Scaling Laws for Ising Spin Systems.¶
Kadanoff, L. P. (1959). Scaling Laws for Ising Spin Systems. Physics of Fluids, 2(12), 1323-1331.
Cited by¶
5 citations across 5 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Renormalization
- In Kadanoff's 1966 treatment,
This sourceIntroduces renormalization group approach to equilibrium critical phenomena; shows that equilibrium phase transitions exhibit emergent scaling and that ensemble-dependent properties vanish only in thermodynamic limit, clarifying finite-size breakdown of equivalence.
- In Kadanoff's 1966 treatment,
- Scale Invariance
- … self-similarity). The power-law form — P(x) ∝ x^−α or f(λx) = λ^α f(x) — is the mathematical signature of scale invariance in both geometric and distributional contexts . The scaling exponent* α (or β, γ, ν in critical phenomena) governs how observables transform and classifies systems into universality classes
This sourceIntroduces renormalization group approach to equilibrium critical phenomena; shows that equilibrium phase transitions exhibit emergent scaling and that ensemble-dependent properties vanish only in thermodynamic limit, clarifying finite-size breakdown of equivalence.
- … self-similarity). The power-law form — P(x) ∝ x^−α or f(λx) = λ^α f(x) — is the mathematical signature of scale invariance in both geometric and distributional contexts . The scaling exponent* α (or β, γ, ν in critical phenomena) governs how observables transform and classifies systems into universality classes
- Scaling and Scale Dependence
- A power law is scale-invariant (the ratio between any two scales holds across the range); the mechanisms causing a power law often are scale-dependent (turbulence in air differs from turbulence in honey), a distinction that Kadanoff (1966) clarified by deriving how block-spin renormalization can preserve a scaling form while the microscopic dynamics change qualitatively across regimes.
This sourceIntroduces renormalization group approach to equilibrium critical phenomena; shows that equilibrium phase transitions exhibit emergent scaling and that ensemble-dependent properties vanish only in thermodynamic limit, clarifying finite-size breakdown of equivalence.
- A power law is scale-invariant (the ratio between any two scales holds across the range); the mechanisms causing a power law often are scale-dependent (turbulence in air differs from turbulence in honey), a distinction that Kadanoff (1966) clarified by deriving how block-spin renormalization can preserve a scaling form while the microscopic dynamics change qualitatively across regimes.
- Thermodynamic Equilibrium
- T4 — Ensemble Equivalence and Finite-Size Effects: Statistical mechanics asserts that microcanonical, canonical, and grand-canonical ensembles are equivalent in the thermodynamic limit (N→∞, V→∞)
This sourceIntroduces renormalization group approach to equilibrium critical phenomena; shows that equilibrium phase transitions exhibit emergent scaling and that ensemble-dependent properties vanish only in thermodynamic limit, clarifying finite-size breakdown of equivalence.
- T4 — Ensemble Equivalence and Finite-Size Effects: Statistical mechanics asserts that microcanonical, canonical, and grand-canonical ensembles are equivalent in the thermodynamic limit (N→∞, V→∞)
- Universality in Critical Phenomena
- T5 — Universality as Deep Insight vs. RG-Scheme Artifact:
This sourceIntroduces renormalization group approach to equilibrium critical phenomena; shows that equilibrium phase transitions exhibit emergent scaling and that ensemble-dependent properties vanish only in thermodynamic limit, clarifying finite-size breakdown of equivalence.
- T5 — Universality as Deep Insight vs. RG-Scheme Artifact:
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