15.4 Phase transitions and critical exponents

Phase transitions and critical exponents are key concepts in understanding critical phenomena. They describe how systems change from one state to another and how physical properties behave near critical points.

These concepts are crucial for grasping the broader topic of critical phenomena and supercritical fluids. They explain why different systems can exhibit similar behavior near phase transitions, despite having different microscopic properties.

Critical Phenomena

Order Parameter and Critical Exponents

Top images from around the web for Order Parameter and Critical Exponents

• 

  Analysis based on scaling relations of critical behaviour at PM–FM phase transition and ... View original

  Is this image relevant?

• 

  Phase transitions – TikZ.net View original

  Is this image relevant?

• 

  Analysis based on scaling relations of critical behaviour at PM–FM phase transition and ... View original

  Is this image relevant?

• 

  Phase transitions – TikZ.net View original

  Is this image relevant?

1 of 2

Top images from around the web for Order Parameter and Critical Exponents

• 

  Analysis based on scaling relations of critical behaviour at PM–FM phase transition and ... View original

  Is this image relevant?

• 

  Phase transitions – TikZ.net View original

  Is this image relevant?

• 

  Analysis based on scaling relations of critical behaviour at PM–FM phase transition and ... View original

  Is this image relevant?

• 

  Phase transitions – TikZ.net View original

  Is this image relevant?

1 of 2

• Order parameter quantifies the degree of order in a system undergoing a phase transition
  • Examples: magnetization in a ferromagnet, density difference between liquid and gas phases
• Critical exponents describe the behavior of physical quantities near the critical point
  • Exponents characterize power-law dependencies of various quantities (order parameter, susceptibility, correlation length) as the system approaches the critical point
• Critical exponents are universal, meaning they are independent of the microscopic details of the system
  • Depend only on the dimensionality of the system and the symmetry of the order parameter

Scaling Laws and Universality Classes

• Scaling laws relate the critical exponents to each other
  • Derived from the assumption that the singular part of the free energy is a homogeneous function near the critical point
• Universality classes group systems with the same critical behavior, regardless of their specific microscopic interactions
  • Systems within the same universality class share the same set of critical exponents
  • Examples: Ising universality class (includes ferromagnets and liquid-gas systems), XY universality class (includes superfluids and superconductors)

Theoretical Approaches

Mean-Field Theory

• Mean-field theory is a simple approximation used to describe phase transitions
  • Assumes that each particle interacts with an average (mean) field created by all other particles
• Neglects fluctuations and correlations between particles
  • Provides qualitatively correct description of phase transitions but fails to predict correct critical exponents
• Examples: Curie-Weiss theory for ferromagnetism, van der Waals equation for liquid-gas transitions

Renormalization Group

• Renormalization group is a powerful method for studying critical phenomena
  • Based on the idea of coarse-graining the system by successively averaging over small-scale fluctuations
• Renormalization group transformations relate the system at different length scales
  • Fixed points of these transformations correspond to critical points
• Allows for the calculation of critical exponents and the understanding of universality
  • Flow of the system under renormalization group transformations determines the universality class

Models and Phase Transitions

Ising Model

• Ising model is a simple lattice model used to study phase transitions
  • Consists of discrete variables (spins) that can take two values (+1 or -1)
  • Spins interact with their nearest neighbors through a coupling constant $J$
• Exhibits a phase transition from a disordered (paramagnetic) phase to an ordered (ferromagnetic) phase as temperature is lowered
  • Critical temperature $T_c$ depends on the lattice dimensionality and the coupling constant
• Exact solution is known in one and two dimensions
  • Provides a benchmark for testing various approximation methods and numerical simulations

Continuous Phase Transitions

• Continuous phase transitions, also known as second-order phase transitions, are characterized by a continuous change in the order parameter across the critical point
  • Examples: ferromagnetic transition, superfluid transition, superconducting transition
• At the critical point, the system exhibits scale invariance and long-range correlations
  • Correlation length diverges, and the system becomes self-similar at all length scales
• Critical behavior is characterized by power-law dependencies and universal critical exponents
  • Fluctuations and correlations dominate the behavior near the critical point, leading to the failure of mean-field theories