10.34 Fall 2015 Metropolis Monte Carlo Algorithm

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1 10.34 Fall 2015 Metropols Monte Carlo Algorthm The Metropols Monte Carlo method s very useful for calculatng manydmensonal ntegraton. For e.g. n statstcal mechancs n order to calculate the prospertes of the system you are requred to use ensemble average. The ensemble average of any property B s gven by: B,,, B r p w r p dr dp w r p dr dp where w s the statstcal weghtng functon, r s confguraton space (a 3 dmensonal vector of spatal coordnates, p s momentum space (a 3 dmensonal vector of momentum. Both the numerators as well as the denomnator have 6 ntegrals to be computed. We can use Monte Carlo method to evaluate these ntegrals. Ths conssts of smply summng over random ponts sampled accordng to the probablty dstrbuton. Rewrtng the above expresson n terms of the probablty dstrbuton P(r, p : where P s defned as:,, B B r p P r p dr dp P r, p w r, p w r, p dr dp If we had a smple explct form for P we could try to sample drectly from that dstrbuton, and then evaluate B at these ponts. However, snce the denomnator of the expresson for P s so dffcult to evaluate, nstead we would le to use a method that wors drectly wth w. ow let us see how to mplement the Metropols Monte Carlo Method to solve the ntegral usng the weghtng functon. Followng s the algorthm: Metropols Monte Carlo Method: We have to generate a random sequence q [1], q [2], q [3] of states to solve for the ntegral. We start wth some value for q and then mae moves to dfferent states. For the th teraton (move, you are at a state q [] and have a scalar Sum defned, where ths scalar holds the value of the ntegral we are attemptng to solve: 1 Randomly generate a step Δq (e.g. usng dm random numbers from rand functon, Δq(n =Δ*(2*rand-1, for n=1..dm, where Δ s the maxmum allowable dsplacement n any of the coordnates n successve teratons 1

2 2 Propose a new state q [proposed] = q [] + Δq 3 Compute w(q [proposed] and w(q [], If w(q [proposed] > w(q [] then q [+1] = q [proposed] else f w(q [proposed] / w(q [] > rand q [+1] = q [proposed] else q [+1] = q [] 4 Sum = Sum + B(q [+1] 5 <B> = Sum/(o. of random ponts ote: rand s a unformly-dstrbuted random number from 0 (zero to 1 (one. The goal of the Metropols MC method s to generate states of q such that: q lm for all where the varable q represents the number of molecules n state q and the weghtng functon w s nown. Two questons that may be on your mnd are: How do we now that the Metropols MC method actually acheves ths goal? Why do we use the partcular acceptance crtera: w(q [proposed] / w(q [] > rand? Dscusson: Suppose we have a state n our system, q. If we wanted to calculate the total number of q states n our system, q, we would need to worry about two terms: (1 The state q moved from another state n our system q j : j P( q q qj j (2 Our system tred to move out of state q but remaned there: 1 P( q q j q j The P(q q j expressons are condtonal probabltes, whch represent the probablty of movng to state q gven that we were n state q j. Summng these two expressons gves us the total number of states n our system n state q : 2

3 where Δ q s defned as: q q P( q q P( q q qj j q j j j j P( q q P( q q qj j q j We now the values of the condtonal probabltes from our acceptance crtera: / j f j Pq ( qj 1 otherwse Loong at one of the other states n our system, q : P( q q P( q q q q q q If w(q < w(q : If w(q < w(q : q q q q q q When Δ q = 0, both of the precedng equatons tell us: q q constant If we choose ths constant to be, the total number of molecules, then we ve reached the goal of the Metropols MC method (as stated approxmately half-way down the second page: q otce, there s nothng specal about states, j, or n ths dervaton. Thus, the precedng equaton s true for all states n our system. Another way to thn about the above expresson s that there wll be no change n the system (Δ q = 0 once the 3

4 relatve populatons of all states ( q / reach ther expected probablty (w q a MC way of sayng system has reached equlbrum. Returnng to our expressons for Δ q : q q q q q q Imagne Δ q > 0: q q 0 q q q q Ths expresson tells us there are more molecules n state q (and less molecules n state q than what we would expect based on the rato of ther probabltes. Thus, Δ q s ncreasng to counter that effect. Smlarly, magne Δ q < 0: q q 0 q q q q In ths case, there are more molecules n the state q (and less n state q than we would expect based on ther probabltes. However, Δ q s decreasng to counter ths. Thus, the system s always tryng to reach equlbrum. Smple Example wth Metropols MC Method: Suppose we want to solve followng ntegral usng Metropols Monte Carlo Method: 4

5 1 0 2 x (1.5 xdx Here, we have f(x = x 2 and weghtng functon w(x = 1.5 x. We have to generate random sequence of x values n (0,1 and accept them on the bass of weghtng functon. Let us start wth x 0 = 0.25 (arbtrarly chosen. Generate random numbers between 0 and 1. For each number x_new, compute (w(x_new. If (w(x_new > =(w(x(-1, then x( = x_new, else (w(x_new/ w(x(-1>rand, then x( = x_new, else x( = x(-1. Sum (f(x( for all and dvde by to get the value of the ntegral. 5

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