Finite size scaling II Surajit Sengupta (IACS+SNBNCBS)

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Presentation transcript:

Finite size scaling II Surajit Sengupta (IACS+SNBNCBS)

The block analysis (coarse-graining) technique ● Simple to use. ● Ordinary NVT MD or MC suffices. ● Get phase boundaries i.e. coexistence densities, compressibilities of coexistent phases. ● Also get surface tensions, Binder cumulants, scaling functions, scaling fields and scaling exponents. ● Can be generalized to obtain elastic constants etc. (generalized susceptibilities).

GAS LIQUID T Density

The Ising model and lattice gas The energy of the Ising model is defined to be: For each pair, if > 0 the interaction is called ferromagnetic < 0 the interaction is called antiferromagnetic = 0 the spins are noninteracting A ferromagnetic interaction tends to align spins, and an antiferromagnetic tends to anti align them.

Ising “lattice” gas

Block analysis for the Ising model Constant magnetization = NVT K. Binder, Z. Phys. B 43, 119 (1981)

Block analysis for the Ising model

But this works only for T << T c In general P(0) is underestimated

Block analysis for the Ising model Near the critical point: Finite size scaling !! Use (1) normalization (2) scaling relations (3) FSS of the Binder cumulant to finally get..

Block analysis for the Ising model From a single simulation you get: (1) coexistence densities (magnetizations). (2) ratio of critical amplitudes (universal values). (3) surface tension.

Block analysis for liq-gas transitions

Roman, White and Velasco Europhys. Lett. 42, 371, (1998)

Block analysis for liq-gas transitions Rovere, Heerman, Binder, Europhys. Lett. 6, 585 (1988)

CC CC THE BLOCK ANALYSIS CC DO 31 I = 1,10 NB = 5 + (I-1) DL = 1.0/DFLOAT(NB) DO 32 J = 1,NB DO 33 K = 1,NB PKNT = 0 DO 34 L = 1,N XT = X(L) YT = Y(L) IF(XT.LT.J*DL.AND.XT.GE.(J-1)*DL)THEN IF(YT.LT.K*DL.AND.YT.GE.(K-1)*DL)THEN PKNT = PKNT + 1 ENDIF 34 CONTINUE BINDEN = PKNT+1 IF(BINDEN.GT.MAXBND)BINDEN=MAXBND IF(BINSPN.GT.MAXBNS)BINSPN=MAXBNS HISDN(I,BINDEN) = HISDN(I,BINDEN) CONTINUE 32 CONTINUE 31 CONTINUE RETURN END

Block analysis for solids ● Solids are very incompressible ● Number fluctuations are (almost) nonexistent ● Block densities suffer from discretization and/or commensurability-incommensurability effects ● Liq-solid density difference is small + it is first order ● No critical scaling properties can be used. ● Block analysis for liq-solid boundary = bad idea

Block analysis (coarse – graining) for solids ● Fluctuations of block strains ● No external stress both NVT and NPT will do ● Can be applied to experiments ● “Model independent” (almost) ● Can be used to get non-local elasticity etc.

Strain-strain correlation functions