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Phys 215A Hw 1, solutions
each problem is 20 points, please format your results as
a weaker proof: ๐๐บ
๐๐ฅ=
๐๐บ
๐๐ฆ=
๐๐บ
๐๐= 0 โผ ๐บ ๐๐ ๐๐๐๐๐๐๐ง๐๐ ๐๐ ๐๐๐ฅ๐๐๐๐ง๐๐
๐๐บ
๐๐= 0 implies ๐น = 0 , the constraint is satisfied
๐๐บ
๐๐ฅ=
๐๐บ
๐๐ฆ= 0 implies โ๐ + ๐ โ๐น = 0 this means, locally at the solution point:
the gradient of field ๐(๐ฅ, ๐ฆ) and ๐น(๐ฅ, ๐ฆ) are parallel โ๐ // โ๐น.
๐๐
๐๐ฅ= 0
๐(๐ฅ) is minimized
or maximized
Landau Book, Page 35
or the gradient of ๐(๐ฅ, ๐ฆ) is perpendicular to the constraint line (or surface) โ๐(๐ฅ, ๐ฆ) โฅ { ๐น(๐ฅ, ๐ฆ) = 0}
Therefore, for under constraint ๐น(๐ฅ(๐ ), ๐ฆ(๐ )) = 0 , the derivative is zero ๐๐(๐ฅ(๐ ),๐ฆ(๐ ))
๐๐ = 0
~ the solution is a local extreme.
Landau Book, 3rd Edition, Page 36, 37:
Examples:
convection This is an example, where the system is not in equilibrium. But you can still maximize the entropy, using some method. And you will see, there is a relative macroscopic motion.
Heat death of the universe? when our universe is in equilibrium no macroscopic motions
Page 43
We can write down the answer using Boltzmann distribution: ๐ธ(๐, ๐) = ๐๐๐
โ ๐
๐๐ต๐+0
๐โ
๐๐๐ต๐+1
= ๐๐
๐
๐๐๐ต๐+1
The distribution ๐(๐ธ) โผ ๐โ
๐ธ
๐๐ต๐ can be derived from the principle of maximum entropy.
However, we can also solve this problem, directly using the principle of maximum entropy.
Suppose the total energy of the system is ๐ธ, ๐ =๐ธ
๐
Numbers of microscopic states = Ways to put ๐ indistinguishable particles in ๐ boxes
ฮฉ = ๐ถ๐๐
Using the Stirling's approximation
๐
๐๐ต= log ฮฉ โ ๐ log ๐ โ ๐ log ๐ โ (๐ โ ๐) log(๐ โ ๐)
Taking the derivative:
๐
๐๐log ฮฉ = โ log ๐ + log(๐ โ ๐) = log
๐ โ ๐
๐= log(
๐
๐โ 1)
Replacing log ฮฉ and ๐ with ๐ and ๐ธ
๐
๐๐ต
๐๐
๐๐ธ= log(
๐๐
๐ธโ 1)
๐๐
๐๐ธ=
1
๐
๐
๐๐ต๐= log(
๐๐
๐ธโ 1)
Finally:
๐ธ(๐) = ๐๐
๐
๐๐๐ต๐ + 1
The total energy is maximized in the limit ๐ โ โ, ๐ธmax =๐๐
2 ๐(๐ ๐ก๐๐ก๐ 1) = ๐(๐ ๐ก๐๐ก๐ 2) = 50%
๐๐ต๐
๐
๐ธ
๐๐