Quantum Monte Carlo Motivation. Computational Physics 2: Variational Monte Carlo methods. Quantum Monte Carlo Motivation

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1 Quatum Mote Carlo Motivatio Computatioal Physics 2: Variatioal Mote Carlo methods Morte Hjorth-Jese Departmet of Physics, Uiversity of Oslo Departmet of Physics ad Astroomy ad Natioal Supercoductig Cyclotro Laboratory, Michiga State Uiversity 2 Feb 5, 208 c , Morte Hjorth-Jese morte.hjorth-jese@fys.uio.o. Released uder CC Attributio-NoCommercial 4.0 licese Give a hamiltoia H ad a trial wave fuctio Ψ T, the variatioal priciple states that the expectatio value of H, defied through drψ E[H] = H = T (R)H(R)Ψ T (R) drψ, T (R)Ψ T (R) is a upper boud to the groud state eergy E 0 of the hamiltoia H, that is E 0 H. I geeral, the itegrals ivolved i the calculatio of various expectatio values are multi-dimesioal oes. Traditioal itegratio methods such as the Gauss-Legedre will ot be adequate for say the computatio of the eergy of a may-body system. Quatum Mote Carlo Motivatio Quatum Mote Carlo Motivatio The trial wave fuctio ca be expaded i the eigestates of the hamiltoia sice they form a complete set, viz., Ψ T (R) = i a i Ψ i (R), ad assumig the set of eigefuctios to be ormalized oe obtais m a ma drψ m (R)H(R)Ψ (R) = a2 E E m a ma drψ 0, m (R)Ψ (R) a2 where we used that H(R)Ψ (R) = E Ψ (R). I geeral, the itegrals ivolved i the calculatio of various expectatio values are multi-dimesioal oes. The variatioal priciple yields the lowest state of a give symmetry. I most cases, a wave fuctio has oly small values i large parts of cofiguratio space, ad a straightforward procedure which uses homogeously distributed radom poits i cofiguratio space will most likely lead to poor results. This may suggest that some kid of importace samplig combied with e.g., the Metropolis algorithm may be a more efficiet way of obtaiig the groud state eergy. The hope is the that those regios of cofiguratios space where the wave fuctio assumes appreciable values are sampled more efficietly. Quatum Mote Carlo Motivatio Quatum Mote Carlo Motivatio The tedious part i a VMC calculatio is the search for the variatioal miimum. A good kowledge of the system is required i order to carry out reasoable VMC calculatios. This is ot always the case, ad ofte VMC calculatios serve rather as the startig poit for so-called diffusio Mote Carlo calculatios (DMC). DMC is a way of solvig exactly the may-body Schroediger equatio by meas of a stochastic procedure. A good guess o the bidig eergy ad its wave fuctio is however ecessary. A carefully performed VMC calculatio ca aid i this cotext. Costruct first a trial wave fuctio ψ T (R, α), for a may-body system cosistig of N particles located at positios R = (R,..., R N ). The trial wave fuctio depeds o α variatioal parameters α = (α,..., α M ). The we evaluate the expectatio value of the hamiltoia H drψ E[H] = H = T (R, α)h(r)ψ T (R, α) drψ. T (R, α)ψ T (R, α) Thereafter we vary α accordig to some miimizatio algorithm ad retur to the first step.

2 Quatum Mote Carlo Motivatio Quatum Mote Carlo Motivatio Basic steps Choose a trial wave fuctio ψ T (R). ψ T (R) 2 P(R) = ψt (R) 2 dr. This is our ew probability distributio fuctio (PDF). The approximatio to the expectatio value of the Hamiltoia is ow drψ E[H(α)] = T (R, α)h(r)ψ T (R, α) drψ. T (R, α)ψ T (R, α) Defie a ew quatity E L (R, α) = ψ T (R, α) Hψ T (R, α), called the local eergy, which, together with our trial PDF yields E[H(α)] = P(R)E L (R)dR N P(R i, α)e L (R i, α) i= with N beig the umber of Mote Carlo samples. Quatum Mote Carlo The Algorithm for performig a variatioal Mote Carlo calculatios rus thus as this Iitialisatio: Fix the umber of Mote Carlo steps. Choose a iitial R ad variatioal parameters α ad calculate ψ α T (R) 2. Iitialise the eergy ad the variace ad start the Mote Carlo calculatio. Calculate a trial positio R p = R + r step where r is a radom variable r [0, ]. Metropolis algorithm to accept or reject this move w = P(R p)/p(r). If the step is accepted, the we set R = R p. Update averages Fiish ad compute fial averages. Observe that the jumpig i space is govered by the variable step. This is Called brute-force samplig. Need importace samplig to get more relevat samplig, see lectures below. Quatum Mote Carlo: hydroge atom The radial Schroediger equatio for the hydroge atom ca be writte as 2 2 ( ) u(r) ke 2 2m r 2 2 l(l + ) r 2mr 2 u(r) = Eu(r), or with dimesioless variables 2 2 u(ρ) ρ 2 with the hamiltoia u(ρ) l(l + ) + ρ 2ρ 2 u(ρ) λu(ρ) = 0, H = 2 2 ρ 2 ρ + l(l + ) 2ρ 2. Use variatioal parameter α i the trial wave fuctio u α T (ρ) = αρe αρ. Quatum Mote Carlo: hydroge atom Quatum Mote Carlo: hydroge atom Isertig this wave fuctio ito the expressio for the local eergy E L gives E L (ρ) = ρ α ( α 2 ). 2 ρ A simple variatioal Mote Carlo calculatio results i α H σ 2 σ/ N E E E E E E E E E E E E E E E E E E E E E E E E E E-0.457E E-03 We ote that at α = we obtai the exact result, ad the variace is zero, as it should. The reaso is that we the have the exact wave fuctio, ad the actio of the hamiltioa o the wave fuctio Hψ = costat ψ, yields just a costat. The itegral which defies various expectatio values ivolvig momets of the hamiltoia becomes the drψ H = T (R)H (R)Ψ T (R) drψ drψ = costat T (R)Ψ T (R) T (R)Ψ T (R) drψ T (R)Ψ T (R) = costat. This gives a importat iformatio: the exact wave fuctio leads to zero variace! Variatio is the performed by miimizig both the eergy ad the variace.

3 Quatum Mote Carlo for bosos Quatum Mote Carlo for bosos For bosos i a harmoic oscillator-like trap we will use is a spherical (S) or a elliptical (E) harmoic trap i oe, two ad fially three dimesios, with the latter give by { V ext(r) = 2 mω2 ho r 2 (S) 2 m[ω2 ho (x 2 + y 2 ) + ωz 2 z 2 ] (E) () We will represet the iter-boso iteractio by a pairwise, repulsive potetial { r V it ( r i r j ) = i r j a 0 r i r j > a (3) where (S) stads for symmetric ad ( ) 2 Ĥ = 2m 2 i + V ext(r i ) + V it (r i, r j ), (2) i i<j where a is the so-called hard-core diameter of the bosos. Clearly, V it ( r i r j ) is zero if the bosos are separated by a distace r i r j greater tha a but ifiite if they attempt to come withi a distace r i r j a. as the two-body Hamiltoia of the system. Quatum Mote Carlo for bosos Quatum Mote Carlo for bosos Our trial wave fuctio for the groud state with N atoms is give by Ψ T (R) = Ψ T (r, r 2,... r N, α, β) = i g(α, β, r i ) f (a, r i r j ), i<j For spherical traps we have β = ad for o-iteractig bosos (a = 0) we have α = /2aho 2. The correlatio wave fuctio is (4) where α ad β are variatioal parameters. The sigle-particle wave fuctio is proportioal to the harmoic oscillator fuctio for the groud state g(α, β, r i ) = exp [ α(x 2 i + y 2 i + βz 2 i )]. (5) { 0 ri r j a f (a, r i r j ) = ( a ri rj ) r i r j > a. (6) Quatum Mote Carlo: the helium atom Quatum Mote Carlo: the helium atom The helium atom cosists of two electros ad a ucleus with charge Z = 2. The cotributio to the potetial eergy due to the attractio from the ucleus is 2ke2 2ke2, r r 2 ad if we add the repulsio arisig from the two iteractig electros, we obtai the potetial eergy V (r, r 2) = 2ke2 r 2ke2 + ke2, r 2 r 2 with the electros separated at a distace r 2 = r r 2. The hamiltoia becomes the Ĥ = 2 2 2m m ad Schroedigers equatio reads 2ke2 r Ĥψ = Eψ. 2ke2 + ke2, r 2 r 2 All observables are evaluated with respect to the probability distributio ψ T (R) 2 P(R) = ψt (R) 2 dr. geerated by the trial wave fuctio. The trial wave fuctio must approximate a exact eigestate i order that accurate results are to be obtaied.

4 Quatum Mote Carlo: the helium atom Choice of trial wave fuctio for Helium: Assume r 0. E L (R) = ψ T (R) Hψ T (R) = ( 2 ) ψ T (R) 2 Zr ψ T (R)+fiite terms. ( E L (R) = d 2 R T (r ) 2 dr 2 d Z ) R T (r ) + fiite terms r dr r For small values of r, the terms which domiate are ( lim E L(R) = d Z ) R T (r r 0 ), R T (r ) r dr r sice the secod derivative does ot diverge due to the fiiteess of Ψ at the origi. Quatum Mote Carlo: the helium atom This results i ad dr T (r ) = Z, R T (r ) dr R T (r ) e Zr. A similar coditio applies to electro 2 as well. For orbital mometa l > 0 we have dr T (r) = Z R T (r) dr l +. Similarly, studyig the case r 2 0 we ca write a possible trial wave fuctio as ψ T (R) = e α(r+r2) e βr2. The last equatio ca be geeralized to ψ T (R) = φ(r )φ(r 2)... φ(r N ) i<j f (r ij ), for a system with N electros or particles. The first attempt at solvig the helium atom The first attempt at solvig the Helium atom Durig the developmet of our code we eed to make several checks. It is also very istructive to compute a closed form expressio for the local eergy. Sice our wave fuctio is rather simple it is straightforward to fid a aalytic expressios. Cosider first the case of the simple helium fuctio Ψ T (r, r 2) = e α(r+r2) The local eergy is for this case ( E L = (α Z) + ) + α 2 r r 2 r 2 which gives a expectatio value for the local eergy give by ( E L = α 2 2α Z 5 ) 6 With closed form formulae we ca speed up the computatio of the correlatio. I our case we write it as ar ij Ψ C = exp + βr ij, i<j which meas that the gradiet eeded for the so-called quatum force ad local eergy ca be calculated aalytically. This will speed up your code sice the computatio of the correlatio part ad the Slater determiat are the most time cosumig parts i your code. We will refer to this correlatio fuctio as Ψ C or the liear Pade-Jastrow. The first attempt at solvig the Helium atom We ca test this by computig the local eergy for our helium wave fuctio ( ) r 2 ψ T (r, r 2) = exp ( α(r + r 2)) exp, 2( + βr 2) with α ad β as variatioal parameters. The local eergy is for this case { α(r + r 2) E L2 = E L + 2( + βr 2) 2 ( rr2 ) r 2 r r 2 2( + βr 2) β r 2 + βr 2 It is very useful to test your code agaist these expressios. It meas also that you do t eed to compute a derivative umerically as discussed i the code example below. The first attempt at solvig the Helium atom For the computatio of various derivatives with differet types of wave fuctios, you will fid it useful to use pytho with symbolic pytho, that is sympy, see olie maual. Usig sympy allows you autogeerate both Latex code as well c++, pytho or Fortra codes. Here you will fid some simple examples. We choose the 2s hydroge-orbital (ot ormalized) as a example with r 2 = x 2 + y 2 + z 2. φ 2s(r) = (Zr 2) exp ( 2 Zr), from sympy import symbols, diff, exp, sqrt x, y, z, Z = symbols( x y z Z ) r = sqrt(x*x + y*y + z*z) r phi = (Z*r - 2)*exp(-Z*r/2) phi diff(phi, x) This does t look very ice, but sympy provides several fuctios that allow for improvig ad simplifyig the output.

5 The first attempt at solvig the Helium atom The first attempt at solvig the Helium atom We ca improve our output by factorizig ad substitutig expressios from sympy import symbols, diff, exp, sqrt, factor, Symbol, pritig x, y, z, Z = symbols( x y z Z ) r = sqrt(x*x + y*y + z*z) phi = (Z*r - 2)*exp(-Z*r/2) R = Symbol( r ) #Creates a symbolic equivalet of r #prit latex ad c++ code prit pritig.latex(diff(phi, x).factor().subs(r, R)) prit pritig.ccode(diff(phi, x).factor().subs(r, R)) We ca i tur look at secod derivatives from sympy import symbols, diff, exp, sqrt, factor, Symbol, pritig x, y, z, Z = symbols( x y z Z ) r = sqrt(x*x + y*y + z*z) phi = (Z*r - 2)*exp(-Z*r/2) R = Symbol( r ) #Creates a symbolic equivalet of r (diff(diff(phi, x), x) + diff(diff(phi, y), y) + diff(diff(phi, z), z)).factor().subs(r, R) # Collect the Z values (diff(diff(phi, x), x) + diff(diff(phi, y), y) +diff(diff(phi, z), z)).factor().collect(z).subs(r, R) # Factorize also the r**2 terms (diff(diff(phi, x), x) + diff(diff(phi, y), y) + diff(diff(phi, z), z)).factor().collect(z).subs(r, R).subs(r**2, R**2).factor() prit pritig.ccode((diff(diff(phi, x), x) + diff(diff(phi, y), y) + diff(diff(phi, z), z)).factor().collect(z).subs(r, R).subs(r**2, R**2 With some practice this allows oe to be able to check oe s ow calculatio ad traslate automatically ito code lies. The first attempt at solvig the Helium atom The c++ code with a VMC Solver class, mai program first #iclude "vmcsolver.h" #iclude <iostream> usig amespace std; it mai() { VMCSolver *solver = ew VMCSolver(); solver->rumotecarloitegratio(); retur 0; The first attempt at solvig the Helium atom The c++ code with a VMC Solver class, the VMCSolver header file #ifdef VMCSOLVER_H #defie VMCSOLVER_H #iclude <armadillo> usig amespace arma; class VMCSolver { public: VMCSolver(); void rumotecarloitegratio(); private: double wavefuctio(cost mat &r); double localeergy(cost mat &r); it Dimesios; it charge; double steplegth; it Particles; double h; double h2; log idum; double alpha; it Cycles; mat rold; mat rnew; ; #edif // VMCSOLVER_H The first attempt at solvig the Helium atom The c++ code with a VMC Solver class, VMCSolver codes, iitialize #iclude "vmcsolver.h" #iclude "lib.h" #iclude <armadillo> #iclude <iostream> usig amespace arma; usig amespace std; VMCSolver::VMCSolver() : Dimesios(3), charge(2), steplegth(.0), Particles(2), h(0.00), h2(000000), idum(-), alpha(0.5*charge), Cycles(000000) { The first attempt at solvig the Helium atom The c++ code with a VMC Solver class, VMCSolver codes void VMCSolver::ruMoteCarloItegratio() { rold = zeros<mat>(particles, Dimesios); rnew = zeros<mat>(particles, Dimesios); double wavefuctioold = 0; double wavefuctionew = 0; double eergysum = 0; double eergysquaredsum = 0; double deltae; // iitial trial positios for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rold(i,j) = steplegth * (ra2(&idum) - 0.5); rnew = rold; // loop over Mote Carlo cycles for(it cycle = 0; cycle < Cycles; cycle++) { // Store the curret value of the wave fuctio wavefuctioold = wavefuctio(rold); // New positio to test for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rnew(i,j) = rold(i,j) + steplegth*(ra2(&idum) - 0.5); // Recalculate the value of the wave fuctio wavefuctionew = wavefuctio(rnew); // Check for step acceptace (if yes, update positio, if o, reset positio) if(ra2(&idum) <= (wavefuctionew*wavefuctionew) / (wavefuctioold*wavefuctioold)) { for(it j = 0; j < Dimesios; j++) { rold(i,j) = rnew(i,j);

6 The first attempt at solvig the Helium atom The c++ code with a VMC Solver class, VMCSolver codes double VMCSolver::localEergy(cost mat &r) { mat rplus = zeros<mat>(particles, Dimesios); mat rmius = zeros<mat>(particles, Dimesios); rplus = rmius = r; double wavefuctiomius = 0; double wavefuctioplus = 0; double wavefuctiocurret = wavefuctio(r); // Kietic eergy, brute force derivatios double kieticeergy = 0; for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rplus(i,j) += h; rmius(i,j) -= h; wavefuctiomius = wavefuctio(rmius); wavefuctioplus = wavefuctio(rplus); retur exp(-argumet * alpha); kieticeergy -= (wavefuctiomius + wavefuctioplus - 2 * wavefuctiocurret); rplus(i,j) = r(i,j); rmius(i,j) = r(i,j); kieticeergy = 0.5 * h2 * kieticeergy / wavefuctiocurret; // Potetial eergy double potetialeergy = 0; double rsigleparticle = 0; for(it i = 0; i < Particles; i++) { rsigleparticle = 0; for(it j = 0; j < Dimesios; j++) { rsigleparticle += r(i,j)*r(i,j); potetialeergy -= charge / sqrt(rsigleparticle); The first attempt at solvig the Helium atom // Cotributio from electro-electro potetial double r2 = 0; The c++ for(it code i with = 0; ai VMC < Particles; Solver class, i++) the { VMCSolver header file for(it j = i + ; j < Particles; j++) { #iclude <armadillo> r2 = 0; #iclude <iostream> for(it k = 0; k < Dimesios; k++) { usig amespace r2 arma; += (r(i,k) - r(j,k)) * (r(i,k) - r(j,k)); usig amespace std; double ra2(log potetialeergy *); += / sqrt(r2); class VMCSolver { retur kieticeergy + potetialeergy; public: VMCSolver(); void rumotecarloitegratio(); private: double wavefuctio(cost mat &r); double localeergy(cost mat &r); it Dimesios; it charge; double steplegth; it Particles; double h; double h2; log idum; double alpha; it Cycles; mat rold; mat rnew; ; VMCSolver::VMCSolver() : Dimesios(3), charge(2), steplegth(.0), The Metropolis Particles(2), algorithm h(0.00), h2(000000), idum(-), alpha(0.5*charge), Cycles(000000) { void VMCSolver::ruMoteCarloItegratio() We { wish to derive the required properties of T ad A such that P ( ) rold = zeros<mat>(particles, Dimesios); i p rnew = zeros<mat>(particles, i so that startig from ay distributio, the method Dimesios); coverges double towavefuctioold the correct distributio. = 0; Note that the descriptio double wavefuctionew = 0; here is double for a discrete eergysum probability = 0; distributio. Replacig probabilities p i with double expressios eergysquaredsum like p(x i )dx= i 0; will take all of these over to the double deltae; correspodig // iitial cotiuum trial positios expressios. for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rold(i,j) = steplegth * (ra2(&idum) - 0.5); rnew = rold; // loop over Mote Carlo cycles for(it cycle = 0; cycle < Cycles; cycle++) { // Store the curret value of the wave fuctio wavefuctioold = wavefuctio(rold); // New positio to test for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rnew(i,j) = rold(i,j) + steplegth*(ra2(&idum) - 0.5); // Recalculate the value of the wave fuctio The first attempt at solvig the Helium atom The c++ code with a VMC Solver class, VMCSolver codes double VMCSolver::waveFuctio(cost mat &r) { double argumet = 0; for(it i = 0; i < Particles; i++) { double rsigleparticle = 0; for(it j = 0; j < Dimesios; j++) { rsigleparticle += r(i,j) * r(i,j); argumet += sqrt(rsigleparticle); The Metropolis algorithm The Metropolis algorithm, see the origial article was iveted by Metropolis et. al ad is ofte simply called the Metropolis algorithm. It is a method to sample a ormalized probability distributio by a stochastic process. We defie P () i to be the probability for fidig the system i the state i at step. The algorithm is the Sample a possible ew state j with some probability T i j. Accept the ew state j with probability A i j ad use it as the ext sample. With probability A i j the move is rejected ad the origial state i is used agai as a sample. The Metropolis algorithm The dyamical equatio for P () i ca be writte directly from the descriptio above. The probability of beig i the state i at step is give by the probability of beig i ay state j at the previous step, ad makig a accepted trasitio to i added to the probability of beig i the state i, makig a trasitio to ay state j ad rejectig the move: P () i = [ ] P ( ) j T j i A j i + P ( ) i T i j ( A i j ). j Sice the probability of makig some trasitio must be, j T i j =, ad the above equatio becomes P () i = P ( ) i + j [ P ( ) j ] T j i A j i P ( ) i T i j A i j.

7 The Metropolis algorithm The Metropolis algorithm For large we require that P ( ) i = p i, the desired probability distributio. Takig this limit, gives the balace requiremet [p j T j i A j i p i T i j A i j ] = 0. j The balace requiremet is very weak. Typically the much stroger detailed balace requiremet is eforced, that is rather tha the sum beig set to zero, we set each term separately to zero ad use this to determie the acceptace probabilities. Rearragig, the result is A j i = p it i j. A i j p j T j i The Metropolis choice is to maximize the A values, that is ( A j i = mi, p ) it i j. p j T j i Other choices are possible, but they all correspod to multilplyig A i j ad A j i by the same costat smaller tha uity. a a The pealty fuctio method uses just such a factor to compesate for pi that are evaluated stochastically ad are therefore oisy. The Metropolis algorithm Havig chose the acceptace probabilities, we have guarateed that if the P () i has equilibrated, that is if it is equal to p i, it will remai equilibrated. Next we eed to fid the circumstaces for covergece to equilibrium. The dyamical equatio ca be writte as P () i = j with the matrix M give by M ij = δ ij [ k M ij P ( ) j T i k A i k ] + T j i A j i. The Metropolis algorithm The Metropolis method is simply the power method for computig the right eigevector of M with the largest magitude eigevalue. By costructio, the correct probability distributio is a right eigevector with eigevalue. Therefore, for the Metropolis method to coverge to this result, we must show that M has oly oe eigevalue with this magitude, ad all other eigevalues are smaller. Summig over i shows that i M ij =, ad sice k T i k =, ad A i k, the elemets of the matrix satisfy M ij 0. The matrix M is therefore a stochastic matrix. Importace samplig We eed to replace the brute force Metropolis algorithm with a walk i coordiate space biased by the trial wave fuctio. This approach is based o the Fokker-Plack equatio ad the Lagevi equatio for geeratig a trajectory i coordiate space. The lik betwee the Fokker-Plack equatio ad the Lagevi equatios are explaied, oly partly, i the slides below. A excellet referece o topics like Browia motio, Markov chais, the Fokker-Plack equatio ad the Lagevi equatio is the text by Va Kampe Here we will focus first o the implemetatio part first. For a diffusio process characterized by a time-depedet probability desity P(x, t) i oe dimesio the Fokker-Plack equatio reads (for oe particle /walker) P t = D x ( x F ) P(x, t), where F is a drift term ad D is the diffusio coefficiet. Importace samplig The ew positios i coordiate space are give as the solutios of the Lagevi equatio usig Euler s method, amely, we go from the Lagevi equatio x(t) = DF (x(t)) + η, t with η a radom variable, yieldig a ew positio y = x + DF (x) t + ξ t, where ξ is gaussia radom variable ad t is a chose time step. The quatity D is, i atomic uits, equal to /2 ad comes from the factor /2 i the kietic eergy operator. Note that t is to be viewed as a parameter. Values of t [0.00, 0.0] yield i geeral rather stable values of the groud state eergy.

8 Importace samplig The process of isotropic diffusio characterized by a time-depedet probability desity P(x, t) obeys (as a approximatio) the so-called Fokker-Plack equatio P t = i D x i ( x i F i ) P(x, t), where F i is the i th compoet of the drift term (drift velocity) caused by a exteral potetial, ad D is the diffusio coefficiet. The covergece to a statioary probability desity ca be obtaied by settig the left had side to zero. The resultig equatio will be satisfied if ad oly if all the terms of the sum are equal zero, 2 P x 2 = P F i + F i P. i x i x i Importace samplig The drift vector should be of the form F = g(x) P x. The, 2 P x 2 = P g i P ( ) P 2 ( ) + Pg 2 P P 2 x i x 2 + g. i x i The coditio of statioary desity meas that the left had side equals zero. I other words, the terms cotaiig first ad secod derivatives have to cacel each other. It is possible oly if g = P, which yields F = 2 Ψ T Ψ T, which is kow as the so-called quatum force. This term is resposible for pushig the walker towards regios of cofiguratio space where the trial wave fuctio is large, icreasig the efficiecy of the simulatio i cotrast to the Metropolis algorithm where the walker has the same probability of movig i every directio. Importace samplig The Fokker-Plack equatio yields a (the solutio to the equatio) trasitio probability give by the Gree s fuctio G(y, x, t) = (4πD t) 3N/2 exp ( (y x D tf (x)) 2 /4D t ) which i tur meas that our brute force Metropolis algorithm A(y, x) = mi(, q(y, x))), with q(y, x) = Ψ T (y) 2 / Ψ T (x) 2 is ow replaced by the Metropolis-Hastigs algorithm as well as Hastig s article, q(y, x) = G(x, y, t) Ψ T (y) 2 G(y, x, t) Ψ T (x) 2 Importace samplig, program elemets Importace samplig, program elemets The full code is this lik. Here we iclude oly the parts pertaiig to the computatio of the quatum force ad the Metropolis update. The program is a modficatio of our previous c++ program discussed previously. Here we display oly the part from the vmcsolver.cpp file. Note the usage of the fuctio GaussiaDeviate. void VMCSolver::ruMoteCarloItegratio() { rold = zeros<mat>(particles, Dimesios); rnew = zeros<mat>(particles, Dimesios); QForceOld = zeros<mat>(particles, Dimesios); QForceNew = zeros<mat>(particles, Dimesios); double wavefuctioold = 0; double wavefuctionew = 0; double eergysum = 0; double eergysquaredsum = 0; double deltae; // iitial trial positios for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rold(i,j) = GaussiaDeviate(&idum)*sqrt(timestep); rnew = rold; Importace samplig, program elemets for(it cycle = 0; cycle < Cycles; cycle++) { // Store the curret value of the wave fuctio wavefuctioold = wavefuctio(rold); QuatumForce(rOld, QForceOld); QForceOld = QForceOld*h/waveFuctioOld; // New positio to test for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rnew(i,j) = rold(i,j) + GaussiaDeviate(&idum)*sqrt(timestep)+QForceOld(i,j)*timestep*D; // for the other particles we eed to set the positio to the old positio sice // we move oly oe particle at the time for (it k = 0; k < Particles; k++) { if ( k!= i) { for (it j=0; j < Dimesios; j++) { rnew(k,j) = rold(k,j); // loop over Mote Carlo cycles // Recalculate the value of the wave fuctio ad the quatum force wavefuctionew = wavefuctio(rnew); QuatumForce(rNew,QForceNew) = QForceNew*h/waveFuctioNew; // we compute the log of the ratio of the grees fuctios to be used i the // Metropolis-Hastigs algorithm GreesFuctio = 0.0; for (it j=0; j < Dimesios; j++) { GreesFuctio += 0.5*(QForceOld(i,j)+QForceNew(i,j))* (D*timestep*0.5*(QForceOld(i,j)-QForceNew(i,j))-rNew(i,j)+rOld(i,j)); GreesFuctio = exp(greesfuctio); // The Metropolis test is performed by movig oe particle at the time if(ra2(&idum) <= GreesFuctio*(waveFuctioNew*waveFuctioNew) / (wavefuctioold*wavefuctioold)) { for(it j = 0; j < Dimesios; j++) { rold(i,j) = rnew(i,j); QForceOld(i,j) = QForceNew(i,j); wavefuctioold = wavefuctionew; else { for(it j = 0; j < Dimesios; j++) { rnew(i,j) = rold(i,j); QForceNew(i,j) = QForceOld(i,j);

9 Importace samplig, program elemets Note umerical derivatives double VMCSolver::QuatumForce(cost mat &r, mat &QForce) { mat rplus = zeros<mat>(particles, Dimesios); mat rmius = zeros<mat>(particles, Dimesios); rplus = rmius = r; double wavefuctiomius = 0; double wavefuctioplus = 0; double wavefuctiocurret = wavefuctio(r); // Kietic eergy double kieticeergy = 0; for(it i = 0; i < Particles; i++) { for(it j = 0; j < Dimesios; j++) { rplus(i,j) += h; rmius(i,j) -= h; wavefuctiomius = wavefuctio(rmius); wavefuctioplus = wavefuctio(rplus); QForce(i,j) = (wavefuctioplus-wavefuctiomius); rplus(i,j) = r(i,j); rmius(i,j) = r(i,j); Importace samplig, program elemets The geeral derivative formula of the Jastrow factor is (the subscript C stads for Correlatio) Ψ C Ψ C x k = k i= g ik x k + i=k+ g ki x k However, with our writte i way which ca be reused later as Ψ C = g(r ij ) = exp f (r ij ), i<j i<j the gradiet eeded for the quatum force ad local eergy is easy to compute. The fuctio f (r ij ) will depeds o the system uder study. I the equatios below we will keep this geeral form. Importace samplig, program elemets I the Metropolis/Hastig algorithm, the acceptace ratio determies the probability for a particle to be accepted at a ew positio. The ratio of the trial wave fuctios evaluated at the ew ad curret positios is give by (OB for the oebody part) R Ψew T Ψ old T = Ψew OB Ψ old OB Ψ ew C Ψ old C Here Ψ OB is our oebody part (Slater determiat or product of boso sigle-particle states) while Ψ C is our correlatio fuctio, or Jastrow factor. We eed to optimize the Ψ T /Ψ T ratio ad the secod derivative as well, that is the 2 Ψ T /Ψ T ratio. The first is eeded whe we compute the so-called quatum force i importace samplig. The secod is eeded whe we compute the kietic eergy term of the local eergy. Ψ Ψ = (Ψ OB Ψ C ) Ψ OB Ψ C = Ψ C Ψ OB + Ψ OB Ψ C Ψ OB Ψ C = Ψ OB Ψ OB + Ψ C Ψ C Importace samplig The expectatio value of the kietic eergy expressed i atomic uits for electro i is ˆK i = 2 Ψ 2 i Ψ, Ψ Ψ ˆK i = 2 i Ψ 2 Ψ. Importace samplig Importace samplig We have defied the correlated fuctio as The secod derivative which eters the defiitio of the local eergy is 2 Ψ Ψ = 2 Ψ OB + 2 Ψ C + 2 Ψ OB Ψ C Ψ OB Ψ C Ψ OB Ψ C We discuss here how to calculate these quatities i a optimal way, Ψ C = N N N g(r ij ) = g(r ij ) = g(r ij ), i<j i<j i= j=i+ with r ij = r i r j = (x i x j ) 2 + (y i y j ) 2 + (z i z j ) 2 i three dimesios or r ij = r i r j = (x i x j ) 2 + (y i y j ) 2 if we work with two-dimesioal systems. I our particular case we have Ψ C = g(r ij ) = exp f (r ij ). i<j i<j

10 Importace samplig Importace samplig The total umber of differet relative distaces r ij is N(N )/2. I a matrix storage format, the relative distaces form a strictly upper triagular matrix 0 r,2 r,3 r,n. 0 r 2,3 r 2,N r rn,n This applies to g = g(r ij ) as well. I our algorithm we will move oe particle at the time, say the kth-particle. This samplig will be see to be particularly efficiet whe we are goig to compute a Slater determiat. We have that the ratio betwee Jastrow factors R C is give by R C = Ψew C Ψ cur C For the Pade-Jastrow form where k U = k = i= g ew ik g cur ik N gki ew g cur. i=k+ ki R C = Ψew C exp Uew Ψ cur = = exp U, C exp U cur i= ( f ew ik fik cur ) N + i=k+ ( f ew ki fki cur ) Importace samplig Importace samplig Oe eeds to develop a special algorithm that rus oly through the elemets of the upper triagular matrix g ad have k as a idex. The expressio to be derived i the followig is of iterest whe computig the quatum force ad the kietic eergy. It has the form i Ψ C = Ψ C, Ψ C Ψ C x i for all dimesios ad with i ruig over all particles. For the first derivative oly N terms survive the ratio because the g-terms that are ot differetiated cacel with their correspodig oes i the deomiator. The, Ψ C k g ik g ki = +. Ψ C x k g ik x k g ki x k i= i=k+ A equivalet equatio is obtaied for the expoetial form after replacig g ij by exp(f ij ), yieldig: Ψ C Ψ C x k = k i= g ik x k + i=k+ g ki x k, with both expressios scalig as O(N). Importace samplig Importace samplig Usig the idetity g ij = g ij, x i x j we get expressios where all the derivatives actig o the particle are represeted by the secod idex of g: Ψ C k g ik = Ψ C x k g ik x k i= ad for the expoetial case: Ψ C Ψ C x k = k i= g ik x k g ki, g ki x i i=k+ i=k+ g ki x i. For correlatio forms depedig oly o the scalar distaces r ij we ca use the chai rule. Notig that we arrive at g ij x j = g ij r ij r ij = x j x i x j r ij g ij r ij, Ψ C k r = ik g ik r ki g ki. Ψ C x k g ik r ik r ik g ki r ki r ki i= i=k+

11 Importace samplig Importace samplig Note that for the Pade-Jastrow form we ca set g ij g(r ij ) = e f (rij ) = e fij ad Therefore, where g ij f ij = g ij. r ij r ij Ψ C k r = ik f ik r ki f ki, Ψ C x k r ik r ik r ki r ki i= i=k+ The secod derivative of the Jastrow factor divided by the Jastrow factor (the way it eters the kietic eergy) is [ 2 ] Ψ C k 2 g ik = 2 Ψ C x x 2 k= i= k ( k g ik + x k k= i= i=k+ ) 2 g ki x i r ij = r j r i = (x j x i )e + (y j y i )e 2 + (z j z i )e 3 is the relative distace. Importace samplig Use the C++ radom class for radom umber geeratios But we have a simple form for the fuctio, amely Ψ C = exp f (r ij ), i<j ad it is easy to see that for particle k we have 2 k Ψ C Ψ C = ij k (r k r i )(r k r j ) f (r ki )f (r kj )+ ( f (r kj ) + 2 ) f (r kj ) r ki r kj r kj j k // Iitialize the seed ad call the Mersiee algo std::radom_device rd; std::mt9937_64 ge(rd()); // Set up the uiform distributio for x \i [[0, ] std::uiform_real_distributio<double> UiformNumberGeerator(0.0,.0); std::ormal_distributio<double> Normaldistributio(0.0,.0); Use the C++ radom class for RNGs, the Mersee twister class Use the C++ radom class for RNGs, the Metropolis test Fidig the ew positio for importace samplig for (it cycles = ; cycles <= NumberMCsamples; cycles++){ // ew positio for (it i = 0; i < NumberParticles; i++) { for (it j = 0; j < Dimesio; j++) { // gaussia deviate to compute ew positios usig a give timestep NewPositio(i,j) = OldPositio(i,j) + Normaldistributio(ge)*sqrt(timestep)+OldQuatumForce(i,j)*timestep*D; Usig the uiform distributio for the Metropolis test // Metropolis-Hastigs algorithm double GreesFuctio = 0.0; for (it j = 0; j < Dimesio; j++) { GreesFuctio += 0.5*(OldQuatumForce(i,j)+NewQuatumForce(i,j))* (D*timestep*0.5*(OldQuatumForce(i,j)-NewQuatumForce(i,j))-NewPositio(i,j)+OldPositio(i,j)); GreesFuctio = exp(greesfuctio); // The Metropolis test is performed by movig oe particle at the time if(uiformnumbergeerator(ge) <= GreesFuctio*NewWaveFuctio*NewWaveFuctio/OldWaveFuctio/OldWaveFuctio ) { for (it j = 0; j < Dimesio; j++) { OldPositio(i,j) = NewPositio(i,j); OldQuatumForce(i,j) = NewQuatumForce(i,j); OldWaveFuctio = NewWaveFuctio;

12 A stochastic process is simply a fuctio of two variables, oe is the time, the other is a stochastic variable X, defied by specifyig the set {x of possible values for X ; the probability distributio, w X (x), over this set, or briefly w(x) The set of values {x for X may be discrete, or cotiuous. If the set of values is cotiuous, the w X (x) is a probability desity so that w X (x)dx is the probability that oe fids the stochastic variable X to have values i the rage [x, x + dx]. A arbitrary umber of other stochastic variables may be derived from X. For example, ay Y give by a mappig of X, is also a stochastic variable. The mappig may also be time-depedet, that is, the mappig depeds o a additioal variable t Y X (t) = f (X, t). The quatity Y X (t) is called a radom fuctio, or, sice t ofte is time, a stochastic process. A stochastic process is a fuctio of two variables, oe is the time, the other is a stochastic variable X. Let x be oe of the possible values of X the y(t) = f (x, t), is a fuctio of t, called a sample fuctio or realizatio of the process. I physics oe cosiders the stochastic process to be a esemble of such sample fuctios. For may physical systems iitial distributios of a stochastic variable y ted to equilibrium distributios: w(y, t) w 0(y) as t. I equilibrium detailed balace costrais the trasitio rates W (y y )w(y) = W (y y)w 0(y), where W (y y) is the probability, per uit time, that the system chages from a state y, characterized by the value y for the stochastic variable Y, to a state y. Note that for a system i equilibrium the trasitio rate W (y y) ad the reverse W (y y ) may be very differet. Cosider, for istace, a simple system that has oly two eergy levels ɛ 0 = 0 ad ɛ = E. For a system govered by the Boltzma distributio we fid (the partitio fuctio has bee take out) We get the W (0 ) exp (ɛ 0/kT ) = W ( 0) exp (ɛ /kt ) W ( 0) = exp ( E/kT ), W (0 ) which goes to zero whe T teds to zero. If we assume a discrete set of evets, our iitial probability distributio fuctio ca be give by w i (0) = δ i,0, ad its time-developmet after a give time step t = ɛ is w i (t) = j The cotiuous aalog to w i (0) is W (j i)w j (t = 0). w(x) δ(x), where we ow have geeralized the oe-dimesioal positio x to a geeric-dimesioal vector x. The Kroeecker δ fuctio is replaced by the δ distributio fuctio δ(x) at t = 0. The trasitio from a state j to a state i is ow replaced by a trasitio to a state with positio y from a state with positio x. The discrete sum of trasitio probabilities ca the be replaced by a itegral ad we obtai the ew distributio at a time t + t as w(y, t + t) = W (y, t + t x, t)w(x, t)dx, ad after m time steps we have w(y, t + m t) = W (y, t + m t x, t)w(x, t)dx. Whe equilibrium is reached we have w(y) = W (y x, t)w(x)dx, that is o time-depedece. Note our chage of otatio for W

13 We ca solve the equatio for w(y, t) by makig a Fourier trasform to mometum space. The PDF w(x, t) is related to its Fourier trasform w(k, t) through w(x, t) = dk exp (ikx) w(k, t), ad usig the defiitio of the δ-fuctio δ(x) = dk exp (ikx), 2π we see that w(k, 0) = /2π. We ca the use the Fourier-trasformed diffusio equatio with the obvious solutio w(k, t) = Dk 2 w(k, t), t w(k, t) = w(k, 0) exp [ (Dk 2 t) ) = 2π exp [ (Dk 2 t) ]. With the Fourier trasform we obtai w(x, t) = dk exp [ikx] 2π exp [ (Dk 2 t) ] = exp [ (x 2 /4Dt) ], 4πDt with the ormalizatio coditio w(x, t)dx =. The solutio represets the probability of fidig our radom walker at positio x at time t if the iitial distributio was placed at x = 0 at t = 0. There is aother iterestig feature worth observig. The discrete trasitio probability W itself is give by a biomial distributio. The results from the cetral limit theorem state that trasitio probability i the limit coverges to the ormal distributio. It is the possible to show that W (il jl, ɛ) W (y, t+ t x, t) = exp [ ((y x) 2 /4D t) ], 4πD t ad that it satisfies the ormalizatio coditio ad is itself a solutio to the diffusio equatio. Let us ow assume that we have three PDFs for times t 0 < t < t, that is w(x 0, t 0), w(x, t ) ad w(x, t). We have the w(x, t) = W (x.t x.t )w(x, t )dx, ad w(x, t) = W (x.t x 0.t 0)w(x 0, t 0)dx 0, ad w(x, t ) = W (x.t x 0, t 0)w(x 0, t 0)dx 0. We ca combie these equatios ad arrive at the famous Eistei-Smolucheski-Kolmogorov-Chapma (ESKC) relatio W (xt x 0t 0) = W (x, t x, t )W (x, t x 0, t 0)dx. We ca replace the spatial depedece with a depedece upo say the velocity (or mometum), that is we have W (v, t v 0, t 0) = W (v, t v, t )W (v, t v 0, t 0)dx.

14 We will ow derive the Fokker-Plack equatio. We start from the ESKC equatio W (x, t x 0, t 0) = W (x, t x, t )W (x, t x 0, t 0)dx. Defie s = t t 0, τ = t t ad t t 0 = s + τ. We have the W (x, s + τ x 0) = W (x, τ x )W (x, s x 0)dx. Assume ow that τ is very small so that we ca make a expasio i terms of a small step xi, with x = x ξ, that is W (x, s x W 0)+ s τ +O(τ 2 ) = W (x, τ x ξ)w (x ξ, s x 0)dx. We assume that W (x, τ x ξ) takes o-egligible values oly whe ξ is small. This is just aother way of statig the Master equatio!! We say thus that x chages oly by a small amout i the time iterval τ. This meas that we ca make a Taylor expasio i terms of ξ, that is we expad ( ξ) W (x, τ x ξ)w (x ξ, s x 0) = [W (x + ξ, τ x)w (x, s x0)].! x =0 We ca the rewrite the ESKC equatio as W s τ = W (x, s x0)+ ( ξ)! =0 x [ W (x, s x 0) We have eglected higher powers of τ ad have used that for = 0 we get simply W (x, s x 0) due to ormalizatio. ] ξ W (x + ξ, τ x)dξ. We say thus that x chages oly by a small amout i the time iterval τ. This meas that we ca make a Taylor expasio i terms of ξ, that is we expad ( ξ) W (x, τ x ξ)w (x ξ, s x 0) = [W (x + ξ, τ x)w (x, s x0)].! x =0 We ca the rewrite the ESKC equatio as W (x, s x 0) s τ = W (x, s x 0)+ =0 ( ξ)! x [ W (x, s x 0) We have eglected higher powers of τ ad have used that for = 0 we get simply W (x, s x 0) due to ormalizatio. ] ξ W (x + ξ, τ x)dξ.

15 We simplify the above by itroducig the momets M = ξ W (x + ξ, τ x)dξ = [ x(τ)], τ τ resultig i W (x, s x 0) s = = ( ξ)! [W (x, s x0)m]. x Whe τ 0 we assume that [ x(τ)] 0 more rapidly tha τ itself if > 2. Whe τ is much larger tha the stadard correlatio time of system the M for > 2 ca ormally be eglected. This meas that fluctuatios become egligible at large time scales. If we eglect such terms we ca rewrite the ESKC equatio as W (x, s x 0) MW (x, s x0) = + 2 M 2W (x, s x 0) s x 2 x 2. I a more compact form we have W s = MW x + 2 M 2W 2 x 2, which is the Fokker-Plack equatio! It is trivial to replace positio with velocity (mometum). Lagevi equatio Cosider a particle suspeded i a liquid. O its path through the liquid it will cotiuously collide with the liquid molecules. Because o average the particle will collide more ofte o the frot side tha o the back side, it will experiece a systematic force proportioal with its velocity, ad directed opposite to its velocity. Besides this systematic force the particle will experiece a stochastic force F(t). The equatios of motio are dr dt = v ad dv dt = ξv + F. Lagevi equatio From hydrodyamics we kow that the frictio costat ξ is give by ξ = 6πηa/m where η is the viscosity of the solvet ad a is the radius of the particle. Solvig the secod equatio i the previous slide we get t v(t) = v 0e ξt + dτe ξ(t τ) F(τ). 0 Lagevi equatio If we wat to get some useful iformatio out of this, we have to average over all possible realizatios of F(t), with the iitial velocity as a coditio. A useful quatity for example is t v(t) v(t) v0 = v ξ2t dτe ξ(2t τ) v 0 F(τ) v0 0 t t + dτ dτe ξ(2t τ τ ) F(τ) F(τ ) v0. 0 0

16 Lagevi equatio I order to cotiue we have to make some assumptios about the coditioal averages of the stochastic forces. I view of the chaotic character of the stochastic forces the followig assumptios seem to be appropriate F(t) = 0, ad F(t) F(t ) v0 = C v0 δ(t t ). We omit the subscript v 0, whe the quatity of iterest turs out to be idepedet of v 0. Usig the last three equatios we get v(t) v(t) v0 = v 2 0 e 2ξt + Cv0 2ξ ( e 2ξt ). For large t this should be equal to 3kT/m, from which it follows that F(t) F(t ) = 6 kt m ξδ(t t ). Lagevi equatio Itegratig t v(t) = v 0e ξt + dτe ξ(t τ) F(τ), 0 we get t τ r(t) = r 0 + v 0 ξ ( e ξt ) + dτ τ e ξ(τ τ ) F(τ ), 0 0 from which we calculate the mea square displacemet (r(t) r 0) 2 v0 = v 2 0 ξ ( e ξt ) 2 + 3kT mξ 2 (2ξt 3 + 4e ξt e 2ξt ). This result is called the fluctuatio-dissipatio theorem. Why blockig? Lagevi equatio For very large t this becomes (r(t) r 0) 2 = 6kT mξ t from which we get the Eistei relatio D = kt mξ where we have used (r(t) r 0) 2 = 6Dt. Statistical aalysis Mote Carlo simulatios ca be treated as computer experimets The results ca be aalysed with the same statistical tools as we would use aalysig experimetal data. As i all experimets, we are lookig for expectatio values ad a estimate of how accurate they are, i.e., possible sources for errors. A very good article which explais blockig is H. Flyvbjerg ad H. G. Peterse, Error estimates o averages of correlated data, Joural of Chemical Physics 9, (989). Why blockig? Statistical aalysis As i other experimets, Mote Carlo experimets have two classes of errors: Statistical errors Systematical errors Statistical errors ca be estimated usig stadard tools from statistics Systematical errors are method specific ad must be treated differetly from case to case. (I VMC a commo source is the step legth or time step i importace samplig) The probability distributio fuctio (PDF) is a fuctio p(x) o the domai which, i the discrete case, gives us the probability or relative frequecy with which these values of X occur: p(x) = prob(x = x) I the cotiuous case, the PDF does ot directly depict the actual probability. Istead we defie the probability for the stochastic variable to assume ay value o a ifiitesimal iterval aroud x to be p(x)dx. The cotiuous fuctio p(x) the gives us the desity of the probability rather tha the probability itself. The probability for a stochastic variable to assume ay value o a o-ifiitesimal iterval [a, b] is the just the itegral: b prob(a X b) = p(x)dx a Qualitatively speakig, a stochastic variable represets the values of umbers chose as if by chace from some specified PDF so that the selectio of a large set of these umbers reproduces this PDF.

17 Also of iterest to us is the cumulative probability distributio fuctio (CDF), P(x), which is just the probability for a stochastic variable X to assume ay value less tha x: x P(x) = Prob(X x) = p(x )dx The relatio betwee a CDF ad its correspodig PDF is the: p(x) = d dx P(x) A particularly useful class of special expectatio values are the momets. The -th momet of the PDF p is defied as follows: x x p(x) dx The zero-th momet is just the ormalizatio coditio of p. The first momet, x, is called the mea of p ad ofte deoted by the letter µ: x = µ xp(x) dx A special versio of the momets is the set of cetral momets, the -th cetral momet defied as: (x x ) (x x ) p(x) dx The zero-th ad first cetral momets are both trivial, equal ad 0, respectively. But the secod cetral momet, kow as the variace of p, is of particular iterest. For the stochastic variable X, the variace is deoted as σx 2 or var(x ): σx 2 = var(x ) = (x x ) 2 = (x x ) 2 p(x) dx (7) = (x 2 2x x 2 + x 2) p(x) dx (8) = x 2 2 x x + x 2 (9) = x 2 x 2 (0) The square root of the variace, σ = (x x ) 2 is called the stadard deviatio of p. It is clearly just the RMS (root-mea-square) value of the deviatio of the PDF from its mea value, iterpreted qualitatively as the spread of p aroud its mea. Aother importat quatity is the so called covariace, a variat of the above defied variace. Cosider agai the set {X i of stochastic variables (ot ecessarily ucorrelated) with the multivariate PDF P(x,..., x ). The covariace of two of the stochastic variables, X i ad X j, is defied as follows: cov(x i, X j ) (x i x i )(x j x j ) = (x i x i )(x j x j ) P(x,..., x ) dx... dx with x i = x i P(x,..., x ) dx... dx () If we cosider the above covariace as a matrix C ij = cov(x i, X j ), the the diagoal elemets are just the familiar variaces, C ii = cov(x i, X i ) = var(x i ). It turs out that all the off-diagoal elemets are zero if the stochastic variables are ucorrelated. This is easy to show, keepig i mid the liearity of the expectatio value. Cosider the stochastic variables X i ad X j, (i j): cov(x i, X j ) = (x i x i )(x j x j ) (2) = x i x j x i x j x i x j + x i x j (3) = x i x j x i x j x i x j + x i x j (4) = x i x j x i x j x i x j + x i x j (5) = x i x j x i x j (6) If X i ad X j are idepedet, we get x i x j = x i x j, resultig i cov(x i, X j ) = 0 (i j). Also useful for us is the covariace of liear combiatios of stochastic variables. Let {X i ad {Y i be two sets of stochastic variables. Let also {a i ad {b i be two sets of scalars. Cosider the liear combiatio: U = i a i X i V = j By the liearity of the expectatio value b j Y j cov(u, V ) = a i b j cov(x i, Y j ) i,j

18 Now, sice the variace is just var(x i ) = cov(x i, X i ), we get the variace of the liear combiatio U = i a ix i : var(u) = i,j a i a j cov(x i, X j ) (7) Ad i the special case whe the stochastic variables are ucorrelated, the off-diagoal elemets of the covariace are as we kow zero, resultig i: var(u) = i var( i a 2 i cov(x i, X i ) = i a i X i ) = i a 2 i var(x i ) a 2 i var(x i ) which will become very useful i our study of the error i the mea value of a set of measuremets. A stochastic process is a process that produces sequetially a chai of values: {x, x 2,... x k,.... We will call these values our measuremets ad the etire set as our measured sample. The actio of measurig all the elemets of a sample we will call a stochastic experimet sice, operatioally, they are ofte associated with results of empirical observatio of some physical or mathematical pheomea; precisely a experimet. We assume that these values are distributed accordig to some PDF p X (x), where X is just the formal symbol for the stochastic variable whose PDF is p X (x). Istead of tryig to determie the full distributio p we are ofte oly iterested i fidig the few lowest momets, like the mea µ X ad the variace σ X. I practical situatios a sample is always of fiite size. Let that size be. The expectatio value of a sample, the sample mea, is the defied as follows: x x k k= The sample variace is: var(x) (x k x ) 2 k= its square root beig the stadard deviatio of the sample. The sample covariace is: cov(x) (x k x )(x l x ) kl Note that the sample variace is the sample covariace without the cross terms. I a similar maer as the covariace i Eq. () is a measure of the correlatio betwee two stochastic variables, the above defied sample covariace is a measure of the sequetial correlatio betwee succeedig measuremets of a sample. These quatities, beig kow experimetal values, differ sigificatly from ad must ot be cofused with the similarly amed quatities for stochastic variables, mea µ X, variace var(x ) ad covariace cov(x, Y ). The law of large umbers states that as the size of our sample grows to ifiity, the sample mea approaches the true mea µ X of the chose PDF: lim x = µ X The sample mea x works therefore as a estimate of the true mea µ X. What we eed to fid out is how good a approximatio x is to µ X. I ay stochastic measuremet, a estimated mea is of o use to us without a measure of its error. A quatity that tells us how well we ca reproduce it i aother experimet. We are therefore iterested i the PDF of the sample mea itself. Its stadard deviatio will be a measure of the spread of sample meas, ad we will simply call it the error of the sample mea, or just sample error, ad deote it by err X. I practice, we will oly be able to produce a estimate of the sample error sice the exact value would require the kowledge of the true PDFs behid, which we usually do ot have. The straight forward brute force way of estimatig the sample error is simply by producig a umber of samples, ad treatig the mea of each as a measuremet. The stadard deviatio of these meas will the be a estimate of the origial sample error. If we are uable to produce more tha oe sample, we ca split it up sequetially ito smaller oes, treatig each i the same way as above. This procedure is kow as blockig ad will be give more attetio shortly. At this poit it is worth while explorig more idirect methods of estimatio that will help us uderstad some importat uderlyig priciples of correlatioal effects.

19 which is othig but the sample covariace divided by the umber of measuremets i the sample. Let us first take a look at what happes to the sample error as the size of the sample grows. I a sample, each of the measuremets x i ca be associated with its ow stochastic variable X i. The stochastic variable X for the sample mea x is the just a liear combiatio, already familiar to us: X = X i i= All the coefficiets are just equal /. The PDF of X, deoted by p X (x) is the desired PDF of the sample meas. The probability desity of obtaiig a sample mea x is the product of probabilities of obtaiig arbitrary values x, x 2,..., x with the costrait that the mea of the set {x i is x : ( p X (x) = p X (x ) p X (x ) δ x x + x2 + + x Ad i particular we are iterested i its variace var(x ). ) dx dx It is geerally ot possible to express p X (x) i a closed form give a arbitrary PDF p X ad a umber. But for the limit it is possible to make a approximatio. The very importat result is called the cetral limit theorem. It tells us that as goes to ifiity, p X (x) approaches a Gaussia distributio whose mea ad variace equal the true mea ad variace, µ X ad σ 2 X, respectively: ( lim p X (x) = 2πvar(X ) ) /2 e (x x)2 2var(X ) (8) The desired variace var(x ), i.e. the sample error squared err 2 X, is give by: err 2 X = var(x ) = 2 cov(x i, X j ) (9) We see ow that i order to calculate the exact error of the sample with the above expressio, we would eed the true meas µ Xi of the stochastic variables X i. To calculate these requires that we kow the true multivariate PDF of all the X i. But this PDF is ukow to us, we have oly got the measuremets of oe sample. The best we ca do is to let the sample itself be a estimate of the PDF of each of the X i, estimatig all properties of X i through the measuremets of the sample. ij Our estimate of µ Xi is the the sample mea x itself, i accordace with the the cetral limit theorem: µ Xi = x i x k = x k= Usig x i place of µ Xi we ca give a estimate of the covariace i Eq. (9) cov(x i, X j ) = (x i x i )(x j x j ) (x i x)(x j x), resultig i ( ) (x k x )(x l x ) = (x k x )(x l x ) = cov(x) l k kl By the same procedure we ca use the sample variace as a estimate of the variace of ay of the stochastic variables X i which is approximated as var(x i ) = x i x i x i x, var(x i ) (x k x ) = var(x) (20) k= Now we ca calculate a estimate of the error err X of the sample mea x : err 2 X = 2 cov(x i, X j ) ij 2 cov(x) = 2 2 cov(x) ij = cov(x) (2)