http://web.ift.uib.no/AMOS/PHYS208/phys208-2005.02.07/2005 boltzmann______________________________________________________________________________

http://web.ift.uib.no/AMOS/PHYS208/2006.02.08/2006 boltzmann______________________________________________________________________________

Boltzmann's factor Heat Capacity- Einstein Model Heat Capacity- Debye Model

Boltzmann's factor --> Derivation OF

(our drawings crashed in the lecture. Here are new ones .....)Boltzmann factor:

Here we have the various oscillators shown. Each has some amountof energy quanta (or coins in the game...)

(there are none with zero here..., but there should be) see below for the red plot

This part illustrates the energy exchanges - two oscillators exchange energy

from one having 3 units and the other one 10 units, they go to 7 units and 6 units.

The "transition probailities are indicated and reversed

Here we have the distributions shown: each af the energy contents has its own redindicator.

each af the energy contents has its own red box indicator. Its length show the count of oscillators having that energy content (ore the amount of moneyin the game)In the panel to the right, after the change, the old counts are indicated by pinkboxesThe proof of Boltzmann distribution: (from older lectures)

when equals

And obviously beta must be negative!

And now trying to repeat yesterday's (new handwritten derivations)

leads to _____________________________________________________

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and it is easy to see that BETA must be negative!(obviously, better readable than last year!)

Heat Capacity- Einstein Model _______________________________________http://web.ift.uib.no/AMOS/fys208/debye/einstein-note30.jpg

Heat Capacity- Debye Modelhttp://web.ift.uib.no/AMOS/fys208/debye/debye30.jpg

This explains why in three dimensions the "mode density" is different from one dimension case.

and one more picture:

Boltzmann game:The Game http://equal-opportunity-game.blogspot.com/ We get 20 thousand ten-cent coins (only $2000 in the game) and collect a group of thousand people who want to play. Each person gets twenty coins.

Now the game starts. It has very simple rules: the players meet person to person, each pair puts all their money together and divide them in any equal opportunity way, for example by using dice, playing cards, small games, or just any guessing-game. Anything goes, as far as it does not favour the stronger, the weaker, the more beautiful or those with social problems: simply equal opportunity, indeed just. The only thing which is not recommended is to agree that "just" is fifty-fifty – then nobody can gain in the game.

So as we start, John and Jane each have 20, they draw cards, Jane leaves with 35 and John now has only five. Jane engages Peter in the game, he already had 30, she 35, they used dice and their calculators on their cellphones, and now Jane has only 5, Peter leaves with 60.Leave Jane, she has too little, follow Peter. He meets Joan, she had 17, Peter had 60, they used again dice and calculators, but now Peter leaves with nothing, Joan has now 77 coins. Let us follow the poor Peter: he meets Mary who still had 22, they use cards, and now by chance, Peter leaves with 11 and Mary also keeps eleven coins.

After 5 hours or so the game ends and everybody can now keep the coins they have left or they won.The question is: what will be the wealth distribution in this equal opportunity game?

In fact, we have a simulation:http://web.ift.uib.no/AMOS/game/http://web.ift.uib.no/AMOS/game/sim1/

To be continued .....