Boltzmann's Dream. Statistical Physics in the Introductory Course*)

Franz J. Vesely

Institute of Experimental Physics, University of Vienna




Outline:
 
1. New Didactic Tools 4. Entropy: Bridging the Gap
2. Exploring Phase Space 5. The Sophomore Phase Transition
3. Adventures in $\mu$-space 6. There is Nothing New ...

 

* This paper was read at the Gordon Research Conference on "Physics in Research and Education",
Plymouth, New Hampshire, June 11-16, 2000, under the title "Statistical Physics for the Sophomores"

 



  1. NEW DIDACTIC TOOLS


Teaching Statistical Physics to 2nd year students:

How to explain physics using little math?


Help comes from Hardware and Theory:


Therefore:

Use Live Simulation and Intuitive Geometry (of phase space) as didactic tools.



To give an example:

Applet Sinai

 


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  2. EXPLORING PHASE SPACE

Josiah Willard Gibbs
 
Simple but relevant model systems:

  • Classical ideal gas
  • Quantum ideal gas
  • Hard discs gas
  • Hard spheres gas
  • Lennard-Jonesium
  • Lattice of harmonic oscillators
  • Lattice of spins

    Most have hyperspherical phase space volumes (except LJ and spin lattice)


    Phase space:

  • Concept of energy surface / microcanonical ensemble
  • Phase space volume and thermodynamic probability

    Explore the simple but surprising properties of n-spheres




    Much physics in a little geometry:
    Let p(x) be the density of x, and let $y=f(x); \; x=f^{-1}(y)$ . Then

     \begin{displaymath}p(y)= p(x) \, \left\vert \frac{dx}{dy}\right\vert
= p[f^{-1}(y)] \, \left\vert \frac{d\,f^{-1}(y)}{dy} \right\vert
\end{displaymath} (1)



    EXAMPLE: Let
    $\phi$ be equidistributed: $p(\phi) = 1/2 \pi$ for $\phi \, \epsilon \, [ 0, 2 \pi ]$. Using cartesian coordinates $x = r \cos \phi$, $y = r \sin \phi$ we find for x (or y ) with $x \, \epsilon \, [\pm r]$ the density (see Fig.
    1)

    \begin{displaymath}p(x) = p(\phi) \left\vert \frac{d \phi}{dx} \right\vert
= \frac{1}{\pi} \frac{1}{\sqrt{r^{2}-x^{2}}}
\end{displaymath} (2)


      
    Figure 1: Projecting an equidistribution on the unit circle onto the x-axis
    \begin{figure}\parbox{12pt}{}
\epsfverbosetrue
\epsfxsize=174 pt
\epsffile{f1tr3...
...psfverbosetrue
\epsfxsize=174 pt
\epsffile{f1tr4.ps}\parbox{12pt}{}
\end{figure}

    But $p(\phi)$ is just the microcanonical distribution for a system with 2 degrees of freedom! Therefore p(x) is the distribution of one d.o.f.

    Simulation: 2-dimensional 1-particle ideal gas (a.k.a. Sinai Billiard)

    Applet Harddisks

    Note the density of the velocity component vx




    From 2 to 1024 dimensions:

    Projecting constant densities on hyperspherical surfaces down onto one axis we find, depending on dimension, (see Fig. 2)

    p2(x) = $\displaystyle \frac{1}{\pi} (1-x^{2})^{-1/2} \;\;\; \; as \; above$  
    p3(x) = $\displaystyle \frac{1}{2} \;\;\;\; constant!$  
    p4(x) = $\displaystyle \frac{2}{\pi} (1-x^{2})^{1/2}$  
    p5(x) = $\displaystyle \frac{3}{4} (1-x^{2})$  
      $\textstyle \dots$    
    p12(x) = $\displaystyle \frac{256}{63 \pi} (1-x^{2})^{9/2}$  
      $\textstyle \dots$    

    $\longrightarrow$ eventually approaches a Gaussian!

    Thus the Maxwell-Boltzmann distribution may be derived solely from the geometry of highdimensional spheres.


      
    Figure 2: Distribution p(x) along one axis of a n-dimensional sphere if its surface is homogeneously covered.

    \epsffile{f1px1.ps}



    Simulation: Gas of N hard discs:

    Applet Harddisks

    Watch the density of the velocity component vx as N is increased.




    Simulation: Gas of N hard spheres:

    Applet Hspheres





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    3. ADVENTURES IN $\mu$ -SPACE


    Ludwig Boltzmann
     

    Instead of Gibbs' 6N dimensions and 1 point per system, consider 6 dimensions and N points with coordinates
    $\{\mbox{\boldmath$\bf r$ }, \mbox{\boldmath$\bf v$ } \}$ : this is Boltzmann's $\mu$ -space. The density $f\left( \mbox{\boldmath$\bf r$ }, \mbox{\boldmath$\bf v$ } ; t \right)$ follows an equation that is too beautiful to be skipped but too heavy yet to be treated in detail:
         
      (3)  

    What we can do is
    • discuss the assumptions that enter
    • discuss its main components
    • give its long-time solution - the Boltzmann distribution again!
    • show a non-equilibrium simulation to demonstrate the scope of Boltzmann's equation

    Simulation: expansion of a packed hard discs system

    Applet Boltzmann


     


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    4. ENTROPY: BRIDGING THE GAP


  • We search for a microscopic correlate to entropy. What about the log-volume in phase space? We have to prove that
    Simulation: Hyper-rectangles and hyper-circles

    Applet Entropy1


     


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    5. THE SOPHOMORE PHASE TRANSITION

    [`moron'=nitwit, `sophos' = the wise one]

    In addition to the above, some further truths are attainable in the course of the third term:



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    6. THERE IS NOTHING NEW UNDER THE SUN

    In the 18th century, higher education, including mathematics, was reserved to men.

    The Italian art critic and `popular science writer' Francesco Algarotti found it unbearable that one half of humanity should not be able to grasp the impact of Newton's revolution.

    So he sat down and wrote a bestselling book in which he sketched Newton's ideas using plain language only. Instead of formulae he invoked similes and intuitive-geometrical arguments.

     

     

     

     

     

     

    Newtonianism for the Ladies:
    How to explain physics using absolutely no math

     

     

     

     

     

     

    Francesco Algarotti, the ''Swan of Padua''


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    Franz J. Vesely
    2000-07-11