Computational Physics - Department of Physics

12.3 Microscopic Derivation of the Diffusion Equation 389 w(i,n+ 1 )−w(i,n) ∆t =D w(i+ 1 ,n)+w(i− 1 ,n)− 2 w(i,n) (∆x)^2 , where ...

390 12 Random walks and the Metropolis algorithm wi(tn) =∑ j Wi j(tn)wj( 0 ), and defining W(il−jl,nε) = (Wn(ε))i j we obtain wi ...

12.3 Microscopic Derivation of the Diffusion Equation 391 Using the binomial formula n ∑ k= 0 ( n k ) aˆkbˆn−k= (a+b)n, ee we ha ...

392 12 Random walks and the Metropolis algorithm w(x,t+ε)−w(x,t) ε = l^2 2 ε w(x+l,t)− 2 w(x,t)+w(x−l,t) l^2 . If we identifyD=l ...

12.3 Microscopic Derivation of the Diffusion Equation 393 W 11 w 1 (t=∞)+W 12 w 2 (t=∞)+W 13 w 3 (t=∞)+W 14 w 4 (t=∞) =w 1 (t=∞) ...

394 12 Random walks and the Metropolis algorithm 12.3.2Continuous Equations Hitherto we have considered discretized versions of ...

12.3 Microscopic Derivation of the Diffusion Equation 395 w(x,t) = ∫∞ −∞ dkexp[ikx]^1 2 π exp [ −(Dk^2 t) ] =√^1 4 πDt exp [ −(x ...

396 12 Random walks and the Metropolis algorithm intmain() { intmax_trials, number_walks; doublemove_probability; // Read in dat ...

12.4 Entropy and Equilibrium Features 397 voidmc_sampling(intmax_trials,intnumber_walks, doublemove_probability,intwalk_cumulati ...

398 12 Random walks and the Metropolis algorithm 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 -20 -15 -10 -5 0 5 10 15 20 w(x,t ...

12.4 Entropy and Equilibrium Features 399 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 S Time steps in units of 10 i Fig. 12.6EntropySjas fun ...

400 12 Random walks and the Metropolis algorithm 12.5 The Metropolis Algorithm and Detailed Balance Let us recapitulate some of ...

12.5 The Metropolis Algorithm and Detailed Balance 401 The algorithm can then be expressed as We make a suggested move to the n ...

402 12 Random walks and the Metropolis algorithm wi=∑ j wjTj→iAj→i=∑ j wjWj→i, which is nothing but the standard equation for a ...

12.5 The Metropolis Algorithm and Detailed Balance 403 evaluate the partition functionZ. For the Boltzmann distribution, detaile ...

404 12 Random walks and the Metropolis algorithm A(j→i) A(i→j) =exp(−β(Ei−Ej)), is that we do not know the acceptance probabilit ...

12.6 Langevin and Fokker-Planck Equations 405 The distributionfis often called the instrumental (we will relate it to the jumpin ...

406 12 Random walks and the Metropolis algorithm Initialize: Establish an initial state, for example a positionx(i) Suggest a mo ...

12.6 Langevin and Fokker-Planck Equations 407 and w(x,t) = ∫∞ −∞ W(x.t|x 0 .t 0 )w(x 0 ,t 0 )dx 0 , and w(x′,t′) = ∫∞ −∞ W(x′.t′ ...

408 12 Random walks and the Metropolis algorithm Mn= 1 τ ∫∞ −∞ ξnW(x+ξ,τ|x)dξ= 〈[∆x(τ)]n〉 τ , resulting in ∂W(x,s|x 0 ) ∂s = ∞ ∑ ...