• Aucun résultat trouvé

We make the following scaling hypothesis for the free energy:

N/A
N/A
Info
Télécharger
Protected

Academic year: 2022

Partager "We make the following scaling hypothesis for the free energy:"

Copied!
1
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)

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.

1

Références

Télécharger maintenant ( PDF - 1 Page - 45.63 KB )

Documents relatifs

Les coccidioses intestinales.

Anesthésie Réanimation Anesthésie Réanimation Anatomie Biochimie-chimie Dermatologie Chirurgie Générale Traumatologie orthopédique Hématologie biologique Chirurgie

On the elastic free energy for smectic-A liquid crystals

The paper is organized as follows: in section 2 we discuss the possible deformations in smectics and the invariant form of the free energy density in terms of the vector normal

Seawards, Landwards. Can we make historical sense of the Borneo as the Homeland of Malay hypothesis?

language, that one Bornean &#34;Malay&#34; kingdom developed not only into a trading power of some consequence but also into an active maritime (shipping) power, in contrast

We prove that the following examples are equivariantly semiprojective

We prove that arbitrary actions of compact groups on finite dimensional C ∗ -algebras are equivariantly semiprojec- tive, that quasifree actions of compact groups on the Cuntz

Free-energy-dissipative schemes for the Oldroyd-B model

Then, Section 3 is devoted to the presentation of a finite element scheme (using piecewise constant approximations of the conformation tensor and its log-formulation, and

We have estimated the following equation:

Holding meduc and f educ xed, by how much does sibs have to increase to reduce expected years of education by one year?. (A noninteger answer is acceptable

The free energy 1

For the study of the stationary Maxwellian solutions of the Vlasov-Poisson system, which are the steady states of the Vlasov-Poisson-Fokker-Planck system, one may refer to

Free energy estimates for the two-dimensional Keller-Segel model

The L 1 -norm is critical in the sense that there exists a critical mass above which all solution blow-up in finite time and below which they globally exist... First