Let us consider a system consisting of three sub-volumes with N Ising spins, s_i

Question

Let us consider a system consisting of three sub-volumes with N Ising spins, s_i, L = plusminus 1, in each (i = 1, ...., N, L = 1, 2, 3). Here we neglect their interaction. The macrostate, M = (m_1, m_2, m_3), shall be defined by the total spins m_L = sigma_i = 1^N s_i, L of each sub-volume. Calculate the Boltzmann entropy S(M). Using this, obtain the equilibrium state M = (m_1, m_2, m_3)and its entropy S(M). Use Stirling's formula. Calculate the average quadratic fluctuations (m_1^2) of the total spin in sub-volume 1. Expand S(M) up to order m_1^2.

Explanation / Answer

a) from Boltzman's formula the entropy and microstates are in relation.

In simple format we can write S=Kln(m) where m is the microstate of the system.

Now considering total number of spins and two fundamental types up and down spins for total N particles we get

microstate m=2^N

considering all types of moments we get the function t(p)= N!/(N+p/2)^2!(N-p/2)^2!

where p is the difference between up and down moments

for large N we get t(p)=2^Nexp(-p^2/2N)

now for macrostate M we get M=2^N

then S= Kln2^N

using stirling formula we write S=NKln2

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