Lecture Monte carlo simulations: Application to lattice models: Part III - TS. Ngô Văn Thanh

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Lecture Monte carlo simulations: Application to lattice models, part III - Finite size effects and reweighting methods. The main contents of this chapter include all of the following: Finite size effects, single histogram method, multiple histogram method, wang-Landau method, the applications.

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Nội dung Text: Lecture Monte carlo simulations: Application to lattice models: Part III - TS. Ngô Văn Thanh

VSOP19, Quy Nhon 3-18/08/2013 Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam.
Part III. Finite size effects and Reweighting methods III.1. Finite size effects III.2. Single histogram method III.3. Multiple histogram method III.4. Wang-Landau method III.5. The applications
A Guide to Monte Carlo Simulations in Statistical Physics D. Landau and K. Binder, (Cambridge University Press, 2009). Monte Carlo Simulation in Statistical Physics: An Introduction K. Binder and D. W. Heermann (Springer-Verlag Berlin Heidelberg, 2010). Understanding Molecular Simulation : From Algorithms to Applications D. Frenkel, (Academic Press, 2002). Frustrated Spin Systems H. T. Diep, 2nd Ed. (World Scientific, 2013). Lecture notes  PDF files : http://iop.vast.ac.vn/~nvthanh/cours/vsop/  Example code : http://iop.vast.ac.vn/~nvthanh/cours/vsop/code/
III.1. Finite size effects Using for determining the properties of the corresponding infinite system distinguish the order of the transition (first or second order transition) test the results of simulation of finite size systems Finite size scaling and critical exponents  Consider a spin model on the lattice of finite size L  Scaling forms of thermodynamic quantities (3.1) : are scaling functions  Scaling relations (3.2)
 The thermodynamic properties at the transition  reduce to proportionality constants  Scaling relations (3.3)  If , then The cumulants  fourth order cumulant of the order parameter (3.4)  As the system size For For
 fourth order cumulant of the energy (3.5)  The logarithmic derivative of the nth power of the magnetization (3.6) For a continuous transition (second order transition) (3.7) where
 Calculate the critical exponents exponent : (3.8) exponent : (3.9) exponent : (3.10) exponent : (3.11) For a discontinuous transition (first order transition)  finite-size scaling laws (3.12)  with
Ferromagnetic Ising spin model on simple cubic lattice
Ferromagnetic Ising spin model on simple cubic lattice
The behavior of VL for the q = 10 Potts model in two dimensions A Guide to Monte Carlo Simulations in Statistical Physics D. Landau and K. Binder
III.2. Single histogram method Introduced by Ferrenberg and Swendsen (1988), Ferrenberg (1991) using histograms to extract information from Monte Carlo simulations Applying to the calculation of critical exponents The theory  Consider a Monte Carlo simulation performed at T = T0  generates system configurations with a frequency proportional to the Boltzmann weight, exp(-E/kBT).  Histogram of energy and magnetization H(E, M)  The probability of simultaneously observing the system (3.13) : is the number of configurations (density of states) with energy E and magnetization M Partition function: (3.14)
 : provides an estimate for over the range of E and M values generated during the simulation N is the number of measurements made  We have (3.15) : is an estimate for the true density of states  From the histogram H(E, M) we can invert Eq. (3.15) to determine (3.16)  Replace in Eq. (3.13) with the expression for , and normalize the distribution. the relationship between the histogram measured at K = K0 and the estimated probability distribution for arbitrary K (3.17)
 The average value of any function of E and M : (3.18) Energy histogram vs energy for ferromagnetic Ising spin model on square lattice of size L = 64, at temperature K = 0.1540
Case study - one dimension single histogram  From the histogram of energy H(E)  The estimated probability distribution for arbitrary K (3.19)  The average value of any function of E (3.20)  For example
The simulations  Find the temperature T0 which is close to transition temperature  Simulate the model by using the standard Monte Carlo simulation (Metropolis algorithm).  Plot specific heat or susceptibility versus temperature, peaking the value of temperature at maxima of or .  Choose the range of energy  T0 = 2.26 Emin = -1.8, Emax = 1.0
 Do the simulation by using the single histogram method at T = T0  Program structure - Gererating an array of N_BIN for energy range of interest - initialize the lattice SUBROUTINE equilibrating ! Equilibrating process DO ieq = 1 to N_eq CALL monte_carlo_step() ENDDO END SUBROUTINE SUBROUTINE averaging ! Averaging process DO iav = 1 to N_av CALL monte_carlo_step() do-analysis ENDDO END SUBROUTINE SUBROUTINE monte_carlo_step generate-one-sweep END SUBROUTINE
Source : single_histogram_ising.f90 PROGRAM SINGLE_HISTOGRAM_ISING IMPLICIT NONE INTEGER, PARAMETER :: n = 40,nms = n*n REAL,DIMENSION(n,n) :: s REAL,ALLOCATABLE,DIMENSION(:) :: e1,e2,m1,m2,he INTEGER :: i,j,ip,im,jp,jm,ie,ia,ms,neq,nav,ib,nbin REAL :: r,emin,emax,t,dt,temp,et1,et2 ! CALL RANDOM_SEED() neq = 200000 nav = 400000 t = 2.3; dt = 4.0 emax = -1.0*float(n*n); emin = -2.0*float(n*n) nbin = INT((emax-emin)/dt) + 1 ALLOCATE(he(nbin),e1(nbin),e2(nbin),m1(nbin),m2(nbin))
CALL spin_conf() ! initial configuration CALL equilibrating() he =0.0; e1 = 0.0; e2 = 0.0; m1 = 0.0; m2 = 0.0 CALL averaging() OPEN(UNIT=12,FILE='sh_ising.dat') i = 0 DO ib = 1,nbin IF (he(ib) > 0) THEN write(12,1) he(ib),e1(ib)/he(ib),e2(ib)/he(ib)& ,m1(ib)/he(ib),m2(ib)/he(ib) i = i + 1 ENDIF ENDDO 1 format(5(F16.6,1x)) CLOSE(12) OPEN(UNIT=12,FILE='sh_ising.info') write(12,*) t,i,n CLOSE(12) DEALLOCATE(he,e1,e2,m1,m2) ! CONTAINS ! list of subroutine/function
SUBROUTINE spin_conf ! CALL RANDOM_NUMBER(s) DO j = 1,n DO i = 1,n IF (s(i,j) > 0.5) THEN s(i,j) = 1.0 ELSE s(i,j) = -1.0 ENDIF ENDDO ENDDO END SUBROUTINE spin_conf ! SUBROUTINE equilibrating DO ie = 1, neq CALL monte_carlo_step() ENDDO END SUBROUTINE equilibrating
SUBROUTINE averaging DO ia = 1, nav CALL monte_carlo_step() temp = 0.0 DO i = 1,n ip = i + 1; im = i - 1 IF (i == n) ip = 1 IF (i == 1) im = n DO j = 1,n jp = j + 1; jm = j - 1 IF (j == n) jp = 1 IF (j == 1) jm = n temp = temp & - s(i,j)*(s(ip,j)+s(im,j)+s(i,jp)+s(i,jm)) ENDDO ENDDO temp = 0.5*temp ib = NINT((temp-emin)/dt) + 1