Chapter 11: Observables in Phase Transition Flashcards Preview

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Flashcards in Chapter 11: Observables in Phase Transition Deck (8):

First-Order Phase Transition

discontinuity in the order parameter


Second-Order Phase Transition

  • critical singularity can be described by critical exponent with respect to t = |1 - T/TC|
    • universality class: important here; relevant parameters
      • spatial dimensionality D
      • symmetry of order parameter


Specific Heat, Magnetization, and Magnetic Susceptibility


Correlation Function

  • correlation length diverges as T → TC
    • details of lattice and short-range behavior become irrelavant → reason for universality


Scaling and Hyperscaling

  • there exist only 2 independent critical exponents 
    • there exist scaling relationships between critical exponents


Finite-Size Scaling Theory 

  • L finite → Z is infinitely differentiable analytic power series → no singularities possible
  • maxima are rounded and displaced
    • max{ζ} ~= L


Finite-Size Scaling Theory 

  • consider susceptibility (below)
  • actually measure scaling function in simulation
    • should get same results regardless of choice of L
    • allows for extrapolation of  γ/ν, 1/ν, Tc


Critical Slowing-Down

  • correlation time τ for loval MC algorithm given below  
    • RecallNeff = N/(2τ)
    • as L increases, τ increases faster, meaning much longer time is needed to each the same level of accuracy
  • can overcome by using clust algorithms (e.g. Wolff algorithm)