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Unformatted text preview: APPM 2350 HOMEWORK ()7 FALL 2010
Due Friday, October 15, 2010 at 4 pm under your TA’s door. 1. Previously in the semester, we derived equations for the distance between a point and a line, and
a point and a plane. Your task is to derive the same results, but using the method of Lagrange
multipliers. (a) Use Lagrange multipliers to establish the formula for the distance D from the point (11:0, yo) to the line can + by = d.
(b) Use Lagrange multipliers to establish the formula limo + 53/0 + 030 — dl
\/ a2 + b2 + c2 for the distance D from the point (:30, yo, 2,0) to the line am + by + oz 2 d. D: 2. Suppose you have three identical boxes, and some total number of particles, say N. Let n1, ng, and
n3 denote the number of particles in each box. Of course, we have that 3
Zn, 2 N. (1)
j=1 3 We can then deﬁne the probability of a particle being in the jth box as = N. Then clearly, P, = 1. (2) 3
=1 .7 Now we deﬁne the entropy, S, of our system of boxes and particles as
3
S=—ZP,1nP,. (3)
j=1 Entropy, roughly speaking, is a measure of the disorder of a system, and it turns out, according to
the Second Law of Thermodynamics, that any closed system wants to maximize its entropy. (3.) Using the method of Lagrange multipliers, prove that the entropy S of our system is maximized
when P1 = P2 = P3 = Thus, to maximize entropy, how should the particles be distributed
across the three boxes? Now, suppose that the three boxes are not exactly alike. Suppose that if a particle is in box 1, then
it has energy E1, if it is in box 2, it has energy E2, etc... Without loss of generality, we assume
0 < E1 < E2 < E3. Finally, suppose that we have a ﬁxed total amount of energy we are trying to
distribute across the three boxes, say E. Therefore, we now have the constraints N, (4) 'M
3.
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Z 3ij ". (7) (b) Using the method of Lagrange multipliers with more than one constraint, ﬁnd the probabilities
P] such that the entropy S is maximized. Let )r denote your Lagrange multiplier for the con—
straint (6) and ,8 denote the Lagrange multiplier for the constraint Your answer for each
P] should be in terms of A, ,8, and E}. (c) Using (6), determine each Pj only in terms of the Lagrange multiplier 3. Assuming B < 0, and
given that 0 < E1 < E2 < E3, which energy state is a particle most likely to be in? Note: when a system has a ﬁxed number of particles and a ﬁxed amount of energy, it is called a canon
ical ensemble. What you have now shown is how the Second Law of Thermodynamics determines
the allocation of a ﬁxed amount of energy for a canonical ensemble. Appm ZESO
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 Fall '07
 ADAMNORRIS
 Thermodynamics, Statistical Mechanics, Second Law Of Thermodynamics, Constraint, lagrange multipliers, Joseph Louis Lagrange

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