math:`k_B` is the Boltzmann constant, :math:`T` is the
user-specified temperature, :math:`U_N` and :math:`U_{N+1}` is the
potential energy of the system with :math:`N` and :math:`N+1` particles.
The *full_energy* option means that the fix calculates the total
potential energy of the entire simulated system, instead of just the

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@ -43,7 +43,7 @@ Style *ufm* computes pairwise interactions using the Uhlenbeck-Ford model (UFM)
where :math:`r_c` is the cutoff, :math:`\sigma` is a distance-scale and
:math:`\epsilon` is an energy-scale, i.e., a product of Boltzmann constant
:math:`k_B`, temperature *T* and the Uhlenbeck-Ford p-parameter which
:math:`k_B`, temperature :math:`T` and the Uhlenbeck-Ford p-parameter which
is responsible
to control the softness of the interactions :ref:`(Paula Leite2017) <PL1>`.
This model is useful as a reference system for fluid-phase free-energy calculations :ref:`(Paula Leite2016) <PL2>`.