Monte Carlo

Monte Carlo /ˈmɒnti ˈkɑːrloʊ/ (Italian pronunciation: [ˈmonte ˈkarlo]; French: Monte-Carlo or colloquially Monte-Carl, Monégasque: Monte-Carlu) officially refers to an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is located. Informally the name also refers to a larger district, the Monte Carlo Quarter (corresponding to the former municipality of Monte Carlo), which besides Monte Carlo/Spélugues also includes the wards of La Rousse/Saint Roman, Larvotto/Bas Moulins, and Saint Michel. The permanent population of the ward of Monte Carlo is about 3,500, while that of the quarter is about 15,000. Monaco has four traditional quarters. From west to east they are: Fontvieille (the newest), Monaco-Ville (the oldest), La Condamine, and Monte Carlo.

Monte Carlo (literally "Mount Charles") is situated on a prominent escarpment at the base of the Maritime Alps along the French Riviera. Near the western end of the quarter is the world-famous Place du Casino, the gambling center which has made Monte Carlo "an international byword for the extravagant display and reckless dispersal of wealth". It is also the location of the Hôtel de Paris, the Café de Paris, and the Salle Garnier (the casino theatre which is the home of the Opéra de Monte-Carlo).

Diffusion Monte Carlo

Diffusion Monte Carlo (DMC) is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system. When actually attempting the calculation, one finds that for bosons, the algorithm scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, DMC employing a clever approximation known as the fixed-node approximation can still yield very accurate results. What follows is an explanation of the basic algorithm, how it works, why fermions cause a problem, and how the fixed-node approximation resolves this problem.

The Projector Method

To motivate the algorithm, let's look at the Schrödinger equation for a particle in some potential in one dimension:

We can condense the notation a bit by writing it in terms of an operator equation, with

David Ceperley

David Ceperley is a theoretical physicist in the physics department at the University of Illinois Urbana-Champaign or UIUC. He is a world expert in the area of Quantum Monte Carlo computations, a method of calculation that is generally recognized to provide accurate quantitative results for many-body problems described by quantum mechanics.

Life, education and career

David Ceperley was born in Charleston, West Virginia USA in 1949, and attended George Washington High School there. He was student at Atlantic College in Wales UK, received a BS degree in physics and mathematics from the University of Michigan, Ann Arbor in 1971 and a PhD in theoretical physics at Cornell University in 1976. His advisors were Geoffrey Chester at Cornell University and Malvin Kalos at the Courant Institute at New York University. He had postdocs in Orsay, France, New York University and Rutgers University where he worked with Joel Lebowitz on the simulation of polymers. He was a staff scientist at the National Resource for Computational Chemistry at Lawrence Berkeley National Laboratory and Lawrence Livermore National Laboratory from 1978-1987. Since 1987, he has been a professor at the University of Illinois Urbana-Champaign and a staff member at the National Center for Supercomputing Applications from 1987-2012. In his development as a leading mathematical and computational physicist David Ceperley had a number of acknowledged mentors, many of whom happen to be his former supervisors such as Geoffrey Chester, Malvin Kalos, Joel Lebowitz and Berni Alder. At UIUC he has been influenced by the physics Nobel Laureates Anthony Leggett and John Bardeen.

Diffusion

Diffusion is the net movement of molecules or atoms from a region of high concentration (or high chemical potential) to a region of low concentration (or low chemical potential). This is also referred to as the movement of a substance down a concentration gradient. A gradient is the change in the value of a quantity (e.g., concentration, pressure, temperature) with the change in another variable (usually distance). For example, a change in concentration over a distance is called a concentration gradient, a change in pressure over a distance is called a pressure gradient, and a change in temperature over a distance is a called a temperature gradient.

The word diffusion is derived from the Latin word, "diffundere", which means "to spread out" (if a substance is “spreading out”, it is moving from an area of high concentration to an area of low concentration). A distinguishing feature of diffusion is that it results in mixing or mass transport, without requiring bulk motion (bulk flow). Thus, diffusion should not be confused with convection, or advection, which are other transport phenomena that utilize bulk motion to move particles from one place to another.