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Wang-Landau Sampling in Statistical Physics

FSU National Magnet Lab Winter School Jan. 12, 2012. D P Landau Center for Simulational Physics The University of Georgia. Wang-Landau Sampling in Statistical Physics. Introduction Basic Method Simple Applications (how well does it work?) 2 nd order transitions – 2d Ising model

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Wang-Landau Sampling in Statistical Physics

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  1. FSU National Magnet Lab Winter School Jan. 12, 2012 D P LandauCenter for Simulational PhysicsThe University of Georgia Wang-Landau Sampling in Statistical Physics • Introduction • Basic Method • Simple Applications (how well does it work?) • 2nd order transitions – 2d Ising model • 1st Order transition – 2d Potts Models • Critical endpoint - triangularIsing model • Some Extensions • Conclusions

  2. Introductory Comments An understanding of materials behavior at non-zero temperature cannot be obtained solely from knowledge of T=0 properties. For this we need to use statistical mechanics/thermodynamics coupled with “knowledge” of the interatomic couplings and interactions with any external fields. This can be a very difficult problem!

  3. Simulation NATURE Experiment Theory

  4. Reminder from Statistical Mechanics ThePartition functioncontains all thermodynamic information: The probability of the nth state appearing is: Thermodynamic properties are then determined from the free energy Fwhere F=-kBTlnZ

  5. Reminder from Statistical Mechanics ThePartition functioncontains all thermodynamic information: Metropolis Monte Carlo approach: sample states via a random walk in probability space The “fruit fly” of statistical physics: The Ising model For a system of N spins have 2N states!

  6. Single Spin-Flip Monte Carlo Method Typical spin configurations for the Ising square lattice with pbc T << Tc T~Tc T>>Tc

  7. Metropolis Monte Carlo simulations of the 2-dim Ising model

  8. Single spin-flip sampling for the Ising model Produce the nth state from the mth state … relative probability is Pn/Pm need only the energy difference,i.e. E=(En-Em)between the states Any transition rate that satisfiesdetailed balanceisacceptable, usually the Metropolis form(Metropolis et al, 1953).W(mn) =  o-1exp (-E/kBT), E > 0=  o-1 , E < 0 where  o is the time required to attempt a spin-flip.

  9. Metropolis Recipe: 1. Choose an initial state 2. Choose a sitei 3. Calculate the energy changeEthat results if the spin at siteiis overturned 4. Generate a random numberrsuch that0 < r < 1 5.If r < exp(-E/kBT), flip the spin 6. Go to 2. This is not a unique solution. An alternative(Glauber, 1963): Wnm=o-1[1+itanh (Ei /kBT)], where iEi is the energy of the ith spin in state n. Both Glauber and Metropolis algorithms are special cases of a general transition rate(Müller-Krumbhaar and Binder, 1973)

  10. Correlation times Define an equilibrium relaxation function(t) and i.e. diverges at Tc!

  11. Problems and Challenges Statics:Monte Carlo methods are valuable, but near Tc critical slowing downfor2nd ordertransitions metastabilityfor1st ordertransitions  Try to reduce characteristic time scales or circumvent them “Dynamics”:stochastic vs deterministic

  12. Multicanonical Sampling The canonical probabilityP(E)may contain multiple maxima, widely spaced in configuration space (e.g. 1st order phase transition, etc.)  Standard methods become “trapped” near one maximum; infrequent transitions between maxima leads to poor relative weights of the maxima and the minima of P(E).  modify the single spin flip probability to enhance the probability of the “unlikely” states between the maxima  accelerates effective sampling! Reformulate the problem an effective Hamiltonian Berg and Neuhaus (1991)

  13. Compare ensembles: Then, Thermodynamic variable

  14. Compare ensembles: But, finding Heff isn’t easy Then, Thermodynamic variable

  15. Parallel tempering (replica exchange) Create multiple systems at differentTi T1 T2 T3 T4 T5 . . . Define  i = 1/Ti Hukushima and Nemoto (1996); Swendsen and Wang (1986)

  16. Parallel tempering (replica exchange) Create multiple systems at differentTi T1 T2 T3 T4 T5 . . . Define  i = 1/Ti Simulate all systems simultaneously Hukushima and Nemoto (1996); Swendsen and Wang (1986)

  17. Parallel tempering (replica exchange) Create multiple systems at differentTi T1 T2 T3 T4 T5 . . . Define  i = 1/Ti Simulate all systems simultaneously At regular intervals interchange configurations at neighboring T with probability P given by: Energy of state i

  18. Parallel tempering (replica exchange) Create multiple systems at differentTi T1 T2 T3 T4 T5 . . . Define  i = 1/Ti But, the Ti must be chosen carefully Simulate all systems simultaneously At regular intervals interchange configurations at neighboring T with probability P given by: Energy of state i

  19. Types of Computer Simulations Deterministic methods . . . (Molecular dynamics) Stochastic methods . . . (Monte Carlo)

  20. Perspective:

  21. Perspective:

  22. Improvements in Performance (Ising model): Computer speed Algorithmic advances - cluster flipping, reweighting. . . 1E10 1E9 1E8 relative performance 1E7 1000000 100000 10000 1000 100 computer speed 10 1 1970 1975 1980 1985 1990 1995 2000

  23. The “Random Walk in Energy Space with a Flat Histogram” method or “Wang-Landau sampling”

  24. A Quite Different Approach Random Walk in Energy Space with a Flat HistogramReminder: Estimate the density of statesg(E) directly –– how?1.Set g(E)=1; choose a modification factor(e.g. f0=e1) 2. Randomly flip a spin with probability: 3.Setg(Ei)  g(Ei)* f H(E) H(E)+1 (histogram) 4. Continue until the histogram is “flat”; decreasef, e.g. f!+1=f 1/25. Repeat steps 2 - 4 untilf= fmin~ exp(10-8)6. Calculate properties using final density of states g(E)

  25. How can we test the method? For a 2nd order transition, study the 2-dim Ising model: Tcis known for an infinite system (Onsager) Bulk properties are known g(E) is known exactly for small systems For a 1st order transition, study the 2-dim Potts model: for q=10 Tcis known for an infinite system (duality) Good numerical values exist for many quantities

  26. Demo: Wang-Landau Sampling for the 2-dim Ising model

  27. Density of States for the 2-dim Ising model Compare exact results with data from random walks in energy space:L L lattices with periodic boundaries  = relative error( exact solution is known for L 64 )

  28. Density of States: Large 2-dim Ising Model • use a parallel, multi-range random walk NO exact solution is available for comparison! Question: • Need to perform a random walk over ALL energies?

  29. Specific Heat of the 2-dim Ising Model = relative error

  30. Free Energy of the 2-dim Ising Model = relative error

  31. How Does fi Affect the Accuracy? error Data for L=32:

  32. Compare Sampling in the 2-dim Ising Model

  33. What About a 1st Order Transition? Look at the q=10 Potts model in 2-dim At Tc coexisting states are separated by an energy barrier

  34. q=10 Potts Model: Determine Tc

  35. q=10 Potts Model: Internal Energy

  36. A “new” old problem: A Critical Endpoint* A schematic view: T is temperature; g is a non-ordering field “Spectator” phase boundary Theory predicts new singularities at (ge,Te) (Fisher and Upton, 1990) *Historical note, this behavior was described but not given a name in: H. W. B. Roozeboom and  E. H. Büchner, Proceedings of the KoninklijkeAcademie der Wetenschappen (1905);The Collected Works of J. W. Gibbs (1906).

  37. A “new” old problem: A Critical Endpoint* A schematic view: T is temperature; g is a non-ordering field “Spectator” phase boundary Theory predicts new singularities at (ge,Te) (Fisher and Upton, 1990) *KritischeEndpunkt:Büchner, Zeit. Phys. Chem. 56 (1906) 257; van der Waals and Kohnstamm, "Lehrbuch der Thermodynamik," Part 2 (1912).

  38. Critical Endpoint: “New” singularities The phase boundary: (Fisher and Upton, 1990)

  39. Critical Endpoint: “New” singularities The phase boundary: (Fisher and Upton, 1990) The coexistence density: (Wilding, 1997)

  40. Critical Endpoint: “New” singularities The phase boundary: (Fisher and Upton, 1990) The coexistence density: For the symmetric case (Wilding, 1997)

  41. Triangular Ising Model with Two-Body and Three-Body Interactions Must search for the critical endpoint (CEP) in (H,T) space! (afterChin and Landau, 1987)

  42. Triangular Ising Model with Two-Body and Three-Body Interactions Order parameter Sublattice #1 Sublattice #2 Sublattice #3 (afterChin and Landau, 1987)

  43. Triangular Ising Model with Two-Body and Three-Body Interactions Order parameter Ordered state has the same symmetry as the 3-state Potts model Sublattice #1 Sublattice #2 Sublattice #3 (afterChin and Landau, 1987)

  44. Triangular Ising Model with Two-Body and Three-Body Interactions W-L sampling: Generate a 2-dim histogram in E-M space  use to determine g(E,M)

  45. Triangular Ising Model with Two-Body and Three-Body Interactions Critical point Magnetization as a function of both temperature and magnetic field

  46. Triangular Ising Model with Two-Body and Three-Body Interactions Magnetization as a function of both temperature and magnetic field Critical endpoint

  47. Triangular Ising Model with Two-Body and Three-Body Interactions Order parameter as a function of both temperature and magnetic field

  48. Triangular Ising Model with Two-Body and Three-Body Interactions Phase diagram in temperature-magnetic field space: Finite size effects  locate the critical endpoint

  49. Triangular Ising Model with Two-Body and Three-Body Interactions Curvature of the 1st order phase boundary near the critical endpoint

  50. Triangular Ising Model with Two-Body and Three-Body Interactions Finite size scaling of the maximum in curvature

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