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fix ti/spring command :h3

[Syntax:]

fix ID group-ID ti/spring K t_switch t_equil keyword value ... :pre

ID, group-ID are documented in "fix"_fix.html command :ulb,l
ti/spring = style name of this fix command :l
K = spring constant (force/distance units) :l
t_switch/t_equil = number of steps of the switching/equilibration procedure :l
zero or more keyword/value pairs may be appended to args :l
keyword = {function} :l
  {function} value = function-ID 
    function-ID = ID of the switching function (1 or 2) :pre
:ule

[Example:]

fix ref all ti/spring 50.0 2000 1000 function 2 :pre

[Description:]

This fix allows you to compute the free energy of solids by performing
a thermodynamic integration between the solid of interest and an
Einstein crystal "(Frenkel)"_#Frenkel. The thermodynamic integration
is performed using the nonequilibrium method of Adiabatic Switching
"(Watanabe,"_#Watanabe "de Koning96)"_#deKoning96.

A spring force is applied independently to each atom in the group to
tether it to its initial position. The initial position for each atom
is its location at the time the fix command was issued. More details
about the springs are available in "fix
spring/self"_fix_spring_self.html. The forces on the atoms are
dynamically scaled during the simulation, the rescaling is done in the
following manner:

:c,image(Eqs/fix_ti_spring_force.jpg)

where F_harm is the force due to the springs, F_solid is the total
force on the atoms due to the interatomic potential and lambda is the
coupling parameter of the thermodynamic integration.

The fix acts as follows: during the first {t_equil} steps after the
fix is defined the value of lambda is zero, this is the period to
equilibrate the system in the lambda = 0 state. After this the value
of lambda changes continuously from 0 to 1 according to the function
defined using the keyword {function} (described below), this is done
in {t_switch} steps. Then comes the second equilibration period of
{t_equil} to equilibrate the system in the lambda = 1 state. After
that the switching back to the lambda = 0 state is made using
{t_switch} timesteps and following the same switching function. After
this period the value of lambda is kept equal to zero and the fix has
no action in the dynamics of the system anymore.

The description of thermodynamic integration in both directions is
done in "de Koning97"_#deKoning97, the main reason is to try to
eliminate the dissipated heat due to the nonequilibrium process.

The {function} keyword allows the use of two different switching
rates, the option {1} results in a constant rescaling where the lambda
parameter changes at a constant rate during the switching time
according to the switching function

:c,image(Eqs/fix_ti_spring_function_1.jpg)

where tau is the scaled time variable t/t_switch. The option number
{2} performs the switching at a rate defined by the following
switching function

:c,image(Eqs/fix_ti_spring_function_2.jpg)

This function has zero slope as lambda approaches its extreme values
(0 and 1), according to ("de Koning96)"_#deKoning96 this results in
smaller fluctuations on the integral to be computed on the
thermodynamic integration.

IMPORTANT NOTE: It is importante to keep the center of mass fixed
during the thermodynamic integration, a non-zero total velocity will
result in divergencies during the integration due to the fact that the
atoms are 'attatched' to its equilibrium positions by the Einstein
crystal. Check the option {zero} of "fix langevin"_fix_langevin_html
and "velocity"_velocity.html. The use of the Nose-Hoover thermostat
("fix nvt"_fix_nvt.html) is NOT recommended due to its well documented
issues with the canonical sampling of harmonic degrees of freedom
(notice that the {chain} option will NOT solve this problem). The
Langevin thermostat ("fix langevin"_fix_langevin.html") works fine.

[Restart, fix_modify, output, run start/stop, minimize info:]

This fix writes the original coordinates of tethered atoms to "binary
restart files"_restart.html, so that the spring effect will be the
same in a restarted simulation. See the "read
restart"_read_restart.html 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.

The "fix modify"_fix_modify.html {energy} option is supported by this
fix to add the energy stored in the per-atom springs to the system's
potential energy as part of "thermodynamic output"_thermo_style.html.

This fix computes a global scalar and a global vector quantities which
can be accessed by various "output
commands"_Section_howto.html#howto_15.  The scalar is an energy which
is the sum of the spring energy for each atom, where the per-atom
energy is 0.5 * K * r^2. The vector has 2 positions, the first one is
the coupling parameter lambda and the second one is the time
derivative of lambda. The scalar and vector values calculated by this
fix are "extensive".

No parameter of this fix can be used with the {start/stop} keywords of
the "run"_run.html command.

The forces due to this fix are imposed during an energy minimization,
invoked by the "minimize"_minimize.html command.

IMPORTANT NOTE: If you want the per-atom spring energy to be included
in the total potential energy of the system (the quantity being
minimized), you MUST enable the "fix modify"_fix_modify.html {energy}
option for this fix.

An example script using this command is provided in the
examples/USER/misc/ti directory.

[Related commands:]

"fix spring"_fix_spring.html, "fix ti/rs"_fix_ti_rs.html

[Restrictions:]

This command is part of the USER-MISC package.  It is only enabled if
LAMMPS was built with those packages.  See the "Making
LAMMPS"_Section_start.html#start_3 section for more info.

[Default:]

The keyword default is function = 1.

:line

:link(Frenkel)
[(Frenkel)] Daan Frenkel and Anthony J. C. Ladd, J. Chem. Phys. 81, 3188
(1984).

:link(Watanabe)
[(Watanabe)] M. Watanabe and W. P. Reinhardt, Phys Rev Lett, 65, 3301 (1990).

:link(deKoning96)
[(de Koning 96)] M. de Koning and A. Antonelli, Phys Rev E, 53, 465 (1996).

:link(deKoning97)
[(de Koning 97)] M. de Koning and A. Antonelli, Phys Rev B, 55, 735 (1997).

