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Problems in Statistical Mechanics

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3. Consider again the lattice of N spin-1/2 particles in an external homogeneous magnetic field, where each particle has two possible states: spin "down" with energy e = 0 and spin "up" with energy e = 1/2". The microstate of the system is specified by the energy states of all the particles, i.e. the list (e1, e2...

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We solve various problems in statistical mechanics pertaining to a system consisting of a finite number of particles with energy independently coupled by spin (up or down).

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(a) We have S = k log W(n) = k log NCn = k log (N!/n!(N-n)!). Using Sterling's approximation to the factorial, we have

S is approximately equal to k log (N^N e^-N/((n^n e^-n)((N-n)^(N-n) e^-(N-n)))
= k (N log N - n log n - (N-n) log(N-n))

Since E(n) = n/2, we have

S(E) = k (N log N - 2E log ...

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