Ensemble Inequivalence of Spin Systems on Graphs

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Ensemble Inequivalence of Spin Systems on Graphs

• Bruno Miguel Tavares Gonçalves
• 06/26/05

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Contents

• Statistical Ensembles
• Microcanonical Ensemble
• Canonical Ensemble
• Ensemble Inequivalence
• Models of Spin Systems
• Graph structures
• K-regular graphs
• Erdös-Renyi
• Small-World Networks
• Different Approaches

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Microcanonical Ensemble

• Fixed Energy
• Direct enumeration of system configurations
  (density of states for a given energy)
• Entropy can be written as
• Value of the entropy can be obtained my
  maximizing over the magnetization
• Remaining thermodynamical quantities of interest
  can be determined from the Entropy

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Canonical Ensemble

• Contact with a head bath at a given temperature
  is assumed
• Energy varies around a mean value U(T) which is a
  function of the temperature
• Partition function is
• Free Energy
• All other quantities can be calculated from the
  Helmoltz Free Energy

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Ensemble Inequivalence

• Microcanonical ensemble exhibits energy ranges
  with negative specific heat
• Non-concave entropy curve
• Present in non-additive systems
• Predicted by theory in
• Astrophysics (self-gravitating systems)
• Spin systems
• 2D Vortices
• Small systems where surface energy is comparable
  to bulk energy
• Observed experimentally in
• Nuclear physics
• Atomic clusters

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Spin Systems

• Ising spins have only two possible directions, UP
  or DOWN.
• Other types of spins (BEG, Potts, etc) are
  allowed more orientations
• Hamiltonian usually contains spin-spin
  interactions and interaction with an external
  field
• System dynamics is due only to spin flips

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K-Regular Graphs

• Start with N nodes
• Add edges in such a way that in the end each node
  has the same number of connections as as all
  other nodes
• Total number of edges is
• Self-Loops and double edges are not allowed
  (unphysical)

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K-Regular Graphs
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Erdös-Renyi Graphs

• Start with N individual nodes
• Create edges between all pairs of nodes with
  probability p
• Connectivity distribution
• K-Regular graph can be considered a mean field
  version

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Watts-Strogatz

• At t0 we have a ring where each node is
  connected to K neighbors.
• - At each step, a connection is modified with
  probability p.

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WS-Small World
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Different Approaches

• Classical results
• Enumeration of configurations in the
  Microcanonical ensemble
• Calculation of the partition function and the
  free energy for the Canonical ensemble
• Flat Histogram computer simulation to determine
  the density of states for all energy values
• Cavity method

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Ising Model

• Microcanonical Ensemble

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MicrocanonicalEntropy as a function of Energy
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MicrocanonicalEntropy as a function of
magnetization
E-0.9
E-1.0
E-1.1
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Microcanonical Entropy as a function of
magnetization at h0
E-0.5,-1
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Microcanonical?(E) at h1
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Ising Model

• Canonical Ensemble

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CanonicalEnergy as a function of ?
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The End
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The End
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Wang-Landau Dynamics

• Perform a random walk in phase space by flipping
  randomly selected spins
• Accept a given spin flip with probability given
  by
• Every time an energy level is visited modify the
  density of states
• When the histogram is flat reduce the
  modification factor
• PRE 64,056101

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Simulation
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Simulation
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Input File

K-Reg Ising model
input file



System definition


Number of spins
in the system N 1000 System connectivity "k" K
4 External field in units of the bond strength
(H/J) H 1.0

Simulation parameters


All energies are assumed to be per spin
and in units of the bond strength (E/NJ)
Minimum energy Emin -1 Maximum energy Emax 1
Size of Energy steps DeltaE 0.002 Density of
states ratio f0 2.78 Minimal value of f. Final
results will be precise to O(ln(ftol)) ftol
1.0000002 Flatness level flat 0.1 Type of
decrease in f n (gt1) -gt f_i1(f_i)(1/n)
others can be added later type 2.0

End input file



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Allowed magnetizations
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Phase Separation

• Non-concave entropy curve
• Separate system in two phases with energy e1 and
  energy e2
• Entropy is now given by straight line
• Non-applicable to non-additive systems

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Cavity Method
J. Stat. Phys 111 (2003) 1
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Cavity Method
J. Stat. Phys 111 (2003) 1
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Scale-Free Networks

• Connectivity distribution obeys a power-law
• Usually include some form of preferential
  attachment during the growth process
• No characteristic value of the degree
• All values of connectivity are possible,
  limited only by finite size effects