Three magnetic systems in thermal contact

[absalonsen]

Homework Statement

"Consider three magnetic systems in thermal contact, each composed of N1, N2, N3 spins, +m or -m. Starting from s1 = s'1 + δ1 etc. calculate the average square deviation <s_i² of each system.

Homework Equations

s'1 is the most probable configuration for s1. N1+N2+N3 = N ; s1+s2+s3 = s

The Attempt at a Solution

I have multiplied each multiplicity function; g1(N1,s1).g2(N2,s2).g3(N3,s3) = g1g2g3exp{-(2s1²/N1)-(2s2²/N2) - (2s3²/N3)}

then i wrote s3 = s - s1 -s2

Then i took log of the equation from both side and took derivative with respect to s1 and s2 and set s1=s'1, s2=s'2. They have to be equal to zero since it is extremum.

I came to this point and i couldn't do anything more.

 

[_coyote_]

2s'1N2N3 + s'1N2 + s'1N3 = sN2N3 + sN2 + sN32s'2N1N3 + s'2N1 + s'2N3 = sN1N3 + sN1 + sN3Can anyone help me?