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Statistical Physics 3 November 24th, 2010

Series 8

Exercise 1:

We make the following scaling hypothesis for the free energy:

f (λ s t, λ r B) = λ d f (t, B) where t = T T

C

− 1, B is an external magnetic field and λ the scaling parameter.

Calculate the critical exponents α , β , γ ,δ as a function of r and s. Verify the following identities:

a) α + 2β + γ = 2 (Rushbrooke) b) β (δ + 1) = 2 − α (Griffith)

Remember that the critical exponents are defined in the following relations:

C(t, 0) ∼ (−t) −α if t < 0 m(t , 0) ∼ (−t) β if t < 0 χ(t, 0) ∼ (−t) −γ if t < 0

m(0, B) ∼ B

δ1

if B > 0

Exercise 2:

We suppose that the gravitational potential U (r 1 , ..., r N ) is an homogeneous function of degree d:

U(λ r 1 , ..., λ r N ) = λ d U (r 1 , ..., r N )

Using scaling arguments derive the exact dependence in r of the gravitational potential from the Kepler law:

r 3 ∼ T 2

Hint: the Newton equation F = ma is invariant under scale change.

Then find the dependence between r and t (analogous to the Kepler law) in the case of a constant force.

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