monte carlo simulation methods

Monte Carlo Methods:- Markov Chain Random movements, one follow another- Importance sampling To sample many points in the region where the Boltzmann factor is

large and few elsewhere- Ergodicity (ensemble average) Such an average over all possible quantum states of a system - Detailed Balance

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Metropolis Method:

]/)(exp[y probabilith accept wit :accept:

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In terms of words:

- Canonical ensemble: N,V,T constant

- Metropolis method: To generate a new configuration with probabilities

Monte Carlo simulation in the canonical ensemble

0. Generate the initial configuration (random or c(2×2))

Beginning of the MC cycle

1. Calculate Eold 2. Randomly select a particle3. Random displacement (or rotation)4. Calculate Enew5. Accept the move with probability (Metropolis method) En<Eo accept (the move) En>Eo accept with probability exp[-(En-Eo)/kT] (Periodic boundary conditions)

End of MC cycle

In terms of diagram:

Monte Carlo simulation in the canonical ensemble

Monte Carlo simulation in the canonical ensembleChoice of the parameters

Initial configuration

do

en1=EN

typ=1call rand_pos(mat,typ,xrp,yrp)x1=xrp, y1=yrp


mat(x1,y1)=0mat(x2,y2)=1


call tot_EN(mat,EN,ei0,ei1)en2=EN

metropolis conditionsacceptance: save new values+ add MC stepRejection: add MC step

If (count.eq.mcs+1) thenexitend ifend do



do

en1=EN








real, parameter:: k=8,617339E-5real *8:: enT,enT2

print *,”temperature:”read(*,*) T

print *,”MC steps:”read(*,*) mcs

print *,”starting configuration, R or C:”read(*,*) ststart=0select case (st)case ( “c”, “C”)start=1end select



do

en1=EN








if (start.eq.1) thendo v=1,10

if (mod(v,2).eq.0) thenmat (u*2,v)=0elsemat (u*2-1,v)=0end if, end if, end do

elsedoxr= int (rand()*10)+1yr= int (rand()*10)+1if (mat(xr,yr).eq.1) thenmat(xr,yr)= 0

do u=1,5

nb=nb+1end iftt= int (10*10*(1-cov))if (tt.eq.nb) thenexitend if, end do, end if

%% Start c(2×2) %%

%% Start random %%



do

en1=EN








if (en2.lt.en1) then

mat(x2,y2)=1

enT=enT+en2enT2=enT2+en2*en2

count=count+1

else

if (exp(-(en2-en1)/(k*T)).gt.rand()) then

mat(x2,y2)=1mat(x1,y1)=0enT=enT+en2enT2=enT2+en2*en2

count=count+1

mat(x1,y1)=0

else

end if, end if

enT=enT+en1enT2=enT2+en1*en1EN=en1

count=count+1

EN=en2

EN=en2

%% Acceptance %%

%% probability %%

%% Rejection %%



do

en1=EN








Subroutine Number of nearest neighbours

NNE=0

id=xn+1if (id.gt.10) then

end if

ig=xn-1if (ig.lt.1) thenig=ig+10end if

jh=yn-1if (jh.gt.10) thenjh=jh-10end if

jb=yn+1if (jb.lt.1) thenjb=jb+10end if

NNE=id + ig + jb + jh

%% periodic boundary conditions %%

id=id-10



do

en1=EN








Subroutine Total energy of the system

EN=0

do ii=1,10do jj=1,10

if mat(ii,jj).eq.1) then

%% periodic boundary conditions %%

xn=iiyn=jj

call near_numb(mat,xn,yn,NNE)

EN=NNE*ei1+ei0



do

en1=EN








Subroutine Random selection

do

xe= int(rand()*10)+1ye= int(rand()*10)+1

if (mat(xe,ye).eq.typ) then

xrp=xeyrp=ye

exit

end if, end do



do

en1=EN








Subroutine Order parameter

matf=mat*2-1

do r = 1,10do q = 1,5

if (mod(r,2).eq.0) then

ma=ma+matf(2*q-1,r)mb=mb+matf(2*q,r)

else

ma=ma+matf(2*q,r)mb=mb+matf(2*q-1,r)

end if, end do, end do

Psi=sqrt((ma-mb)*(ma-mb)/100

Subroutine Heat capacity parameter

cv = ((enT2/mcs)-(enT*enT) / (mcs*mcs))) / (k*T*T)

Order-disorder transition Cv parameter

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500

T(K)

Cv/N

0 (m

eV/K

)

Tc=1350K

Simulation: Simple cubic surface (100) plan- Θ=0.5ML

- Cell :10*10

- Start: random configuration

- Number of MC steps: 10 million

c(2×2)

Critical phase

Full disorder state

- Nearest neighbours interaction energy:

Eint= +0.1eV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500

T(K)

Order-disorder transition:Ψ parameter

Simulation: Simple cubic surface (100) plan- Θ=0.5ML

- Cell :10*10

- Start: random configuration

- Number of MC steps: 10 million

Tc=1350KC(2×2)

Critical phase

Full disorder state

- Nearest neighbours interaction energy:

Eint= +0.1eV

Ψ=F(T(K))

monte carlo simulation methods

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