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:

• Hardware: PCs now powerful enough for nontrivial simulations within 1-2 minutes
  $\longrightarrow$ Live lecture hall simulations are possible

• Theory: Chaos in small, low-dimensional systems
  $\longrightarrow$Phase space can be drawn

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

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

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

  

Figure 1: Projecting an equidistribution on the unit circle onto the x-axis

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 v_x

From 2 to 10²⁴ dimensions:

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

$$\frac{1}{\pi} (1-x^{2})^{-1/2} \;\;\; \; as \; above$$ p₂(x) =

$$\frac{1}{2} \;\;\;\; constant!$$ p₃(x) =

$$\frac{2}{\pi} (1-x^{2})^{1/2}$$ p₄(x) =

$$\frac{3}{4} (1-x^{2})$$ p₅(x) =

$$\dots$$

$$\frac{256}{63 \pi} (1-x^{2})^{9/2}$$ p₁₂(x) =

$$\dots$$

$\longrightarrow$ 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.

Simulation: Gas of N hard discs:

Applet Harddisks

Watch the density of the velocity component v_x 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:

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

• log V is additive;
• d log V/dE (also called temperature) governs the flow of energy between systems in contact.

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:

• Other ensembles: the exp(-E/kt) law again
• Equidistribution law (kT/2-law)
• Partition functions and free energies
• Simple approximations to transport coefficients
• How quantum counting modifies Boltzmann's distribution
• Principles of MC and MD simulation

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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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