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.TXT 3 21 -1074292889 0 0
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\plain\cf1\fs24 \pard {\cf2\f1\fs36 Ideal Gas Simulation}}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard \tab In this computer 
lab we will simulate a two-dimensional ideal gas. An example of such a 
system is helium adsorbed on evaporated silver films (see for example: 
T.W.Kenny, P.L.Richards, Phys.Rev.Lett.64,2386(1990).  An ideal gas is 
a sufficiently dilute gas so that the forces between molecules are 
negligible.  We are going to use animation to visualise the motion of 
the molecules.  Then we will illustrate the 2'nd law of 
thermodynamics: natural systems evolve to more disorder (higher 
entropy). We will also explore the influence of boundary conditions on 
small systems.\par \tab We start by setting the number of molecules 
and their initial positions.}
.TXT 25 2 -1074292951 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard The number of 
molecules is:}
.EQN 0 32 -1074292953 0 0
{0:N}NAME:49
.TXT 3 -32 -1074292950 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard We use the index i to 
label the molecules:}
.EQN 0 50 -1074292943 0 0
{0:i}NAME:0;{0:N}NAME-1
.TXT 3 -50 -1074292942 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
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\plain\cf1\fs24 \pard {\cf2\f1  }}
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\plain\cf1\fs24 \pard The time label is:{\cf2\f1   }}
.EQN 0 23 -1074292880 0 0
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.TXT 6 -25 -1074292916 0 0
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\plain\cf1\fs24 \pard {\cf2\f1 \tab }{\cf2 We arrange the N molecules 
in an ordered square array.  We chose this ordered configuration to 
better show that the system evolves to a disorderd configuration, thus 
demonstrating the essence of the second law of thermodynamics.}}
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\plain\cf1\fs24 \pard {\cf2\f1 \tab }{\cf2 We set limits to the outer 
region |x| < lim, |y| < lim.  The constant limg sets the margins for 
the graph which shows the positions of the N molecules.}}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard The initial 
configuration is shown below:}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard COPYRIGHT MIRON 
KAUFMAN 1997}
.EQN 2 0 -1074292991 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;\red0\green64\blue128;}{
\fonttbl{\f0\fcharset0\fnil Courier;}{\f1\fcharset0\fnil Arial;}{\f2
\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard {\cf2\f1 \tab }{\cf2 
The new position of a particule is:  }{\cf2 x}{\dn new}{\cf2  = x}{\dn old}{
\cf2  + }{\cf2\f2 D}{\cf2 x and y}{\dn new}{\cf2  = y}{\dn old}{\cf2  
+ }{\f2 D}{\cf2 y. Now }{\f2 D}{\cf2 x=vx}{\f2 D}{\cf2 t and }{\f2 D}{
\cf2 y=vy}{\f2 D}{\cf2 t. The velocities vx and vy}{\dn  }{\cf2 are 
random variables each normally distributed: vx~}{\cf2\b N(meanvx,sx) 
and }{\cf2 vy~}{\cf2\b N(meanvy,sy}){\cf2 . Most gases and liquids 
(not liquid crystals) are }{\cf2\i isotropic, }{\cf2 i.e. all 
directions are equivalent. Then the two variances are equal. The 
standard deviation is determined by two physical parameters: the 
temperature and the mass of the molecule,  }{\f2 s}{\cf2  = (k}{\dn B}{
\cf2 T/m)}{\up 1/2}{\cf2 .  In the absence of a driving force the mean 
velocity is zero.}}
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\plain\cf1\fs24 \pard {\cf2\f1  \tab }{\cf2 Now we have to decide what 
happens when the molecules get to the walls.  We are going to assume 
periodic bondary conditions: if the molecules wander off to the left, 
they reappear at the right, and }{\cf2\i vice versa}{\cf2 ; if the 
molecules get to the top wall they reappear at the bottom wall, and }{
\cf2\i vice versa.}}
.EQN 13 0 33 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard COPYRIGHT MIRON 
KAUFMAN 1997}
.EQN 1 0 -1074293101 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard The graph above shows 
the last configuration.\par \par {\b ASSIGNMENT1}: Use the Animation 
option of MathCad 6 to create a movie (*.mov file) to visualise the 
molecular motion.\par First set the FRAME constant: T = FRAME +1. Then 
select the graph. Click on VIEW and then on ANIMATE.  Give the upper 
bound for the FRAME, by choosing TO: 49, for example.  That means that 
your movie will have 49+1 = 50 frames. Then select the graph again.  
Click on SAVE AS. Give a name to the file. Then SAVE.   For more 
information go to HELP ANIMATION.  If you want to play your animation 
go to VIEW and then pick PLAYBACK.\par \pard {\b \par ASSIGNMENT2}: 
Run the ideal gas simulation with different boundary conditions: when 
a molecule goes beyond a wall it is set back on the same wall.  Which 
boundary conditions, these or the periodic ones, give the faster 
equilibration? }
.TXT 61 0 -1074292898 0 0
Cg a73.000000,73.000000,23
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard COPYRIGHT MIRON KAUFMAN}
.TXT 4 -1 -1074292896 0 0
Cg a74.000000,74.000000,919
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard The gas started in an 
ordered configuration and evolved to a disordered configuration.  This 
is the essence of the Second Law of Thermodynamics. Nature obeys thius 
law, but if you would like to violate it you should play the movie 
file backwards.  This is the analog of pushing back the toothpaste 
into the tube. \par \tab We are going next to make this visual 
impression  quantitative by using the concept of {\i entropy}.  
Entropy is the measure of disorder (or lack of information).  This 
concept was invented in the context of thermal physics by Carnot, 
Boltzmann and others.  Entropy is used by electrical engineers and 
computer scientistists to measure the amount of information.  It is 
also use by cognitive scientists to quantify memory.  Entropy is also 
used in Biology to quantify evolution and diversity of species. \par 
\tab We start by calculating the number of molecules in each of the 
four regions we divided the box into.   }
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0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.TXT 34 -39 -1074292894 0 0
Cg a73.000000,73.000000,23
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard COPYRIGHT MIRON KAUFMAN}
.TXT 4 0 -1074292892 0 0
Cg a73.000000,73.000000,81
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard We calculate next the 
frequency (probability) for visiting each of the 4 regions.}
.EQN 7 2 -1074293117 0 0
({0:P1}NAME)[({0:j}NAME):(({0:n1}NAME)[({0:j}NAME))/({0:N}NAME)
.EQN 0 14 -1074293039 0 0
({0:P2}NAME)[({0:j}NAME):(({0:n2}NAME)[({0:j}NAME))/({0:N}NAME)
.EQN 0 15 -1074293038 0 0
({0:P3}NAME)[({0:j}NAME):(({0:n3}NAME)[({0:j}NAME))/({0:N}NAME)
.EQN 0 17 -1074292887 0 0
({0:P4}NAME)[({0:j}NAME):(({0:n4}NAME)[({0:j}NAME))/({0:N}NAME)
.EQN 3 -48 -1074292888 0 0
&&(_n_u_l_l_&_n_u_l_l_)&({0:P1}NAME)[({0:j}NAME),({0:P2}NAME)[({0:j}NAME),({0:P3}NAME)[({0:j}NAME),({0:P4}NAME)[({0:j}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&{0:j}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 1 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 1 2 0 1 1 NO-TRACE-STRING
0 1 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 63 25 10 0 3 
.TXT 34 0 -1074292890 0 0
Cg a72.000000,72.000000,243
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard Note that P1 starts at 
one (because we started with all molecules in region 1) and evolves in 
time to about 0.25.  The other three probabilities start at zero and 
evolve to 0.25 each.\par \tab The entropy associated with the spatial 
configurations is:}
.EQN 10 4 -1074292886 0 0
({0:entropy}NAME)[({0:j}NAME):-(({0:P1}NAME)[({0:j}NAME)*{0:ln}NAME(({0:P1}NAME)[({0:j}NAME))+({0:P2}NAME)[({0:j}NAME)*{0:ln}NAME(({0:P2}NAME)[({0:j}NAME))+({0:P3}NAME)[({0:j}NAME)*{0:ln}NAME(({0:P3}NAME)[({0:j}NAME))+({0:P4}NAME)[({0:j}NAME)*{0:ln}NAME((
{0:P4}NAME)[({0:j}NAME)))
.EQN 2 -4 -1074292885 0 0
&&(_n_u_l_l_&_n_u_l_l_)&({0:entropy}NAME)[({0:j}NAME),{0:ln}NAME(4)@&0&(_n_u_l_l_&_n_u_l_l_)&{0:j}NAME
0 1 1 1 1 0 0 1 1 
0 1 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 61 22 10 0 3 
.TXT 31 0 -1074292884 0 0
Cg a73.000000,73.000000,370
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard Note that the entropy 
starts at zero, meaning order or complete information, and increases 
in time to ln(4), which corresponds to disorder or total lack of 
information.  This is the Second Law of thermodynamics: physical 
systems evolve in time to the state of maximum posssible entropy. The 
system equilibrates (reaches the maximum entropy) in a time of about 
100 units.}
.TXT 13 0 -1074292878 0 0
Cg a73.000000,73.000000,23
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Courier;}}\plain\cf1\fs24 \pard COPYRIGHT MIRON KAUFMAN}