diff --git a/doc/src/Howto_bioFF.rst b/doc/src/Howto_bioFF.rst index 7418a43861..5aa151d5e7 100644 --- a/doc/src/Howto_bioFF.rst +++ b/doc/src/Howto_bioFF.rst @@ -73,7 +73,7 @@ with additional switching or shifting functions that ramp the energy and/or force smoothly to zero between an inner :math:`(a)` and outer :math:`(b)` cutoff. The older styles with *charmm* (not *charmmfsw* or *charmmfsh*\ ) in their name compute the LJ and Coulombic interactions -with an energy switching function (esw) S(r) which ramps the energy +with an energy switching function (esw) :math:`S(r)` which ramps the energy smoothly to zero between the inner and outer cutoff. This can cause irregularities in pairwise forces (due to the discontinuous second derivative of energy at the boundaries of the switching region), which diff --git a/doc/src/Howto_diffusion.rst b/doc/src/Howto_diffusion.rst index 5ee40d2cd3..f4af73ff4c 100644 --- a/doc/src/Howto_diffusion.rst +++ b/doc/src/Howto_diffusion.rst @@ -1,8 +1,8 @@ Calculate diffusion coefficients ================================ -The diffusion coefficient D of a material can be measured in at least -2 ways using various options in LAMMPS. See the examples/DIFFUSE +The diffusion coefficient :math:`D` of a material can be measured in at least +2 ways using various options in LAMMPS. See the ``examples/DIFFUSE`` directory for scripts that implement the 2 methods discussed here for a simple Lennard-Jones fluid model. @@ -12,7 +12,7 @@ of the MSD versus time is proportional to the diffusion coefficient. The instantaneous MSD values can be accumulated in a vector via the :doc:`fix vector ` command, and a line fit to the vector to compute its slope via the :doc:`variable slope ` function, and -thus extract D. +thus extract :math:`D`. The second method is to measure the velocity auto-correlation function (VACF) of the system, via the :doc:`compute vacf ` @@ -20,4 +20,4 @@ command. The time-integral of the VACF is proportional to the diffusion coefficient. The instantaneous VACF values can be accumulated in a vector via the :doc:`fix vector ` command, and time integrated via the :doc:`variable trap ` function, -and thus extract D. +and thus extract :math:`D`. diff --git a/doc/src/Howto_kappa.rst b/doc/src/Howto_kappa.rst index ca9a418d5e..d4dc5e92a8 100644 --- a/doc/src/Howto_kappa.rst +++ b/doc/src/Howto_kappa.rst @@ -1,20 +1,22 @@ Calculate thermal conductivity ============================== -The thermal conductivity kappa of a material can be measured in at -least 4 ways using various options in LAMMPS. See the examples/KAPPA +The thermal conductivity :math:`\kappa` of a material can be measured in at +least 4 ways using various options in LAMMPS. See the ``examples/KAPPA`` directory for scripts that implement the 4 methods discussed here for a simple Lennard-Jones fluid model. Also, see the :doc:`Howto viscosity ` page for an analogous discussion for viscosity. -The thermal conductivity tensor kappa is a measure of the propensity +The thermal conductivity tensor :math:`\mathbf{\kappa}` is a measure of the propensity of a material to transmit heat energy in a diffusive manner as given by Fourier's law -J = -kappa grad(T) +.. math:: -where J is the heat flux in units of energy per area per time and -grad(T) is the spatial gradient of temperature. The thermal + J = -\kappa \cdot \text{grad}(T) + +where :math:`J` is the heat flux in units of energy per area per time and +:math:`\text{grad}(T)` is the spatial gradient of temperature. The thermal conductivity thus has units of energy per distance per time per degree K and is often approximated as an isotropic quantity, i.e. as a scalar. @@ -49,7 +51,7 @@ details. The fourth method is based on the Green-Kubo (GK) formula which relates the ensemble average of the auto-correlation of the heat flux -to kappa. The heat flux can be calculated from the fluctuations of +to :math:`\kappa`. The heat flux can be calculated from the fluctuations of per-atom potential and kinetic energies and per-atom stress tensor in a steady-state equilibrated simulation. This is in contrast to the two preceding non-equilibrium methods, where energy flows continuously diff --git a/doc/src/Howto_structured_data.rst b/doc/src/Howto_structured_data.rst index 8b9c3dbc80..03f20e471a 100644 --- a/doc/src/Howto_structured_data.rst +++ b/doc/src/Howto_structured_data.rst @@ -341,7 +341,12 @@ data files and obtain a list of dictionaries. .. code-block:: - [{'timestep': 0, 'pe': -6.773368053259247, 'ke': 4.498875000000003}, {'timestep': 50, 'pe': -4.80824944183232, 'ke': 2.5257981827119798}, {'timestep': 100, 'pe': -4.787560887558151, 'ke': 2.5062598821985103}, {'timestep': 150, 'pe': -4.747103368600548, 'ke': 2.46609592554545}, {'timestep': 200, 'pe': -4.750905285854413, 'ke': 2.4701136792591694}, {'timestep': 250, 'pe': -4.777432735632181, 'ke': 2.4962152903997175}] + [{'timestep': 0, 'pe': -6.773368053259247, 'ke': 4.498875000000003}, + {'timestep': 50, 'pe': -4.80824944183232, 'ke': 2.5257981827119798}, + {'timestep': 100, 'pe': -4.787560887558151, 'ke': 2.5062598821985103}, + {'timestep': 150, 'pe': -4.747103368600548, 'ke': 2.46609592554545}, + {'timestep': 200, 'pe': -4.750905285854413, 'ke': 2.4701136792591694}, + {'timestep': 250, 'pe': -4.777432735632181, 'ke': 2.4962152903997175}] Line Delimited JSON (LD-JSON) ----------------------------- @@ -352,7 +357,8 @@ Each line represents one JSON object. .. code-block:: LAMMPS - fix extra all print 50 """{"timestep": $(step), "pe": $(pe), "ke": $(ke)}""" title "" file output.json screen no + fix extra all print 50 """{"timestep": $(step), "pe": $(pe), "ke": $(ke)}""" & + title "" file output.json screen no .. code-block:: json :caption: output.json diff --git a/doc/src/Howto_viscosity.rst b/doc/src/Howto_viscosity.rst index 3c97628179..2d61d782f5 100644 --- a/doc/src/Howto_viscosity.rst +++ b/doc/src/Howto_viscosity.rst @@ -1,22 +1,24 @@ Calculate viscosity =================== -The shear viscosity eta of a fluid can be measured in at least 6 ways -using various options in LAMMPS. See the examples/VISCOSITY directory +The shear viscosity :math:`\eta` of a fluid can be measured in at least 6 ways +using various options in LAMMPS. See the ``examples/VISCOSITY`` directory for scripts that implement the 5 methods discussed here for a simple Lennard-Jones fluid model and 1 method for SPC/E water model. Also, see the :doc:`page on calculating thermal conductivity ` for an analogous discussion for thermal conductivity. -Eta is a measure of the propensity of a fluid to transmit momentum in +:math:`\eta` is a measure of the propensity of a fluid to transmit momentum in a direction perpendicular to the direction of velocity or momentum flow. Alternatively it is the resistance the fluid has to being sheared. It is given by -J = -eta grad(Vstream) +.. math:: -where J is the momentum flux in units of momentum per area per time. -and grad(Vstream) is the spatial gradient of the velocity of the fluid + J = -\eta \cdot \text{grad}(V_{\text{stream}}) + +where :math:`J` is the momentum flux in units of momentum per area per time. +and :math:`\text{grad}(V_{\text{stream}})` is the spatial gradient of the velocity of the fluid moving in another direction, normal to the area through which the momentum flows. Viscosity thus has units of pressure-time. @@ -38,11 +40,11 @@ velocity to prevent the fluid from heating up. In both cases, the velocity profile setup in the fluid by this procedure can be monitored by the :doc:`fix ave/chunk ` -command, which determines grad(Vstream) in the equation above. -E.g. the derivative in the y-direction of the Vx component of fluid -motion or grad(Vstream) = dVx/dy. The Pxy off-diagonal component of +command, which determines :math:`\text{grad}(V_{\text{stream}})` in the equation above. +E.g. the derivative in the y-direction of the :math:`V_x` component of fluid +motion or :math:`\text{grad}(V_{\text{stream}}) = \frac{\text{d} V_x}{\text{d} y}`. The :math:`P_{xy}` off-diagonal component of the pressure or stress tensor, as calculated by the :doc:`compute pressure ` command, can also be monitored, which -is the J term in the equation above. See the :doc:`Howto nemd ` page for details on NEMD simulations. +is the :math:`J` term in the equation above. See the :doc:`Howto nemd ` page for details on NEMD simulations. The third method is to perform a reverse non-equilibrium MD simulation using the :doc:`fix viscosity ` command which implements @@ -55,7 +57,7 @@ See the :doc:`fix viscosity ` command for details. The fourth method is based on the Green-Kubo (GK) formula which relates the ensemble average of the auto-correlation of the -stress/pressure tensor to eta. This can be done in a fully +stress/pressure tensor to :math:`\eta`. This can be done in a fully equilibrated simulation which is in contrast to the two preceding non-equilibrium methods, where momentum flows continuously through the simulation box.