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Unformatted text preview: Homework #3 Solutions Question 1a) Let us start by calculating the entropy using the given mul tiplicity function σ ( N,U ) = log[ g ( N,U )] = 3 N 2 log( U ) + log( C ) The fundamental temperature τ is defined by 1 τ = parenleftbigg ∂σ ∂U parenrightbigg N = 3 N 2 U (1) where we have used the fact that the derivative of log( U ) equals 1/U. We can solve this equation for U to get U = 3 Nτ 2 = 3 Nk B T 2 1b) Take the second derivative of the entropy parenleftbigg ∂ 2 σ ∂U 2 parenrightbigg N = ∂ ∂U parenleftbigg 3 N 2 U parenrightbigg N = − 3 N 2 U 2 < This tells us that the entropy versus energy curve is concave down, suggesting that the temperature of the system τ (inverse slope of σ vs. U ) will increase, with increasing U . Question 2) Let us start by following the hint. The energy is related to the spin excess by U = − −→ B · −→ M = − 2 smB . Using this relation we can write the entropy as a function of energy as σ ( U ) = σ − U 2 2 m 2 B 2 N , with σ = log[ g ( N, 0)]. Next use the relation between entropy and temperature given by the first equality in equation ( ?? ) above to get 1 τ = − U m 2 B 2 N In order to get the magnetization, recall that U is related to it by U = − MB (considering only the component of magnetic field along magnetization).(considering only the component of magnetic field along magnetization)....
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 Fall '11
 Anlage
 Thermodynamics, mechanics, Derivative, Normal Distribution, Work, Statistical Mechanics, Entropy, σ, G1 G2, 1018 seconds

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