Developments in the theory of universality

Rome University

Description

Developments in the theory of universality Vieri Mastropietro Rome University ”Tor Vergata”

• tor vergata


• vieri mastropietro


• precise test


• shuttle experiments


• physics says


• universality allow testable

Sujets

• University
• University of Rome Tor Vergata

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 18
• Langue: English

Extrait

Developments in the theory universality

Vieri Mastropietro

Rome University Tor Vergata” ”

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The universality hypothesis (well confirmed by experiments) in statistical physics says that a number of macroscopic critical properties are largely independent of the microscopic structure, at least inside a universality class of systems.

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Universality in stastistical physics

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Universality in stastistical physics

The universality hypothesis (well confirmed by experiments) in statistical physics says that a number of macroscopic critical properties are largely independent of the microscopic structure, at least inside a universality class of systems. Precise test for universality in shuttle experiments for helium (2000); agreement with several digits.

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The universality hypothesis (well confirmed by experiments) in statistical physics says that a number of macroscopic critical properties are largely independent of the microscopic structure, at least inside a universality class of systems. Precise test for universality in shuttle experiments for helium (2000); agreement with several digits. Universality allow testable predictions even if we do not known the details of the microscopic model.

Universality in stastistical physics

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Universality in stastistical physics

The universality hypothesis (well confirmed by experiments) in statistical physics says that a number of macroscopic critical properties are largely independent of the microscopic structure, at least inside a universality class of systems. Precise test for universality in shuttle experiments for helium (2000); agreement with several digits. Universality allow testable predictions even if we do not known the details of the microscopic model. Connections between universality and renormalization. Deep connections bewteen QFT and statistical physics (statistical field theory).

The Ising model

The paradigmatic model for statistical mechanics is the 2D Ising model

H=J X X σ x σ x+e j ≡ X h x j=0,1 x∈Λx∈Λ

σ x =±, Λ is a square lattice,x∈Λ,e 0 = (0,1),e 1 = (1,0).

phase

Thepartition functionisZ= P σ e −βH(σ) and transitions appear as non-analyticity points of −1 f β =−βlim |Λ | →∞| Λ 1 | logZ.

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Ising model

The

pendentfi.e.inde β >ed x h y hβ=< c ro6βananerthfastcays > y h x h< 2 C+| c β−β|g efilwh | 2 −y|x ∼ C β c cilaniidT.ehrcti renumber ces arepu | 1− ξ|y−x wopy fo re |T∼C c |−Tit,w −1 hξ caltritiratuempe athn ersi 2 √ 1− c β=J On(91(regascehT))44 ve)andthecorrelaitno C s ( v )β−∼ 1 Col dtanspheifieceaches(tdnoc ireditav

The Ising model

Thepartition functionisZ= P σ e −βH(σ) and phase transitions appear as non-analyticity points of f β =−β −1 lim |Λ| | →∞ Λ 1 | logZ. (Onsager (1944)) The critical temperature is tanhβ c J= √ 2−1 and the specific heat (second derivative) and the correlations

C v (β)∼ −C 1 log|β−β c |+C 2 <h x h y > β c | ∼ x− C y| 2

while forβ6=β c <h x h y > β decays faster than any power ofξ −1 |x−y|, withξ −1 ∼C|T−T c |. Thecritical indices arepure numberi.e. independent fromJ.