.MCAD 304020000 1 74 461 0
.CMD PLOTFORMAT
0 0 1 1 1 0 0 1 1 
0 0 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 21 15 0 0 3 
.CMD FORMAT  rd=d ct=10 im=i et=3 zt=40 pr=3 mass length time charge temperature tr=0 vm=0
.CMD SET ORIGIN 0
.CMD SET TOL 0.001000000000000
.CMD SET PRNCOLWIDTH 8
.CMD SET PRNPRECISION 4
.CMD PRINT_SETUP 1.200000 1.218750 1.200000 1.200000 0
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.CMD DEFAULT_TEXT_PARPROPS 0 0 0
.CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables
.CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants
.CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text
.CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1
.CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2
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.CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5
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.CMD UNITS U=1
.CMD DIMENSIONS_ANALYSIS 0 0
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.TXT 2 1 7 0 0
Cg b73.000000,73.000000,475
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset0\fnil Century Gothic;}{\f2
\fcharset2\fnil Symbol;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}\par {\b \par               }{\f1\fs36\ul STATISTICAL 
PHYSICS COMPUTER LAB }{\f1\fs36\ul #5:\par \tab }{\f1\fs36        }{\f1
\fs36\ul  Bose-Einstein Ideal Gas}{\f1\fs32 \par }{\fs28 \par }We 
study the Bose-Einstein ideal gas of  {\f2 ¥}ltra-relativistic 
particles, {\f2 e} = pc, in three dimensions.  First we define the 
constants.  Then we define the set of F functions that are used in the 
Bose-Einstein theory.  {\object{\*\objclass \eqn}\rsltpict{\*\objdata 
.EQN 24 37 440 0 0
{0:F}NAME({0:m}NAME,{0:\x}NAME):(1)/({0:\G}NAME({0:m}NAME))*(0&{0:\¥}NAME`(({0:x}NAME)^({0:m}NAME-1))/(({0:e}NAME)^({0:x}NAME)*({0:\x}NAME)^(-1)-1)&{0:x}NAME)
}}  where {\f2 x} < 1 \par It can be shown that: {\object{\*\objclass 
\eqn}\rsltpict{\*\objdata 
.EQN 33 18 447 0 0
{0:F}NAME({0:m}NAME,{0:\x}NAME):((1,{0:\¥}NAME,{0:i}NAME,(({0:\x}NAME)^({0:i}NAME))/(({0:i}NAME)^({0:m}NAME))){64})
}} We start with the energy density U(T, {\f2 x})/V and the density of 
bosons N(T{\f2 ,}{\f2 x})/V{\f2  }where {\f2 x  }is the fugacity. }
.EQN 40 0 119 0 0
{0:c}NAME:3*(10)^(8)
.EQN 0 8 120 0 0
{0:k.B}NAME:1.381*(10)^(-23)
.EQN 0 13 121 0 0
{0:hbar}NAME:1.05457*(10)^(-34)
.EQN 0 18 193 0 0
{0:h}NAME:{0:hbar}NAME*2*{0:\p}NAME
.EQN 0 17 334 0 0
{0:g}NAME:1
.EQN 6 -56 349 0 0
{0:F}NAME({0:m}NAME,{0:\x}NAME):((1,100,{0:i}NAME,(({0:\x}NAME)^({0:i}NAME))/(({0:i}NAME)^({0:m}NAME))){64})
.TXT 6 0 264 0 0
Cg a72.000000,72.000000,38
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The energy per unit 
volume u = U/V is:}
.EQN 0 35 455 0 0
{0:u}NAME({0:T}NAME,{0:\x}NAME):(24*{0:\p}NAME*({0:k.B}NAME)^(4))/(({0:c}NAME)^(3)*({0:h}NAME)^(3))*{0:g}NAME*({0:T}NAME)^(4)*{0:F}NAME(4,{0:\x}NAME)
.TXT 10 -34 367 0 0
Cg a72.000000,72.000000,31
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Since U = 3pV, the 
pressure is:}
.EQN 0 33 456 0 0
{0:p}NAME({0:T}NAME,{0:\x}NAME):(8*{0:\p}NAME*({0:k.B}NAME)^(4))/(({0:c}NAME)^(3)*({0:h}NAME)^(3))*{0:g}NAME*({0:T}NAME)^(4)*{0:F}NAME(4,{0:\x}NAME)
.TXT 9 -33 343 0 0
Cg a72.000000,72.000000,48
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard The number of bosons per 
unit volume n = N/V is:}
.EQN 0 45 457 0 0
{0:n}NAME({0:T}NAME,{0:\x}NAME):(8*{0:\p}NAME*({0:k.B}NAME)^(3))/(({0:c}NAME)^(3)*({0:h}NAME)^(3))*{0:g}NAME*({0:T}NAME)^(3)*{0:F}NAME(3,{0:\x}NAME)
.TXT 13 -44 374 0 0
Cg a71.000000,71.000000,131
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs24 
\pard The chemical potential is obtained from: exp({\f1 m/}k{\dn B}T) 
= {\f1 x}:\par \tab \tab \tab  {\object{\*\objclass \eqn}\rsltpict{
\*\objdata 
.EQN 88 21 385 0 0
{0:\m}NAME({0:T}NAME,{0:\x}NAME):{0:k.B}NAME*{0:T}NAME*{0:ln}NAME({0:\x}NAME)
}}\par \pard The entropy per volume s = S/V is obtained from the Euler 
equation: }
.EQN 12 23 458 0 0
{0:s}NAME({0:T}NAME,{0:\x}NAME):({0:u}NAME({0:T}NAME,{0:\x}NAME))/({0:T}NAME)+({0:p}NAME({0:T}NAME,{0:\x}NAME))/({0:T}NAME)-({0:\m}NAME({0:T}NAME,{0:\x}NAME)*{0:n}NAME({0:T}NAME,{0:\x}NAME))/({0:T}NAME)
.TXT 10 -23 429 0 0
Cg a71.000000,71.000000,100
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Consider a gas of 
density 10{\up 10} particles per cubic meter and calculate the 
condensation temperature:}
.EQN 6 8 420 0 0
{0:T}NAME:10.
.EQN 4 -1 419 0 0
{0:T.C}NAME:{0:root}NAME({0:n}NAME({0:T}NAME,1)-(10)^(10),{0:T}NAME)
.EQN 4 2 421 0 0
{0:T.C}NAME={0}?_n_u_l_l_
.TXT 3 -8 430 0 0
Cg a70.000000,70.000000,114
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs24 
\pard We compute the fugacity: for T > T{\dn C} it is the root of the 
equation n(T, {\f1 x})  = 10{\up 10} and for T <T{\dn C} it is equal 
to 1.}
.EQN 6 9 431 0 0
{0:\x}NAME:.9
.EQN 3 -3 398 0 0
{0:\x}NAME({0:T}NAME):{0:if}NAME({0:T}NAME>{0:T.C}NAME,{0:root}NAME({0:n}NAME({0:T}NAME,{0:\x}NAME)-(10)^(10),{0:\x}NAME),1)
.EQN 4 3 404 0 0
{0:T}NAME:0.1,0.2;20
.EQN 4 0 405 0 0
{0:\m}NAME({0:T}NAME):{0:k.B}NAME*{0:T}NAME*{0:ln}NAME({0:\x}NAME({0:T}NAME))
.EQN 4 -10 407 0 0
1*(10)^(-21)&&(_n_u_l_l_&_n_u_l_l_)&{0:\m}NAME({0:T}NAME)@20&&({0:T.C}NAME&_n_u_l_l_)&{0:T}NAME
0 0 1 1 1 0 1 1 1 
0 0 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 56 15 10 0 3 Chemical Potential vs Temperature
.EQN 25 1 408 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:p}NAME({0:T}NAME,{0:\x}NAME({0:T}NAME)),(10)^(10)*{0:k.B}NAME*{0:T}NAME@20&&({0:T.C}NAME&_n_u_l_l_)&{0:T}NAME
0 0 1 1 1 0 1 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 3 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 56 15 10 0 3 Equation of State: p vs T for fixed n
.TXT 28 -1 432 0 0
Cg a71.000000,71.000000,68
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Note that at high T the 
equation of state is the classical: p = nk{\dn B}T}
.TXT 3 -2 427 0 0
Cg b73.000000,73.000000,29
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}}
.EQN 6 3 410 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:s}NAME({0:T}NAME,{0:\x}NAME({0:T}NAME))@20&&({0:T.C}NAME&_n_u_l_l_)&{0:T}NAME
0 0 1 1 1 0 1 1 1 
0 0 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 56 15 10 0 3 Entropy vs Temperature
.TXT 24 -1 459 0 0
Cg a71.000000,71.000000,83
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Note that at absolute 
zero s = 0, in agreement with the 3'rd law of thermodynamics.}
.EQN 3 1 415 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:u}NAME({0:T}NAME,{0:\x}NAME({0:T}NAME)),3*{0:k.B}NAME*{0:T}NAME*(10)^(10)@20&0&({0:T.C}NAME&_n_u_l_l_)&{0:T}NAME
0 0 1 1 1 0 1 1 1 
0 0 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 56 15 10 0 3 Energy vs Temperature
.TXT 28 1 433 0 0
Cg a69.000000,69.000000,59
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard Note that at high T the 
energy is the classical: u = 3k{\dn B}Tn }
.EQN 4 -1 437 0 0
{0:c.V}NAME({0:T}NAME):{0:T}NAME"{0:u}NAME({0:T}NAME,{0:\x}NAME({0:T}NAME))
.EQN 6 0 434 0 0
&&(_n_u_l_l_&_n_u_l_l_)&{0:c.V}NAME({0:T}NAME),3*{0:k.B}NAME*(10)^(10)@20&&({0:T.C}NAME&_n_u_l_l_)&{0:T}NAME
0 0 1 1 1 0 1 1 1 
0 0 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 56 15 10 0 3 Isochoric Heat Capacity vs Temperature
.TXT 28 1 435 0 0
Cg a69.000000,69.000000,156
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset2\fnil Symbol;}}\plain\cf1\fs24 
\pard Note the singularity of C{\dn V }at the condensation 
temperature.  It looks like the greek letter {\f1\b l} so this phase 
transition is sometimes called the {\f1\b l} transition.}
.TXT 6 -4 298 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997\par }}
.TXT 647 0 282 0 0
Cg b73.000000,73.000000,30
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs24 \pard {\fs16 COPYRIGHT MIRON 
KAUFMAN, 1997}{\fs20 \par }}