From Newton to Mandelbrot by Dietrich Stauffer H. Eugene Stanley & Annick Lesne

Author:Dietrich Stauffer, H. Eugene Stanley & Annick Lesne
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

So N / V and P decrease proportionally to . Since mgh is the potential energy, this result agrees with the exponential function of (4.1).

A second argument is more formal: water in a two-litre flask behaves as it does in two separate one-litre flasks, as regards its internal properties, since the contribution of the surface to its total energy is negligible. The product of the probability for the left-hand litre and the probability for the right-hand litre is therefore equal to the probability for the two-litre flask. The energy of the two-litre flask is equal to the sum of the one-litre energies. Accordingly, since the probability depends only on the energy E, the equality must hold. This, however, is a characteristic of the exponential function. From this argument one also learns that (4.1) is valid only for large numbers of particles; for small numbers of molecules the surface effects neglected here become important.

In this argument we applied a fundamental result of probability calculations, that statistically independent probabilities multiply each other. Thus if half of the students have blue eyes and the other half have brown eyes, and if ten per cent of the students fail the examinations in Theoretical Physics at the end of the year, then the probability that a student has blue eyes and has failed is %. For according to modern knowledge Theoretical Physics has nothing to do with eye colour. If half of the students work harder than average, and the other half less hard than average, then the probability that a student is working harder than average and nevertheless fails in the examinations is significantly smaller than . The two probabilities are now no longer independent, but correlated: try it out!

In addition to these two traditional arguments, now a modern one (and not normal examination material). We let the computer itself calculate the probability. We take the Ising model from Sect. 2.​2.​2 on electrodynamics (in matter). Atomic spins were there oriented either upwards or downwards, which was simulated in that program by or , respectively. If all the spins are parallel to each other, then the energy is zero; each neighbouring pair of antiparallel spins makes a constant contribution to the energy. Therefore if, in the square lattice, a spin is surrounded by k anti-parallel neighbours, the resulting energy associated with this is proportional to k, and the probability is proportional to . On pure geometry there are just as many possibilities for as for (we need only reverse the spin in the middle), and also there are just as many for as for . Accordingly, if we determine the number , telling how often k anti-parallel spins occur in the simulation, then in equilibrium the ratio must correspond to and the ratio to if the axiom (4.1) and the program are correct: . In fact one finds this confirmed, provided one lets the computer run long enough for the initial deviations from equilibrium to become unimportant. (Still more impressive is the calculation in a cubic lattice, where we then find that , and .

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.