.ig
:!cat %|tbl|eqn|troff |dpost >%.ps;ghostview %.ps&
:!cat %|tbl|eqn|troff |dpost >%.ps
..
.\" setup for mm document
.\".nr S 14	\" point
.nr S 12	\" point
.nr S 14
.nr N 2		\" no page number on first page
.\".nr P 2	\" passing strange - but page # low by 1 (or 2?) if P -ne 2
.so /user2/addon/usr/lib/tmac/tmac.m
.so /user2/addon/usr/lib/tmac/tmac.pictures
.SA 0		\" ragged right
.nr Pt 2	\" indent paragraphs except after header, etc
.BS
'sp 0.425i	\" compensate for builtin vertical 0.425i offset (NEC feature?)
.BE		\" alternatively include sp -.425i in PH, but more complicated
.\"
.EQ
delim $$
.EN
.EQ
define qi	% q sub i %
define vi	% q dot sub i %
define ai	% q dotdot sub i %
.EN
.ce
\fBNOTES ON\fP
.ce
\fBMOLECULAR DYNAMICS SIMULATION\fP
.SP 2
.BP md_strategy.eps 4i 6i
.ce
SCHEME I -- STRATEGY
.SP 2
At each time step new values of the positions $qi$ and 
velocities $vi$ for each particle in the
system are calculated from the current values of the positions and
velocities.
.SP
The sequence of values of $qi$ and $vi$ define the functions
$qi ( t )$ and $vi ( t )$ and constitute the trajectory of the
system, i.e., the description of the motion of the system,
of how it changes and fluctuates in time.
.SP
The procedure for updating the $qi$ and $vi$ is
outlined in Scheme I.
Estimation of new values of the positions and velocities
requires, in addition to the current values of the positions and velocities,
values of the accelerations, $ai ~=~ d q dot sub i / d t$, for each 
particle.
.SP
The acceleration for the ith particle is calculated from Newton's equation
of motion,
.SP
.EQ (1)
ai ~=~ F sub i over m sub i
.EN
.SP
where $m sub i$ is the mass of the ith particle and $F sub i$ is
the force on it.
$F sub i$ is calculated as the gradient of the potential energy,
.SP
.EQ (2)
F sub i ~=~ - {{ partial V sub p }
over { partial q sub i } }
.EN
.SP
The potential function $V sub p$ takes into account interactions
between the various atoms or groups of the macromolecule.
It is a function of the coodinates, $q sub i$.
Typical potential functions are discussed in Chapter III of BKP and
on page 999 of vanG&B.
Because the positions $qi$ are functions of the time, $V sub p$, $F sub i$,
and $ai$ are also functions of the time and must be computed for each
new set of values of $qi$, i.e., for each time step.
.SP
There are various algorithms for calculating new values of the
positions and velocities from the current positions and velocities and
the accelerations.  For a brief discussion see page 51 of BKP and
page 1010 of vanG&B.
The leap-frog scheme is relatively simple (vanG&B, page 1010): 
the position at the end of the next time step is calculated as
.SP
.EQ (3)
q sub i ( t + DELTA t ) ~=~ q sub i ( t ) + 
q dot sub i ( t + {DELTA t} over 2 ) ~ DELTA t
.EN
.SP
The second term of the rhs of eqn (3) 
contains the new estimate of the velocity,
at the midpoint of the time step, calculated as
.SP
.EQ (4)
q dot sub i ( t + {DELTA t} over 2 ) ~=~ 
q dot sub i ( t - {DELTA t} over 2 ) + q dotdot sub i ( t ) ~ DELTA t
.EN
.SP
The first term on the rhs of eqn (4) is the current estimate of
the velocity, at the midpoint of the last time step;
the second rhs term contains the current estimate of the
acceleration calculated as described above.
.SP
Thus one has integrated Newton's equation of motion,
calculating new values for $qi$ and $vi$ for times
$DELTA t$ and $DELTA t / 2$ beyond the current time.
To obtain an accurate integration,
the time step $DELTA t$ is kept small, ca. $10 sup -15$ second,
which is small compared to the characteristic times for
atom motions.
.SP 3
The relationship between the kinetic energy of the system at time $t$,
$T sub k ( t )$,
the masses and velocities of the particles, $m sub i$ and $q dot sub i ( t )$,
and the temperature of the system at time $t$, $T ( t )$,
is discussed in BKP(page 34) and vanG&B (page 1008) and is
given by
.SP
.EQ (5)
T sub k ( t ) ~=~ 
sum from i=1 to N 1 over 2 m sub i q dot sub i sup 2 ( t ) ~=~
3 over 2 ~ N ~ k sub B ~ T ( t )
.EN
.SP
where N is the number of particles in the system,
or the number of degrees of freedom if there are constraints on the
motion.
.SP
The initial conditions chosen for the coordinates and velocities of each
atom are unlikely to be near the equilibrium values for the temperature
desired.
Equilibration of the system by interation for a number of time steps allows
calculation of the temperature corresponding to the initial conditions.
The velocities can then be scaled appropriately for bringing the system to
within some range of a specified temperature.
