CP 68: An M6800 Operating System (A Software source book) by Jack E Hemenway

By Jack E Hemenway

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Stanley [229]. According to him, the terminology refers only to the classical thermodynamic variables, disregarding the magnetic ones. , of a magnetic gas or fluid), and what we now call volume remains a particle number. Lattice gases and spin systems 36 without it, and the simple paradigmatic models it provoked (such as the Ising model), most of the progress of the last 80 years would not have been possible. Before we turn, in the next chapter, to the rigorous probabilistic setup of Gibbs measures for lattice spin systems, we will look at two singular situations that were studied in the early days of these models, and that gave rise to some confusion.

Assume that there are N1 , N2 molecules in each partition (possibly of different types of gases). Assume that the piston allows for energy to pass from one part to the other. Then the total energy E = E 1 + E 2 is conserved. We would like to know the probability distribution of the value E 1 . 33) In other words, the two systems are in equilibrium when the partial derivatives with respect to the energy of their entropies, the inverse temperatures, are the same. This is sometimes called the zeroth law of thermodynamics.

V dN p N i=1 pi2 2m 2 N d N q exp −β 0 V 1 p = dq exp −β dp N! 2m 0 1 N = V [2π m/β] N /2 N! 61) We see that this computation does not require knowledge of the formula for the surface area of the N -dimensional sphere, which we used in the micro-canonical formula. Stirling’s formula states that N! 63) Exercise: Compute the entropy for the one-dimensional gas from this formula. Compute the entropy directly from the micro-canonical partition function and compare. Do the same for the three-dimensional ideal gas.

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