Applying a constraint in the calculus of variations

[Philip Koeck]

I have an analytical function F of the discrete variables n_i, which are natural numbers. I also know that the sum of all n_i is constant and equal to N.
N also appears explicitly in F, but F is not a function of N. F exists in a coordinate system given by the n_i only.
Should I carry out the variation as if N would vary when I vary any of the n_i and then apply the constant N as a constraint with a Lagrange multiplier or is it correct to leave out the variation of N with n_i from the beginning.
As an example you can look at: F = N + ∑g_i ln n_i
The g_i are just weights, which can be different for every i.

 

[stevendaryl]

Lagrange multipliers are explicitly for the sort of problem you're talking about. However, if the constraint is that $\sum_j n_j = N$, then it leads to a boring result: $\frac{\partial F}{\partial n_i} = \lambda$ (where $\lambda$ is the lagrange multiplier, some constant).

However, if the $n_j$ are supposed to all be natural numbers, then taking partial derivatives isn't going to minimize $F$, because the values of $n_j$ that minimize $F$ might not be natural numbers. I suppose you could use that answer as a starting place, and then search nearby for integer values that minimize $F$?

 

[Philip Koeck]

Which of the following two solutions is correct?
1 + g_i/n_i - λ = 0
or g_i/n_i - λ = 0

 

[stevendaryl]

Which of the following two solutions is correct?
1 + g_i/n_i - λ = 0
or g_i/n_i - λ = 0

Since $\lambda$ isn't a fixed number, there is no difference which of those you use. They lead to the same answer for $n_i$, once you eliminate $\lambda$.

 

[Philip Koeck]

I'm still mystified.
Can we look at the actual problem instead? A bit more complicated, I'm afraid.
I want to maximize F given below under the constraints ∑ n_i = N and ∑ n_i u_i = U, with constant N and U. I'll use α and β for the multipliers. The g_i and u_i are given parameters.

F = N ln N - N + ∑ ( n_i ln g_i - n_i ln n_i + n_i )

Are both the following solutions correct, would you say?

ln N + ln g_i - ln n_i - α - β u_i = 0

ln g_i - ln n_i - α - β u_i = 0​

 

[BvU]

Looks like you are trying to derive the Boltzmann distribution. Google is your friend.

 

[stevendaryl]

I'm still mystified.
Can we look at the actual problem instead? A bit more complicated, I'm afraid.
I want to maximize F given below under the constraints ∑ n_i = N and ∑ n_i u_i = U, with constant N and U. I'll use α and β for the multipliers. The g_i and u_i are given parameters.

F = N ln N - N + ∑ ( n_i ln g_i - n_i ln n_i + n_i )

Are both the following solutions correct, would you say?

ln N + ln g_i - ln n_i - α - β u_i = 0

ln g_i - ln n_i - α - β u_i = 0​

Those lead to the exact same answers for $n_i$. They lead to different values for $\alpha$, but you don't care about the value of $\alpha$.

 

[Philip Koeck]

Those lead to the exact same answers for $n_i$. They lead to different values for $\alpha$, but you don't care about the value of $\alpha$.

Isn't α given by ∂F/∂N ? How can it be different for the two solutions?
Shouldn't it be completely defined by F?
Yes I am looking at the derivation of Boltzmann statistics, but without the correction for indistinguishability.
That's why F contains two terms that depend only on N.

 

[stevendaryl]

Isn't α given by ∂F/∂N ? How can it be different for the two solutions?
Shouldn't it be completely defined by F?

Do you agree that the two solutions lead to the same values for $n_i$?

In both cases, the solution is: $n_i = N g_i e^{-\beta u_i}/\sum_i (g_i e^{-\beta u_i})$

The two different values for $\alpha$ differ by $\ln N$.

What's important is $n_i$, not $\alpha$.

 

[Philip Koeck]

Do you agree that the two solutions lead to the same values for $n_i$?

In both cases, the solution is: $n_i = N g_i e^{-\beta u_i}/\sum_i (g_i e^{-\beta u_i})$

The two different values for $\alpha$ differ by $\ln N$.

What's important is $n_i$, not $\alpha$.

I would write the solutions as follows:
n_i = N g_i exp(- α - β u_i)
and
n_i = g_i exp(- α - β u_i)
I see that you remove the α from the solutions by introducing the normalization sum.

I agree that if the α differ by ln N the two results are the same, but how do you argue that the α should be different?