Introduction to Phase Transitions and Critical Phenomena¶
Stanley, H. E. (1971). Introduction to Phase Transitions and Critical Phenomena. Oxford University Press.
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9 citations across 9 artifacts.
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Primes¶
- Asymptotic Behavior
- Not `scale_invariance`. Scale invariance is a symmetry — the system looks the same under rescaling
This sourceDevelops scale invariance and self-similarity at critical points — a symmetry under rescaling, distinct from dominant-term asymptotics.
- Not `scale_invariance`. Scale invariance is a symmetry — the system looks the same under rescaling
- Criticality
- The canonical reference is the second-order phase transition of statistical mechanics, where as a control parameter is tuned to a critical value, the correlation length of fluctuations diverges, the susceptibility to external fields diverges, and event-size distributions become power laws rather than exponentials — a regime first systematically characterized in the modern sense by Stanley's (1971) treatise on critical phenomena.
This sourceInternational Series of Monographs on Physics. Oxford University Press. Foundational textbook treatment of critical phenomena (order parameters, critical exponents, scaling, universality classes), systematically characterizing the second-order-transition regime the prime names as criticality.
- The canonical reference is the second-order phase transition of statistical mechanics, where as a control parameter is tuned to a critical value, the correlation length of fluctuations diverges, the susceptibility to external fields diverges, and event-size distributions become power laws rather than exponentials — a regime first systematically characterized in the modern sense by Stanley's (1971) treatise on critical phenomena.
- Phase Diagram
This sourceFoundational treatment of critical phenomena: develops the structural picture of an order parameter that is negligible below a critical value x_c, rises across a transition region, and assumes a different power-law regime above x_c, with sharpness governed by the universality class.
- Scale Invariance
- The 2D Ising model of spins on a square lattice undergoes a second-order phase transition at critical temperature T_c ≈ 2.27 J/k_B
This sourceFoundational treatment of critical phenomena: develops the structural picture of an order parameter that is negligible below a critical value x_c, rises across a transition region, and assumes a different power-law regime above x_c, with sharpness governed by the universality class.
- The 2D Ising model of spins on a square lattice undergoes a second-order phase transition at critical temperature T_c ≈ 2.27 J/k_B
- Scaling and Scale Dependence
- A small fire spreads proportionally to fuel and wind (linear); a large fire exhibits critical cascade behavior and firestorm dynamics (nonlinear, scale-dependent), a regime change paralleling the qualitative transitions Stanley (1971) catalogued for thermodynamic systems crossing critical points.
This sourceFoundational treatment of critical phenomena: develops the structural picture of an order parameter that is negligible below a critical value x_c, rises across a transition region, and assumes a different power-law regime above x_c, with sharpness governed by the universality class.
- A small fire spreads proportionally to fuel and wind (linear); a large fire exhibits critical cascade behavior and firestorm dynamics (nonlinear, scale-dependent), a regime change paralleling the qualitative transitions Stanley (1971) catalogued for thermodynamic systems crossing critical points.
- Threshold
- The sharpness of the transition—idealized as a step but typically smoother—is a second-order property characterizing the threshold's precision and is determined by the underlying mechanism's cooperativity and the observation scale, a structural picture Stanley (1971) develops in detail in his foundational treatment of phase transitions and critical phenomena.
This sourceFoundational treatment of critical phenomena: develops the structural picture of an order parameter that is negligible below a critical value x_c, rises across a transition region, and assumes a different power-law regime above x_c, with sharpness governed by the universality class.
- The sharpness of the transition—idealized as a step but typically smoother—is a second-order property characterizing the threshold's precision and is determined by the underlying mechanism's cooperativity and the observation scale, a structural picture Stanley (1971) develops in detail in his foundational treatment of phase transitions and critical phenomena.
- Threshold-Driven Order Emergence
- The system persists in a disordered or fluid state despite continuous variation in conditions, until
This sourceFoundational treatment of critical phenomena: develops the structural picture of an order parameter that is negligible below a critical value x_c, rises across a transition region, and assumes a different power-law regime above x_c, with sharpness governed by the universality class.
- The system persists in a disordered or fluid state despite continuous variation in conditions, until
- Universality
- Statistical physics: critical exponents at continuous phase transitions depend only on dimensionality and order-parameter symmetry; a fluid and a magnet at their critical points share scaling laws despite utterly different underlying physics, because both sit in the same universality class.
This sourceStandard text establishing universality classes, shared critical exponents across unlike systems (fluids and magnets), and the Ising model as a canonical representative.
- Statistical physics: critical exponents at continuous phase transitions depend only on dimensionality and order-parameter symmetry; a fluid and a magnet at their critical points share scaling laws despite utterly different underlying physics, because both sit in the same universality class.
- Universality in Critical Phenomena
This sourceFoundational treatment of critical phenomena: develops the structural picture of an order parameter that is negligible below a critical value x_c, rises across a transition region, and assumes a different power-law regime above x_c, with sharpness governed by the universality class.
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