P T A typical experiment in a real (not virtual) space 1.Some material is put in a container at fixed T & P. 2.The material is in a thermal fluctuation, producing lots of different configurations (a set of microscopic states) for a given amount of time. It is the Mother Nature who generates all the microstates. 3.An apparatus is plugged to measure an observable (a macroscopic quantity) as an average over all the microstates produced from thermal fluctuation. P T How do we mimic the Mother Nature in a virtual space to realize lots of microstates, all of which correspond to a given macroscopic state? How do we mimic the apparatus in a virtual space to obtain a macroscopic quantity (or property or observable) as an average over all the microstates? P T

microscopic states (microstates) or microscopic configurations under external constraints (N or , V or P, T or E, etc.)  Ensemble (micro-canonical, canonical, grand canonical, etc.) Average over a collection of microstates Macroscopic quantities (properties, observables) thermodynamic –  or N, E or T, P or V, C v, C p, H,  S,  G, etc. structural – pair correlation function g(r), etc. dynamical – diffusion, etc. These are what are measured in true experiments. they’re generated naturally from thermal fluctuation In a real-space experimentIn a virtual-space simulation How do we mimic the Mother Nature in a virtual space to realize lots of microstates, all of which correspond to a given macroscopic state? By MC or MD method! it is us who needs to generate them by MC or MD methods. t1t1 t2t2 t3t3 ~10 23 particles

Molecular Dynamics (MD) vs. Monte Carlo (MC) Molecular Dynamics Simulation (Deterministic) Starts from the initial microstate (a collection of positions & velocities) Solves the Newton equation of motion under the inter-particle potential V (and force F) Microstates generated by integration over time Time evolution (trajectory), dynamic behavior over time Gives a direct connection with true experiments Gives both equilibrium properties and dynamic properties Monte Carlo Simulation (Stochastic = random or based on random numbers) Microstates generated by stochastic sampling involving a random number generator Gives equilibrium properties only (no time dependence, no dynamics!) Solves mathematical problems using stochastic sampling (like rolling a dice) Performs simulation of any process whose development is influenced by random factors, but also the method enables artificial construction of a probabilistic model Randomly selects values to fit a probability distribution (e.g. bell curve, linear distribution, etc.) to create scenarios of a problem Apt to consume large computing resources (“method of last resort”) Historically executed on the fastest computers available at the time V

“Monte Carlo” Casino

John von Neumann, Stan Ulam, and Nick Metropolis are considered to found the method from the collaboration at Los Alamos on the Manhattan project during the World War II. The name "Monte Carlo" comes from the Monte Carlo Casino (gambling house) in Monaco and first appeared in the article "The Monte Carlo Method" by Metropolis and Ulam (1949). Well before 1949, certain problems in statistics were solved by means of random sampling. Buffon experimentally determined a value of  by casting a needle on a ruled grid (1768) and Fredericks & Levy showed how it can be used to solve boundary value problems (1928). Kelvin used random sampling techniques to initialize trajectories of particles undergoing elastic collision with container walls (1901). This led to the failure of the equi-partition law and the to foundation of statistical mechanics. Fermi used this method in the calculation of neutron diffusion in nuclear reactors (1930's). A formal foundation for the method was developed by von Neumann (PDE) (1940’s). However, simulation of random variables by hand was a laborious process. Stan Ulam realized the importance of the computer in the implementation of the approach. Using MC as a universal numerical technique became practical only with the advent of computers (ENIAC, MANIAC, etc.) and high-quality pseudorandom number generators. History of Monte Carlo Simulation

Various Applications of Monte Carlo Techniques Integration (especially of high-dimensional functions) System simulation Physical phenomena – nuclear power, radiation, thermodynamics, etc. (The use of MC in the area of nuclear power has undergone an important evolution.) Quantum Monte Carlo – wave functions and expectation values (QMC gives most accurate method for general quantum many-body systems.) Simulation of games (bingo, solitaire, etc.) Weather, Equipment Productivity, Risk Analysis and Management VLSI designs - tolerance analysis Computer graphics – rendering Projects are often associated with a high degree of uncertainty and complexity resulting from the unpredictable nature of events and the multi-dimensionality. MC generates multiple scenarios depending upon the assumptions fed into the model. MC calculates multiple scenarios by repeatedly inserting different sampling values from probability distribution for the uncertain variables into the computerized spread-sheet.

1.Analytical integration, if possible Not for many functions (very limited) 2.Numerical integration (summation) Rectangular/trapezoidal/parabolic rules 3.Numerical integration (Monte Carlo) Random sampling of the area enclosed by a<x<b and 0<y<f max (x) MC Application No. 1. How to evaluate integrals f(x) ab x A 0

3.Numerical integration (Hit-and-Miss Monte Carlo) Random sampling of the area enclosed by a<x<b and 0<y<f max (x) MC Application No. 1. How to evaluate integrals f(x) ab x A f max (x) f(x) ab f max (x) x  Choose at random M points in x  [a,b]. Designate the number of points lying under the curve y = f(x) by M'. It is geometrically obvious that the area of A is approximately equal to the ratio M'/M. The greater the number of drawings or trials (M), the greater the accuracy of this estimate.

 = 4 A where A = area of the first quadrant of a circle of the radius r = 1 1st example of MC: Let’s calculate  ! Hit-and-Miss (or Rejection) MC Method Equivalent to integrating the equation of the circle x y A O (x i,y i )

 = … 1st example of MC: Let’s calculate  ! Hit-and-Miss (or Rejection) MC Method

N = 10,000Pi= N = 100,000Pi= N = 1,000,000Pi= N = 10,000,000Pi= … 1st example of MC: Let’s calculate  ! Hit-and-Miss (or Rejection) MC Method