Reptation Monte Carlo is a

quantum Monte Carlo Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the ...

method. It is similar to

Diffusion Monte Carlo Diffusion Monte Carlo (DMC) or diffusion quantum Monte Carlo is a quantum Monte Carlo method that uses a Green's function to calculate low-lying energies of a quantum many-body Hamiltonian. Introduction and motivation of the algorithm Diffusion ...

, except that it works with paths rather than points. This has some advantages relating to calculating certain properties of the system under study that diffusion Monte Carlo has difficulty with. In both diffusion Monte Carlo and reptation Monte Carlo, the method first aims to solve the time-dependent Schrödinger equation in the

imaginary time Imaginary time is a mathematical representation of time that appears in some approaches to special relativity and quantum mechanics. It finds uses in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a ...

direction. When you propagate the Schrödinger equation in time, you get the dynamics of the system under study. When you propagate it in imaginary time, you get a system that tends towards the

ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...

of the system. When substituting

it

in place of

t

, the Schrodinger equation becomes identical with a

diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...

. Diffusion equations can be solved by imagining a huge population of particles (sometimes called "walkers"), each

diffusing Molecular diffusion is the motion of atoms, molecules, or other particles of a gas or liquid at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid, size and density (or their product, ...

in a way that solves the original equation. This is how diffusion Monte Carlo works. Reptation Monte Carlo works in a very similar way, but is focused on the paths that the walkers take, rather than the

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

of walkers. In particular, a path may be mutated using a

Metropolis algorithm A metropolis () is a large city or conurbation which is a significant economic, political, and cultural area for a country or region, and an important hub for regional or international connections, commerce, and communications. A big city b ...

which tries a change (normally at one end of the path) and then accepts or rejects the change based on a

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

calculation. The update step in diffusion Monte Carlo would be moving the walkers slightly, and then duplicating and removing some of them. By contrast, the update step in reptation Monte Carlo mutates a path, and then accepts or rejects the mutation.

References

* * * Quantum chemistry Quantum Monte Carlo {{Quantum-chemistry-stub