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<div class="section" id="fix-ttm-command">
<span id="index-0"></span><h1>fix ttm command<a class="headerlink" href="#fix-ttm-command" title="Permalink to this headline"></a></h1>
</div>
<div class="section" id="fix-ttm-mod-command">
<h1>fix ttm/mod command<a class="headerlink" href="#fix-ttm-mod-command" title="Permalink to this headline"></a></h1>
<div class="section" id="syntax">
<h2>Syntax<a class="headerlink" href="#syntax" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre>fix ID group-ID ttm seed C_e rho_e kappa_e gamma_p gamma_s v_0 Nx Ny Nz T_infile N T_outfile
fix ID group-ID ttm/mod seed init_file Nx Ny Nz T_infile N T_outfile
</pre></div>
</div>
<ul class="simple">
<li>ID, group-ID are documented in <a class="reference internal" href="fix.html"><em>fix</em></a> command</li>
<li>style = <em>ttm</em> or <em>ttm_mod</em></li>
<li>seed = random number seed to use for white noise (positive integer)</li>
<li>remaining arguments for fix ttm:</li>
</ul>
<div class="highlight-python"><div class="highlight"><pre>C_e = electronic specific heat (energy/(electron*temperature) units)
<HR>
<H3>fix ttm command
</H3>
<H3>fix ttm/mod command
</H3>
<P><B>Syntax:</B>
</P>
<PRE>fix ID group-ID ttm seed C_e rho_e kappa_e gamma_p gamma_s v_0 Nx Ny Nz T_infile N T_outfile
fix ID group-ID ttm/mod seed init_file Nx Ny Nz T_infile N T_outfile
</PRE>
<UL><LI>ID, group-ID are documented in <A HREF = "fix.html">fix</A> command
<LI>style = <I>ttm</I> or <I>ttm_mod</I>
<LI>seed = random number seed to use for white noise (positive integer)
<LI>remaining arguments for fix ttm:
<PRE> C_e = electronic specific heat (energy/(electron*temperature) units)
rho_e = electronic density (electrons/volume units)
kappa_e = electronic thermal conductivity (energy/(time*distance*temperature) units)
gamma_p = friction coefficient due to electron-ion interactions (mass/time units)
@ -152,55 +37,54 @@ fix ID group-ID ttm/mod seed init_file Nx Ny Nz T_infile N T_outfile
Nz = number of thermal solve grid points in the z-direction (positive integer)
T_infile = filename to read initial electronic temperature from
N = dump TTM temperatures every this many timesteps, 0 = no dump
T_outfile = filename to write TTM temperatures to (only needed if N &gt; 0)
</pre></div>
</div>
<ul class="simple">
<li>remaining arguments for fix ttm/mod:</li>
</ul>
<div class="highlight-python"><div class="highlight"><pre>init_file = file with the parameters to TTM
T_outfile = filename to write TTM temperatures to (only needed if N > 0)
</PRE>
<LI>remaining arguments for fix ttm/mod:
<PRE> init_file = file with the parameters to TTM
Nx = number of thermal solve grid points in the x-direction (positive integer)
Ny = number of thermal solve grid points in the y-direction (positive integer)
Nz = number of thermal solve grid points in the z-direction (positive integer)
T_infile = filename to read initial electronic temperature from
N = dump TTM temperatures every this many timesteps, 0 = no dump
T_outfile = filename to write TTM temperatures to (only needed if N &gt; 0)
</pre></div>
</div>
</div>
<div class="section" id="examples">
<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre>fix 2 all ttm 699489 1.0 1.0 10 0.1 0.0 2.0 1 12 1 initialTs 1000 T.out
T_outfile = filename to write TTM temperatures to (only needed if N > 0)
</PRE>
</UL>
<P><B>Examples:</B>
</P>
<PRE>fix 2 all ttm 699489 1.0 1.0 10 0.1 0.0 2.0 1 12 1 initialTs 1000 T.out
fix 2 all ttm 123456 1.0 1.0 1.0 1.0 1.0 5.0 5 5 5 Te.in 1 Te.out
fix 2 all ttm/mod 34277 parameters.txt 5 5 5 T_init 10 T_out
</pre></div>
</div>
</div>
<div class="section" id="description">
<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline"></a></h2>
<p>Use a two-temperature model (TTM) to represent heat transfer through
fix 2 all ttm/mod 34277 parameters.txt 5 5 5 T_init 10 T_out
</PRE>
<P><B>Description:</B>
</P>
<P>Use a two-temperature model (TTM) to represent heat transfer through
and between electronic and atomic subsystems. LAMMPS models the
atomic subsystem as usual with a molecular dynamics model and the
classical force field specified by the user, but the electronic
subsystem is modeled as a continuum, or a background &#8220;gas&#8221;, on a
subsystem is modeled as a continuum, or a background "gas", on a
regular grid. Energy can be transferred spatially within the grid
representing the electrons. Energy can also be transferred between
the electronic and the atomic subsystems. The algorithm underlying
this fix was derived by D. M. Duffy and A. M. Rutherford and is
discussed in two J Physics: Condensed Matter papers: <a class="reference internal" href="#duffy"><span>(Duffy)</span></a>
and <a class="reference internal" href="#rutherford"><span>(Rutherford)</span></a>. They used this algorithm in cascade
discussed in two J Physics: Condensed Matter papers: <A HREF = "#Duffy">(Duffy)</A>
and <A HREF = "#Rutherford">(Rutherford)</A>. They used this algorithm in cascade
simulations where a primary knock-on atom (PKA) was initialized with a
high velocity to simulate a radiation event.</p>
<p>The description in this sub-section applies to both fix ttm and fix
high velocity to simulate a radiation event.
</P>
<P>The description in this sub-section applies to both fix ttm and fix
ttm/mod. Fix ttm/mod adds options to account for external heat
sources (e.g. at a surface) and for specifying parameters that allow
the electronic heat capacity to depend strongly on electronic
temperature. It is more expensive computationally than fix ttm
because it treats the thermal diffusion equation as non-linear. More
details on fix ttm/mod are given below.</p>
<p>Heat transfer between the electronic and atomic subsystems is carried
details on fix ttm/mod are given below.
</P>
<P>Heat transfer between the electronic and atomic subsystems is carried
out via an inhomogeneous Langevin thermostat. This thermostat differs
from the regular Langevin thermostat (<a class="reference internal" href="fix_langevin.html"><em>fix langevin</em></a>) in three important ways. First, the
from the regular Langevin thermostat (<A HREF = "fix_langevin.html">fix
langevin</A>) in three important ways. First, the
Langevin thermostat is applied uniformly to all atoms in the
user-specified group for a single target temperature, whereas the TTM
fix applies Langevin thermostatting locally to atoms within the
@ -211,57 +95,65 @@ reservoir, whereas the heat reservoir for fix TTM is finite and
represents the local electrons. Third, the TTM fix allows users to
specify not just one friction coefficient, but rather two independent
friction coefficients: one for the electron-ion interactions
(<em>gamma_p</em>), and one for electron stopping (<em>gamma_s</em>).</p>
<p>When the friction coefficient due to electron stopping, <em>gamma_s</em>, is
(<I>gamma_p</I>), and one for electron stopping (<I>gamma_s</I>).
</P>
<P>When the friction coefficient due to electron stopping, <I>gamma_s</I>, is
non-zero, electron stopping effects are included for atoms moving
faster than the electron stopping critical velocity, <em>v_0</em>. For
further details about this algorithm, see <a class="reference internal" href="#duffy"><span>(Duffy)</span></a> and
<a class="reference internal" href="#rutherford"><span>(Rutherford)</span></a>.</p>
<p>Energy transport within the electronic subsystem is solved according
faster than the electron stopping critical velocity, <I>v_0</I>. For
further details about this algorithm, see <A HREF = "#Duffy">(Duffy)</A> and
<A HREF = "#Rutherford">(Rutherford)</A>.
</P>
<P>Energy transport within the electronic subsystem is solved according
to the heat diffusion equation with added source terms for heat
transfer between the subsystems:</p>
<img alt="_images/fix_ttm.jpg" class="align-center" src="_images/fix_ttm.jpg" />
<p>where C_e is the specific heat, rho_e is the density, kappa_e is the
thermal conductivity, T is temperature, the &#8220;e&#8221; and &#8220;a&#8221; subscripts
transfer between the subsystems:
</P>
<CENTER><IMG SRC = "Eqs/fix_ttm.jpg">
</CENTER>
<P>where C_e is the specific heat, rho_e is the density, kappa_e is the
thermal conductivity, T is temperature, the "e" and "a" subscripts
represent electronic and atomic subsystems respectively, g_p is the
coupling constant for the electron-ion interaction, and g_s is the
electron stopping coupling parameter. C_e, rho_e, and kappa_e are
specified as parameters to the fix. The other quantities are derived.
The form of the heat diffusion equation used here is almost the same
as that in equation 6 of <a class="reference internal" href="#duffy"><span>(Duffy)</span></a>, with the exception that the
as that in equation 6 of <A HREF = "#Duffy">(Duffy)</A>, with the exception that the
electronic density is explicitly reprensented, rather than being part
of the the specific heat parameter.</p>
<p>Currently, fix ttm assumes that none of the user-supplied parameters
will vary with temperature. Note that <a class="reference internal" href="#duffy"><span>(Duffy)</span></a> used a tanh()
of the the specific heat parameter.
</P>
<P>Currently, fix ttm assumes that none of the user-supplied parameters
will vary with temperature. Note that <A HREF = "#Duffy">(Duffy)</A> used a tanh()
functional form for the temperature dependence of the electronic
specific heat, but ignored temperature dependencies of any of the
other parameters. See more discussion below for fix ttm/mod.</p>
<p>These fixes require use of periodic boundary conditions and a 3D
other parameters. See more discussion below for fix ttm/mod.
</P>
<P>These fixes require use of periodic boundary conditions and a 3D
simulation. Periodic boundary conditions are also used in the heat
equation solve for the electronic subsystem. This varies from the
approach of <a class="reference internal" href="#rutherford"><span>(Rutherford)</span></a> where the atomic subsystem was
approach of <A HREF = "#Rutherford">(Rutherford)</A> where the atomic subsystem was
embedded within a larger continuum representation of the electronic
subsystem.</p>
<p>The initial electronic temperature input file, <em>T_infile</em>, is a text
subsystem.
</P>
<P>The initial electronic temperature input file, <I>T_infile</I>, is a text
file LAMMPS reads in with no header and with four numeric columns
(ix,iy,iz,Temp) and with a number of rows equal to the number of
user-specified grid points (Nx by Ny by Nz). The ix,iy,iz are node
indices from 0 to nxnodes-1, etc. For example, the initial electronic
temperatures on a 1 by 2 by 3 grid could be specified in a <em>T_infile</em>
as follows:</p>
<div class="highlight-python"><div class="highlight"><pre>0 0 0 1.0
temperatures on a 1 by 2 by 3 grid could be specified in a <I>T_infile</I>
as follows:
</P>
<PRE>0 0 0 1.0
0 0 1 1.0
0 0 2 1.0
0 1 0 2.0
0 1 1 2.0
0 1 2 2.0
</pre></div>
</div>
<p>where the electronic temperatures along the y=0 plane have been set to
0 1 2 2.0
</PRE>
<P>where the electronic temperatures along the y=0 plane have been set to
1.0, and the electronic temperatures along the y=1 plane have been set
to 2.0. The order of lines in this file is no important. If all the
nodal values are not specified, LAMMPS will generate an error.</p>
<p>The temperature output file, <em>T_oufile</em>, is created and written by
nodal values are not specified, LAMMPS will generate an error.
</P>
<P>The temperature output file, <I>T_oufile</I>, is created and written by
this fix. Temperatures for both the electronic and atomic subsystems
at every node and every N timesteps are output. If N is specified as
zero, no output is generated, and no output filename is needed. The
@ -271,66 +163,83 @@ The next Nx*Ny*Nz columns contain the temperatures for the atomic
subsystem, and the final Nx*Ny*Nz columns contain the temperatures for
the electronic subsystem. The ordering of the Nx*Ny*Nz columns is
with the z index varing fastest, y the next fastest, and x the
slowest.</p>
<p>These fixes do not change the coordinates of their atoms; they only
scales their velocities. Thus a time integration fix (e.g. <a class="reference internal" href="fix_nve.html"><em>fix nve</em></a>) should still be used to time integrate the affected
slowest.
</P>
<P>These fixes do not change the coordinates of their atoms; they only
scales their velocities. Thus a time integration fix (e.g. <A HREF = "fix_nve.html">fix
nve</A>) should still be used to time integrate the affected
atoms. The fixes should not normally be used on atoms that have their
temperature controlled by another fix - e.g. <a class="reference internal" href="fix_nh.html"><em>fix nvt</em></a> or
<a class="reference internal" href="fix_langevin.html"><em>fix langevin</em></a>.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">The current implementations of these fixes create a
temperature controlled by another fix - e.g. <A HREF = "fix_nh.html">fix nvt</A> or
<A HREF = "fix_langevin.html">fix langevin</A>.
</P>
<P>IMPORTANT NOTE: The current implementations of these fixes create a
copy of the electron grid that overlays the entire simulation domain,
for each processor. Values on the grid are summed across all
processors. Thus you should insure that this grid is not too large,
else your simulation could incur high memory and communication costs.</p>
</div>
<hr class="docutils" />
<p><strong>Additional details for fix ttm/mod</strong></p>
<p>Fix ttm/mod uses the heat diffusion equation with possible external
heat sources (e.g. laser heating in ablation simulations):</p>
<img alt="_images/fix_ttm_mod.jpg" class="align-center" src="_images/fix_ttm_mod.jpg" />
<p>where theta is the Heaviside step function, I_0 is the (absorbed)
laser pulse intensity for ablation simulations, l_skin is the depth
of skin-layer, and all other designations have the same meaning as in
else your simulation could incur high memory and communication costs.
</P>
<HR>
<P><B>Additional details for fix ttm/mod</B>
</P>
<P>Fix ttm/mod uses the heat diffusion equation with possible external
heat sources (e.g. laser heating in ablation simulations):
</P>
<CENTER><IMG SRC = "Eqs/fix_ttm_mod.jpg">
</CENTER>
<P>where theta is the Heaviside step function, I_0 is the (absorbed)
laser pulse intensity for ablation simulations, l_skin is the depth
of skin-layer, and all other designations have the same meaning as in
the former equation. The duration of the pulse is set by the parameter
<em>tau</em> in the <em>init_file</em>.</p>
<p>Fix ttm/mod also allows users to specify the dependencies of C_e and
<I>tau</I> in the <I>init_file</I>.
</P>
<P>Fix ttm/mod also allows users to specify the dependencies of C_e and
kappa_e on the electronic temperature. The specific heat is expressed
as</p>
<img alt="_images/fix_ttm_ce.jpg" class="align-center" src="_images/fix_ttm_ce.jpg" />
<p>where <em>X</em> = T_e/1000, and the thermal conductivity is defined as
as
</P>
<CENTER><IMG SRC = "Eqs/fix_ttm_ce.jpg">
</CENTER>
<P>where <I>X</I> = T_e/1000, and the thermal conductivity is defined as
kappa_e = D_e*rho_e*C_e, where D_e is the thermal diffusion
coefficient.</p>
<p>Electronic pressure effects are included in the TTM model to account
coefficient.
</P>
<P>Electronic pressure effects are included in the TTM model to account
for the blast force acting on ions because of electronic pressure
gradient (see <a class="reference internal" href="#chen"><span>(Chen)</span></a>, <a class="reference internal" href="#norman"><span>(Norman)</span></a>). The total force
acting on an ion is:</p>
<img alt="_images/fix_ttm_blast.jpg" class="align-center" src="_images/fix_ttm_blast.jpg" />
<p>where F_langevin is a force from Langevin thermostat simulating
gradient (see <A HREF = "Chen">(Chen)</A>, <A HREF = "#Norman">(Norman)</A>). The total force
acting on an ion is:
</P>
<CENTER><IMG SRC = "Eqs/fix_ttm_blast.jpg">
</CENTER>
<P>where F_langevin is a force from Langevin thermostat simulating
electron-phonon coupling, and nabla P_e/n_ion is the electron blast
force.</p>
<p>The electronic pressure is taken to be P_e = B*rho_e*C_e*T_e</p>
<p>The current fix ttm/mod implementation allows TTM simulations with a
force.
</P>
<P>The electronic pressure is taken to be P_e = B*rho_e*C_e*T_e
</P>
<P>The current fix ttm/mod implementation allows TTM simulations with a
vacuum. The vacuum region is defined as the grid cells with zero
electronic temperature. The numerical scheme does not allow energy
exchange with such cells. Since the material can expand to previously
unoccupied region in some simulations, the vacuum border can be
allowed to move. It is controlled by the <em>surface_movement</em> parameter
in the <em>init_file</em>. If it is set to 1, then &#8220;vacuum&#8221; cells can be
changed to &#8220;electron-filled&#8221; cells with the temperature <em>T_e_min</em> if
allowed to move. It is controlled by the <I>surface_movement</I> parameter
in the <I>init_file</I>. If it is set to 1, then "vacuum" cells can be
changed to "electron-filled" cells with the temperature <I>T_e_min</I> if
atoms move into them (currently only implemented for the case of
1-dimensional motion of flat surface normal to the X axis). The
initial borders of vacuum can be set in the <em>init_file</em> via <em>lsurface</em>
and <em>rsurface</em> parameters. In this case, electronic pressure gradient
is calculated as</p>
<img alt="_images/fix_ttm_blast1.jpg" class="align-center" src="_images/fix_ttm_blast1.jpg" />
<p>where lambda is the electron mean free path (see <a class="reference internal" href="#norman"><span>(Norman)</span></a>,
<a class="reference internal" href="#pisarev"><span>(Pisarev)</span></a>)</p>
<p>The fix ttm/mod parameter file <em>init_file</em> has the following syntax/
initial borders of vacuum can be set in the <I>init_file</I> via <I>lsurface</I>
and <I>rsurface</I> parameters. In this case, electronic pressure gradient
is calculated as
</P>
<CENTER><IMG SRC = "Eqs/fix_ttm_blast1.jpg">
</CENTER>
<P>where lambda is the electron mean free path (see <A HREF = "#Norman">(Norman)</A>,
<A HREF = "#Pisarev">(Pisarev)</A>)
</P>
<P>The fix ttm/mod parameter file <I>init_file</I> has the following syntax/
Every line with the odd number is considered as a comment and
ignored. The lines with the even numbers are treated as follows:</p>
<div class="highlight-python"><div class="highlight"><pre>a_0, energy/(temperature*electron) units
ignored. The lines with the even numbers are treated as follows:
</P>
<PRE>a_0, energy/(temperature*electron) units
a_1, energy/(temperature^2*electron) units
a_2, energy/(temperature^3*electron) units
a_3, energy/(temperature^4*electron) units
@ -351,27 +260,31 @@ B, dimensionless
lambda, length units
n_ion, ions/volume units
surface_movement: 0 to disable tracking of surface motion, 1 to enable
T_e_min, temperature units
</pre></div>
</div>
</div>
<hr class="docutils" />
<div class="section" id="restart-fix-modify-output-run-start-stop-minimize-info">
<h2>Restart, fix_modify, output, run start/stop, minimize info<a class="headerlink" href="#restart-fix-modify-output-run-start-stop-minimize-info" title="Permalink to this headline"></a></h2>
<p>These fixes write the state of the electronic subsystem and the energy
exchange between the subsystems to <a class="reference internal" href="restart.html"><em>binary restart files</em></a>. See the <a class="reference internal" href="read_restart.html"><em>read_restart</em></a> command
T_e_min, temperature units
</PRE>
<HR>
<P><B>Restart, fix_modify, output, run start/stop, minimize info:</B>
</P>
<P>These fixes write the state of the electronic subsystem and the energy
exchange between the subsystems to <A HREF = "restart.html">binary restart
files</A>. See the <A HREF = "read_restart.html">read_restart</A> command
for info on how to re-specify a fix in an input script that reads a
restart file, so that the operation of the fix continues in an
uninterrupted fashion.</p>
<p>Because the state of the random number generator is not saved in the
restart files, this means you cannot do &#8220;exact&#8221; restarts with this
uninterrupted fashion.
</P>
<P>Because the state of the random number generator is not saved in the
restart files, this means you cannot do "exact" restarts with this
fix, where the simulation continues on the same as if no restart had
taken place. However, in a statistical sense, a restarted simulation
should produce the same behavior.</p>
<p>None of the <a class="reference internal" href="fix_modify.html"><em>fix_modify</em></a> options are relevant to these
fixes.</p>
<p>Both fixes compute 2 output quantities stored in a vector of length 2,
which can be accessed by various <a class="reference internal" href="Section_howto.html#howto-15"><span>output commands</span></a>. The first quantity is the
should produce the same behavior.
</P>
<P>None of the <A HREF = "fix_modify.html">fix_modify</A> options are relevant to these
fixes.
</P>
<P>Both fixes compute 2 output quantities stored in a vector of length 2,
which can be accessed by various <A HREF = "Section_howto.html#howto_15">output
commands</A>. The first quantity is the
total energy of the electronic subsystem. The second quantity is the
energy transferred from the electronic to the atomic subsystem on that
timestep. Note that the velocity verlet integrator applies the fix ttm
@ -381,101 +294,57 @@ Consequently, the change in the atomic subsystem energy is lagged by
half a timestep relative to the change in the electronic subsystem
energy. As a result of this, users may notice slight fluctuations in
the sum of the atomic and electronic subsystem energies reported at
the end of the timestep.</p>
<p>The vector values calculated are &#8220;extensive&#8221;.</p>
<p>No parameter of the fixes can be used with the <em>start/stop</em> keywords
of the <a class="reference internal" href="run.html"><em>run</em></a> command. The fixes are not invoked during
<a class="reference internal" href="minimize.html"><em>energy minimization</em></a>.</p>
</div>
<div class="section" id="restrictions">
<h2>Restrictions<a class="headerlink" href="#restrictions" title="Permalink to this headline"></a></h2>
<p>Fix <em>ttm</em> is part of the MISC package. It is only enabled if LAMMPS
was built with that package. Fix <em>ttm/mod</em> is part of the USER-MISC
the end of the timestep.
</P>
<P>The vector values calculated are "extensive".
</P>
<P>No parameter of the fixes can be used with the <I>start/stop</I> keywords
of the <A HREF = "run.html">run</A> command. The fixes are not invoked during
<A HREF = "minimize.html">energy minimization</A>.
</P>
<P><B>Restrictions:</B>
</P>
<P>Fix <I>ttm</I> is part of the MISC package. It is only enabled if LAMMPS
was built with that package. Fix <I>ttm/mod</I> is part of the USER-MISC
package. It is only enabled if LAMMPS was built with that package.
See the <a class="reference internal" href="Section_start.html#start-3"><span>Making LAMMPS</span></a> section for more
info.</p>
<p>These fixes can only be used for 3d simulations and orthogonal
See the <A HREF = "Section_start.html#start_3">Making LAMMPS</A> section for more
info.
</P>
<P>These fixes can only be used for 3d simulations and orthogonal
simlulation boxes. You must also use periodic
<a class="reference internal" href="boundary.html"><em>boundary</em></a> conditions.</p>
</div>
<div class="section" id="related-commands">
<h2>Related commands<a class="headerlink" href="#related-commands" title="Permalink to this headline"></a></h2>
<p><a class="reference internal" href="fix_langevin.html"><em>fix langevin</em></a>, <a class="reference internal" href="fix_dt_reset.html"><em>fix dt/reset</em></a></p>
<p><strong>Default:</strong> none</p>
<hr class="docutils" />
<p id="duffy"><strong>(Duffy)</strong> D M Duffy and A M Rutherford, J. Phys.: Condens. Matter, 19,
016207-016218 (2007).</p>
<p id="rutherford"><strong>(Rutherford)</strong> A M Rutherford and D M Duffy, J. Phys.:
Condens. Matter, 19, 496201-496210 (2007).</p>
<p id="chen"><strong>(Chen)</strong> J Chen, D Tzou and J Beraun, Int. J. Heat
Mass Transfer, 49, 307-316 (2006).</p>
<p id="norman"><strong>(Norman)</strong> G E Norman, S V Starikov, V V Stegailov et al., Contrib.
Plasma Phys., 53, 129-139 (2013).</p>
<p id="pisarev"><strong>(Pisarev)</strong> V V Pisarev and S V Starikov, J. Phys.: Condens. Matter, 26,
475401 (2014).</p>
</div>
</div>
<A HREF = "boundary.html">boundary</A> conditions.
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "fix_langevin.html">fix langevin</A>, <A HREF = "fix_dt_reset.html">fix dt/reset</A>
</P>
<P><B>Default:</B> none
</P>
<HR>
<A NAME = "Duffy"></A>
</div>
</div>
<footer>
<P><B>(Duffy)</B> D M Duffy and A M Rutherford, J. Phys.: Condens. Matter, 19,
016207-016218 (2007).
</P>
<A NAME = "Rutherford"></A>
<hr/>
<P><B>(Rutherford)</B> A M Rutherford and D M Duffy, J. Phys.:
Condens. Matter, 19, 496201-496210 (2007).
</P>
<A NAME = "Chen"></A>
<div role="contentinfo">
<p>
&copy; Copyright .
</p>
</div>
Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/snide/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
<P><B>(Chen)</B> J Chen, D Tzou and J Beraun, Int. J. Heat
Mass Transfer, 49, 307-316 (2006).
</P>
<A NAME = "Norman"></A>
</footer>
<P><B>(Norman)</B> G E Norman, S V Starikov, V V Stegailov et al., Contrib.
Plasma Phys., 53, 129-139 (2013).
</P>
<A NAME = "Pisarev"></A>
</div>
</div>
</section>
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<P><B>(Pisarev)</B> V V Pisarev and S V Starikov, J. Phys.: Condens. Matter, 26,
475401 (2014).
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