Introduction Classical Monte Carlo Quantum Monte Carlo Summary Introduction to Monte Carlo Methods Tim Schoof 3rd Graduate Summer Institute “Complex Plasmas” August 6, 2012 Introduction Classical Monte Carlo Quantum Monte Carlo Summary Applications of Monte Carlo Introduction Classical Monte Carlo Quantum Monte Carlo Summary Outline 1 Introduction History Monte Carlo integration 2 Classical Monte Carlo Classical thermodynamics Metropolis algorithm Details 3 Quantum Monte Carlo Quantum thermodynamics Path integral Sign problem 4 Summary Introduction Classical Monte Carlo Quantum Monte Carlo Summary Table of contents 1 Introduction History Monte Carlo integration 2 Classical Monte Carlo Classical thermodynamics Metropolis algorithm Details 3 Quantum Monte Carlo Quantum thermodynamics Path integral Sign problem 4 Summary Introduction Classical Monte Carlo Quantum Monte Carlo Summary Monte Carlo Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results Invented by Stanislaw Ulam and John von Neumann Named after the Monte Carlo Casino, where Ulam’s uncle often gambled away his money Introduction Classical Monte Carlo Quantum Monte Carlo Summary The basic idea In 1946, being ill and playing solitaire, Ulam was interested in the chances to win the game. After spending a lot of time trying to estimate them by pure combinatorial calculation, he came up with the idea to simply lay out the cards 100 times and count the number of successful plays. Soon he realizes that this approach can be applied to the problem of neutron transport Stanislaw Ulam with FERMIAC Introduction Classical Monte Carlo Quantum Monte Carlo Summary The beginnings Together with von Neumann the first calculations were performed for the JOURNAL OF THE AMERICAN Manhatten Project at Los Alamos on the STATISTICAL ASSOCIATION ENIAC Number 167 SEPTEMBER 1949 Volume 44 The ENIAC was the first electronic THE MONTE CARL0 METHOD NICHOLASMETROPOLIS AND S. ULAM Los Alamos Laboratory general-purpose computer We shall present here the motivation and a general descrip- tion of a method dealing with a class of problems in mathe- matical physics. The method is, essentially, a statistical approach to the study of differential equations, or more The first unclassified paper on Monte Carlo generally, of integro-differential equations that occur in various branches of the natural sciences. was published in 1949 LREADY in the nineteenth century a sharp distinction began to ap- A pear between two different mathematical methods of treating physical phenomena. Problems involving only a few particles were studied in classical mechanics, through the study of systems of ordinary differential equations. For the description of systems with very many particles, an entirely different technique was used, namely, the method of statistical mechanics. In this latter approach, one does not concen- trate on the individual particles but studies the properties of sek of particles. In pure mathematics an intensive study of the properties of sets of points was the subject of a newTfield. This is the so-called theory of sets, the basic theory of integration, and the twentieth century de- velopment of the theory of probabilities prepared the formal apparatus for the use of such models in theoretical physics, i.e., description of properties of aggregates of points rather than of individual points and their coordinates. Soon after khe development of the calculus, the mathematical ap- paratus of partial differential equations was used for dealing with the problems of the physics of the continuum. Hydrodynamics is the most widely known field formulated in this fashion. A little later came the treatment of the problems of heat conduction and still later the field theories, like the electromagnetic theory of Maxwell. All this is very well known. It is of course important to remember that the study of the Introduction Classical Monte Carlo Quantum Monte Carlo Summary Basic example: Area of the unit circle Choose M random points that are uniformly distributed within the enclosing square Count the number of points N that are within the unit circle An estimation for the area of the circle is given by N A ≈ 22 M In the left picture it is A ≈ 3:15 Introduction Classical Monte Carlo Quantum Monte Carlo Summary Area of the m-dimensional unit hypersphere m Exact 2 3.1415 3 4.1887 4 4.9348 5 5.2637 6 5.1677 7 4.7247 8 4.0587 Exact result m m md π 2 r V = m Γ( 2 + 1) Introduction Classical Monte Carlo Quantum Monte Carlo Summary Area of the m-dimensional unit hypersphere m Exact quad. time result 2 3.1415 0.00 3:13 3 4.1887 1:0 · 10−4 4:21 4 4.9348 1:2 · 10−3 4:97 5 5.2637 0.03 5:29 6 5.1677 0.62 5:20 7 4.7247 14.9 4:77 8 4.0587 369 4:09 Numerical integration by quadrature, e.g., for 3d Z q V 3d = 2 dx dy z(x; y) with z(x; y) = 1 − (x 2 + y 2) x 2+y 2≤1 Introduction Classical Monte Carlo Quantum Monte Carlo Summary Area of the m-dimensional unit hypersphere m Exact quad. time result MC time result 2 3.1415 0.00 3:13 0.07 3:14 3 4.1887 1:0 · 10−4 4:21 0.09 4:19 4 4.9348 1:2 · 10−3 4:97 0.12 4:93 5 5.2637 0.03 5:29 0.14 5:27 6 5.1677 0.62 5:20 0.17 5:17 7 4.7247 14.9 4:77 0.19 4:72 8 4.0587 369 4:09 0.22 4:07 Monte Carlo integration m m-dimensional points ~xi are uniformly sampled in volume V = 2 M 1 X V md = 2m Θ(~x ) with Θ(~x ) = 1 if ~x · ~x ≤ 1 M i i i i i=1 Introduction Classical Monte Carlo Quantum Monte Carlo Summary Area of the m-dimensional unit hypersphere m Exact quad. time result MC time result 2 3.1415 0.00 3:13 0.07 3:14 3 4.1887 1:0 · 10−4 4:21 0.09 4:19 4 4.9348 1:2 · 10−3 4:97 0.12 4:93 5 5.2637 0.03 5:29 0.14 5:27 6 5.1677 0.62 5:20 0.17 5:17 7 4.7247 14.9 4:77 0.19 4:72 8 4.0587 369 4:09 0.22 4:07 =) Monte Carlo integration is faster for m > 6 Conclusion Monte Carlo integration is efficient for high dimensional integrals Introduction Classical Monte Carlo Quantum Monte Carlo Summary Monte Carlo integration Straightforward sampling Random points fxi g are chosen uniformly M Z b b − a X I = f (x)dx ≈ f (x ) M i a i=1 Importance sampling fxi g are chosen with the probability p(x) Z b M f (x) 1 X f (xi ) I = p(x)dx ≈ p(x) M p(xi ) a i=1 p(x) should be close to jf (x)j Introduction Classical Monte Carlo Quantum Monte Carlo Summary Table of contents 1 Introduction History Monte Carlo integration 2 Classical Monte Carlo Classical thermodynamics Metropolis algorithm Details 3 Quantum Monte Carlo Quantum thermodynamics Path integral Sign problem 4 Summary Introduction Classical Monte Carlo Quantum Monte Carlo Summary Thermodynamic expectation values In thermodynamic equilibrium classical expectation values of observables O are given by Z dPdR hOi = O(R; P)p(R; P) h3N N! with R = ~r1;:::;~rN ; P = ~p1;:::; ~pN and the integral is over the whole phase space In the canonical ensemble for fixed T ; V ; N the probability p(R; P) of a microstate (R; P) is given by 1 1 p(R; P) = e−βH(R;P); β = Z kBT Z dPdR Z(T ; V ; N) = e−βH(R;P) h3N N! It is Z dPdR p(R; P) = 1; p(R; P) ≥ 0 8 R; P h3N N! Introduction Classical Monte Carlo Quantum Monte Carlo Summary Metropolis algorithm Developed by Marshall N. Rosenbluth THE 0 R Y 0 F T RAe KEF FEe T SIN R A D I 0 L Y SIS 0 F W ATE R 1087 instead, only water molecules with different amounts of paths (a) and (b) can be designated H 20* and those excitation energy. These may follow any of three paths: following path (c) can be designated H 20t. It seems Named after Nicholas Metropolis the head reasonable to assume for the purpose of these calcula­ (a) The excitation energy is lost without dissociation tions that the ionized H 0 molecules will become the into radicals (by collision, or possibly radiation, as in 2 H 20t molecules, but this is not likely to be a complete aromatic hydrocarbons). correspondence. (b) The molecules dissociate, but the resulting radi­ designer of the MANIAC In conclusion we would like to emphasize that the cals recombine without escaping from the liquid cage. qualitative result of this section is not critically de­ (c) The molecules dissociate and escape from the pendent on the exact values of the physical parameters cage. In this case we would not expect them to move used. However, this treatment is classical, and a correct more than a few molecular diameters through the dense treatment must be wave mechanical; therefore the medium before being thermalized. result of this section cannot be taken as an a priori First paper on 2d rigid spheres published in 1953 theoretical prediction. The success of the radical diffu­ In accordance with the notation introduced by Burton, Magee, and Samuel,22 the molecules following sion model given above lends some plausibility to the occurrence of electron capture as described by this One of the most important algorithms of the 20th 22 Burton, Magee, and Samuel, J.