diff --git a/doc/src/compute_pressure.rst b/doc/src/compute_pressure.rst index 9b086abbee..c1a9e3d2ec 100644 --- a/doc/src/compute_pressure.rst +++ b/doc/src/compute_pressure.rst @@ -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 ` that impose constraints (e.g., the :doc:`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 ` +that impose constraints (e.g., the :doc:`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 diff --git a/doc/src/fix_eos_table_rx.rst b/doc/src/fix_eos_table_rx.rst index 52ffa3494f..c9f134f985 100644 --- a/doc/src/fix_eos_table_rx.rst +++ b/doc/src/fix_eos_table_rx.rst @@ -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 diff --git a/doc/src/fix_gcmc.rst b/doc/src/fix_gcmc.rst index e748f17ae7..62b858a0af 100644 --- a/doc/src/fix_gcmc.rst +++ b/doc/src/fix_gcmc.rst @@ -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 diff --git a/doc/src/fix_langevin.rst b/doc/src/fix_langevin.rst index 5979155799..50c1489d7c 100644 --- a/doc/src/fix_langevin.rst +++ b/doc/src/fix_langevin.rst @@ -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) `, diff --git a/doc/src/fix_nphug.rst b/doc/src/fix_nphug.rst index b45cfe964f..1cedfa909a 100644 --- a/doc/src/fix_nphug.rst +++ b/doc/src/fix_nphug.rst @@ -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. ---------- diff --git a/doc/src/fix_nve_dotc_langevin.rst b/doc/src/fix_nve_dotc_langevin.rst index c2835aaccd..e47e8f4a2a 100644 --- a/doc/src/fix_nve_dotc_langevin.rst +++ b/doc/src/fix_nve_dotc_langevin.rst @@ -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) `, diff --git a/doc/src/fix_viscous.rst b/doc/src/fix_viscous.rst index 45c31e10cf..6c48cb5f12 100644 --- a/doc/src/fix_viscous.rst +++ b/doc/src/fix_viscous.rst @@ -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. diff --git a/doc/src/fix_widom.rst b/doc/src/fix_widom.rst index 7c431a2c36..5be93100b3 100644 --- a/doc/src/fix_widom.rst +++ b/doc/src/fix_widom.rst @@ -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 diff --git a/doc/src/pair_ufm.rst b/doc/src/pair_ufm.rst index 22438d559b..5de0b91ee7 100644 --- a/doc/src/pair_ufm.rst +++ b/doc/src/pair_ufm.rst @@ -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) `. This model is useful as a reference system for fluid-phase free-energy calculations :ref:`(Paula Leite2016) `.