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 -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
Live lecture hall simulations are possible

• Theory: Chaos in small, low-dimensional systems
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 . Then

 (1)

EXAMPLE: Let
be equidistributed: for . Using cartesian coordinates , we find for x (or y ) with the density (see Fig.
1)

 (2)

But 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) = p3(x) = p4(x) = p5(x) = p12(x) =

eventually approaches a Gaussian!

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

 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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Ludwig Boltzmann

Instead of Gibbs' 6N dimensions and 1 point per system, consider 6 dimensions and N points with coordinates
: this is Boltzmann's -space. The density 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
• 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''

Franz J. Vesely
2000-07-11