Computational Physics

In General > s.a. programming languages; random and stochastic process.
* History: The original idea of performing numerical experiments was Fermi's, and was first used in the Fermi-Pasta-Ulam model.
* Status: 1994, Typically, use × 10⁶ data elements, but × 10⁸ are possible; The cost may be $1000/hr though.
* Method: Reduce pdes to finite difference equations (can be done in different ways); Use combinatorial methods.
* Remark: It is useful to practice with wave equations on a PC and to do related plots.
@ Intros, books: Koonin 85 [BASIC]; Hogg & Huberman PRP(87); DeVries 94 [FORTRAN, r PT(94)aug], AJP(96)apr-RL; Fosdick et al 96; Giordano 97; Weissert 97 [history]; Wong 97 [methods]; Vesely 01; Giordano & Nakanishi 05; Yevick 05; Gibbs 06; Klein & Godunov 06; Pang 06; Landau et al 07; Thijssen 07; Landau AJP(08)apr-RL and issue AJP(08)apr; Hoover a0812 [personal view]; Hartmann 09 [practical guide].
@ And teaching: Spencer AJP(05)feb [as undergraduate lab sequence]; Chabay & Sherwood AJP(08)apr [in calculus-based physics].
@ Physics and computation: Feynman IJTP(82); Toffoli IJTP(82); Geroch & Hartle FP(86); Landauer FP(86); Langer PT(99)jul [comments]; Rossi ht/06-in [challenges and opportunities].
@ Theoretical physics and PCs: Schmid et al 90; Stauffer & Stanley 95.
@ Visualization: Earnshaw & Wiseman 92; PW(93)sep, 48; Hammond PW(96); Hege & Polthier 97; Sanders et al NJP(09) [focus]; Farr a0905/JVWR [self-gravitating systems, using virtual worlds].
@ Supercomputers: Kaufmann & Smarr 93 [I].

Special Techniques
* Monte Carlo method: A statistical method to calculate quantities that are too difficult to compute analytically, generating random events in a computer; Versions are the random walk (Metropolis) and the Hamiltonian ones.
* Metropolis algorithm: Different random configurations of a system are generated by small variations as in a random walk or Markov chain, which are then given a probability of being accepted; It fails in systems on the verge of a phase transition.
* Mesh enhancement: The process in which an existing mesh is modified to better meet the requirements of the system.
@ Texts: Binder & Heermann 92 [Montecarlo]; MacKeown 97 [stochastic]; Mitzenmacher & Upfal 05 [probabilistic].
@ Monte Carlo method: Kosztin et al AJP(96)may [diffusion method for minima]; Binder RPP(97) [in statistical physics]; Doye & Wales PRL(98) [optimization and thermodynamics]; Janke PhyA(98) [disordered systems]; Jadach phy/99 [guide], CPC(00)phy/99 [self-adapting simplicial grid]; Talbot et al JPA(03) [exact results for sho]; Landau et al AJP(04)oct [Wang-Landau sampling in statistical mechanics]; Hajian PRD(07)ap/06 [Hamiltonian version, and cosmology]; > s.a. composite systems; diffusion; lattice field theory; observational cosmology; stochastic process [Markov].
@ Monte Carlo, quantum: Suzuki ed-93 [condensed matter]; Rombouts et al PRL(06) [new updating scheme]; Anderson 07; Pollet et al JCP(07) [optimality].
@ Monte Carlo, for fermions: Corney & Drummond PRL(04)qp, PRB(06)cm/04; Assaraf et al JPA(07).
@ Metropolis algorithm: Bhanot RPP(88); Berg PRL(03) [for rugged dynamical variables].
@ Other numerical methods: Knuth 69-73; Acton 70; Dahlquist & Bjoerck 74; MacKeown & Newman 87; Gould & Tobochnik 88; Heermann 90; MacDonald 94; García 00 [Matlab]; Enns & McGuire 01 [II]; Press et al 07.
@ Symplectic integration: Fleck et al ApplP(76); Suzuki PLA(90), PLA(92); B K Berger et al.
@ Minimization / optimization: Contucci et al mp/04 [annealing procedure].
@ Mesh adjustment: Pretorius & Lehner JCP(04) [adaptive]; Choi et al JCP(04) [refinement boundaries]; Anderson et al JCP(05) [unstructured simplices]; Baker & van Meter PRD(05)gq [in general relativity, reflections from interfaces]; Pretorius & Choptuik JCP(06)gq/05 [adaptive, coupled elliptic-hyperbolic systems].
@ Related topics: Fulling qp/99, qp/99 [large integers and remainders]; Steeb 99 [non-linear]; Hansen et al 05 [mesh enhancement]; Acebrón & Spigler JCP(05) [quasi-random numbers for stochastic systems]; Billo 07 [Excel].

Specific Areas > s.a. Boltzmann Equation; BRST; combinatorics; formulations of classical mechanics; integral equations.
@ Classical mechanics: Hoover 99 [irreversibility, chaos]; Wilkins 99; Timberlake & Hasbun AJP(08)apr.
@ Wave equations: van Putten PRD(97)gq [non-linear, general relativity]; Iriondo & Reula PRD(02)gq/01 [spherical scalar]; Rossmanith et al JCP(04) [hyperbolic systems on curved manifolds]; Visher et al JCP(04) [1+1, stable high-order discretization]; Anderson & Kimn CP(07) [spacetime finite-element approach]; VanWyk 08 [and differential equations in general]; Bernardini & Pirozzoli JCP(09) [Runge-Kutta method]; > s.a. computing languages [Matlab].
@ Astrophysics: Spurzem ap/97-in [N-body]; Kalashnikov gq/01 [Maple]; Evans et al PRD(05) [relativistic, adaptive mesh].
@ Fluids: Zheng et al JCP(05) [multiphase flow, adaptive]; > s.a. H-Theorem.
@ Thermodyamics and statistical mechanics: Tobochnik et al AJP(05)aug [T and chemical potential, MonteCarlo]; Landau & Binder 05 [Monte Carlo]; Krauth 06; Tobochnik & Gould AJP(08)apr.
@ For quantum theory: Ishikawa JPA(02) [accurate method]; Dowling et al JCP(07)qp/05 [quantum mechanics]; Hirayama & Holdom ht/05, et al hl/05 [classical simulation]; Latorre JPA(07) [entanglement entropy and simulating quantum mechanics]; Hastings PRB(07)-a0706 [1D systems at finite temperature]; De Raedt a0712-in [event-by-event simulation]; Schuch et al NJP(08) [simulating quantum evolution]; Dakic et al PRL(08) [simulating quantum measurements with hidden-variable states]; De Raedt et al PhyE(09)-a0908 [event-by-event simulations]; Anastassi & Simos PRP(09) [multistep methods]; > s.a. pilot-wave interpretation; quantum mechanics [texts]; schrödinger equation.
@ For field theory: Ehrlich 95; Peeters ht/07, Brewin a0903 [Cadabra, symbolic and tensor algebra]; > s.a. coordinates; dirac fields; lattice theory.
@ For Maxwell theory: Cockburn et al JCP(04) [Galerkin method]; Rieben et al JCP(05) [unstructured grid]; Collino et al JCP(06) [mesh refinement for FDTD solution]; > s.a. electromagnetism in curved spacetime, magnetism [magnetohydrodynamics].
@ For gravitation / cosmology: MacCallum IJMPA(02) [computer algebra]; Vulcanov & Vulcanov cs/04-in [Maple + GrTensorII libraries for cosmology]; Moylan et al gq/05-in [GRworkbench]; Tertychniy a0704 [GR_EC]; Shirokov a0711 [GRACOS code]; > s.a. numerical relativity.
@ For quantum gravity: Ambjørn et al 97; Hamber gq/98; > s.a. regge calculus and quantum regge calculus.

Specific Packages > s.a. programming languages [e.g., Mathematica].
@ Interactive Physics: Schwarz 96 [Mac & Windows].

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send feedback and suggestions to bombelli at olemiss.edu – modified 8 nov 2009