00:04
Now we're asked to answer a question about the boltzmin distribution.
00:10
So this is actually a generalization of the previous exercise, exercise 51 to n variables.
00:21
So we want to show that there is a constant mu, such that the maximum of the entry function, s, subject to the constraints that the sum of x1 through xn is equal to n that the sum of ej xj is equal to e occurs for xi equal to a inverse times e to the mu ei, where a is equal to n to the negative first times the sum of e to the mu ei.
01:09
Well, we have that our constraint equations are g of x1 through xn equals, on the one hand, we just have the sum for components, x1 through xn minus n equals 0.
01:31
And we have the other constrained equation, h of x1 through xn.
01:37
This is e x1, sorry, e1x1 summed up through en xn minus e equals 0.
01:50
And i want to find lagrange equations.
01:53
As in the previous problem, we have the gradient of this is the vector.
01:58
1 plus natural log of x1, 1 plus natural log of x2, all the way up to 1 plus natural log of xn.
02:08
The gradient of g is the vector 1 -1 -1, with n -1s, and the gradient of h is the vector e1, e2, 2, e -n.
02:26
And the lagrange condition, which is that the gradient of s is equal to lambda times the gradient of g plus mu times the gradient of h gives us the lagrange equations...