Monte Carlo Method

In General > s.a. computational physics; integration.
* Idea: A statistical method used to calculate quantities that are too difficult to compute analytically, in which one generates random events in a computer; Versions are the random walk (Metropolis) and the Hamiltonian ones.
@ Texts and reviews: Jadach phy/99 [guide]; Newman & Barkema 99, Krauth 06 [in statistical physics]; Binder & Heermann 19; > s.a. specific areas.
@ General references: Kosztin et al AJP(96)may [diffusion method for minima]; Binder RPP(97) [in statistical physics]; Doye & Wales PRL(98) [optimization and thermodynamics], CPC(00)phy/99 [self-adapting simplicial grid]; Landau et al AJP(04)oct [Wang-Landau sampling in statistical mechanics]; Kendall et al 05; Ambegaokar & Troyer AJP(10)feb [error estimation]; During & Kurchan EPL(10)-a1004 [statistical mechanics of Monte Carlo sampling].

Types of Algorithms > s.a. markov process; Simulated Annealing.
* Markov Chain Monte Carlo method: Different random configurations of a system are generated by small variations as in a Markov chain (for example, a random walk), and are then given a probability of being accepted; Two versions are the Metropolis algorithm and Hamiltonian Monte Carlo.
* Metropolis algorithm: A version of the MCMC method which applies to a thermal system, for which the probability of acceptance depends on the temperature; The algorithm fails in systems on the verge of a phase transition.
@ Markov Chain Monte Carlo method: Ottosen a1206 [rev]; Alexandru et al PRL(16)-a1605 [real-time dynamics on the lattice using the Schwinger-Keldysh formalism]; Betancourt a1706 [history]; Hanada a1808 [intro]; Joseph book(20)-a1912-ln [in quantum field theories].
@ Metropolis algorithm: Bhanot RPP(88); Berg PRL(03) [for rugged dynamical variables]; Moussa a1903-conf [quantum]; > s.a. path integrals.
@ Other algorithms: Suwa & Todo PRL(10)-a1007 [without detailed balance]; Jansen et al JPCS(13)-a1211, CPC(14)-a1302 [quasi-Monte Carlo method, and lattice field theories]; Herdeiro & Doyon PRE(16)-a1605 [method for critical systems in infinite volume]; Cai et al a1811 [inchworm Monte Carlo method, open quantum systems]; Edwards et al AP(19)-a1903 [worldline Monte Carlo]; > s.a. Glauber Dynamics.
@ Quantum Monte Carlo: Suzuki ed-93 [condensed matter]; Rombouts et al PRL(06) [new updating scheme]; Anderson 07; Pollet et al JCP(07) [optimality]; Temme et al Nat(11)mar-a0911 [sampling from Gibbs distribution]; Destainville et al PRL(10); Fantoni & Moroni JChemP(14)-a1408 [for quantum Gibbs ensemble]; Zen et al PRB(16)-a1605 [improved accuracy and speed]; Gubernatis et al 16 [pedagogical overview]; Becca & Sorella 17 [for correlated systems]; Hangleiter et al SciAdv(20)-a2001 [easing the sign problem]; Mareschal a2103, a2103 [history].

Applications > s.a. computational physics by areas [including statistical mechanics, field theory, gravity, quantum mechanics].
@ For fermions: Corney & Drummond PRL(04)qp, PRB(06)cm/04; Assaraf et al JPA(07).
@ Astrophysics and cosmology: Hajian PRD(07)ap/06 [Hamiltonian version, and cosmology]; > s.a. black-hole formation; observational cosmology.
@ Other systems: Janke PhyA(98) [disordered systems]; Talbot et al JPA(03) [exact results for simple harmonic oscillator]; Lahbabi & Legoll JSP(13) [multiscale systems in time]; Pavlovsky et al a1410-conf [path integral for relativistic quantum systems]; Silva Fernandes & Fartaria AJP(15)sep [gas-liquid coexistence].
> Other systems: see Chemical Potential; composite systems; diffusion; lattice field theory; Mean-Field Method; schrödinger equation.