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The article I will be editing is Computational physics, with intent to up the quality to about that of computational chemistry, a B-class article. Anyone who sees this, feel free to send me advice. The sources are lacking and I am currently looking for more

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Computational physics is the study and use of numerical analysis to solve problems in physics for which a quantitative theory already exists.^[1] Computational physics does not necessarily depend on the usage of computers, but rather on the usage of numerical methods.^[2] While theories based on mathematical models provide very precise predictions on how systems behave, an exact solution may not always be feasible, requiring an approximated solution.^[3] Such a solution would be written as a finite (and typically large) number of mathematical operations (algorithm), and a computer would be used to perform these operations and compute an approximated solution and respective error.^[1]

Computational physics is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics^[4], but others consider it an intermediate branch between theoretical and experimental physics, a third way that supplements theory and experiment.^[3]^[5]

Historically, computational physics as a concept has existed since the late 1600's in Isaac Newton's work,Principia, but such methods were largely considered impractical until the arrival of modern computers.^[2]

Challenges in computational physics

Physics problems are in general very difficult to solve exactly. This is due to several (mathematical) reasons: lack of algebraic and/or analytic solubility, complexity, and chaos. This can occur, for instance, when the solution does not have a closed-form expression, or is too complicated. In such cases, numerical approximations are often required to come to a meaningful conclusion.

For example, - even apparently simple problems, such as calculating the wavefunction of an electron orbiting an atom in a strong electric field (Stark effect), may require great effort to formulate a practical algorithm (if one can be found); other cruder or brute-force techniques, such as graphical methods or root finding, may be required. On the more advanced side, mathematical perturbation theory is also sometimes used (a working is shown for this particular example here).

In addition, the computational cost and computational complexity for many-body problems (and their classical counterparts) tend to grow quickly. Solving quantum mechanical problems is generally of exponential order in the size of the system ^[3] and for classical N-body it is of order N-squared. For example, macroscopic system may have a size of the order of $10^{23}$ constituent particles, with the interactions between particles being even more numerous. Methods such as Monte Carlo integration allow averaging the properties of the particles instead of looking at each one individually, reducing the necessary number of computations.^[3]

Finally, many physical systems are inherently nonlinear at best, and at worst chaotic: this means it can be difficult to ensure any numerical errors do not grow to the point of causing major deviations in the data.^[7] One way around this problem is to take the average properties over a long period of time, assuming that the system reaches a relatively stable equilibrium where the deviations average out.^[3]

Methods and algorithms

Because computational physics uses a broad class of problems, it is generally divided amongst the different mathematical problems it numerically solves, or the methods it applies. Between them, one can consider:

root finding (using e.g. Newton-Raphson method)
system of linear equations (using e.g. LU decomposition)
ordinary differential equations (using e.g. Runge–Kutta methods)
integration (using e.g. Romberg method and Monte Carlo integration)
partial differential equations (using e.g. finite difference method and relaxation method)
matrix eigenvalue problem (using e.g. Jacobi eigenvalue algorithm and power iteration)

All these methods (and several others) are used to calculate physical properties of the modeled systems.

Computational physics also shares a number of ideas with computational chemistry - for example, the density functional theory used by computational solid state physicists to calculate the properties of solids is also used by chemists to calculate the properties of molecules.

Furthermore, computational physics encompasses the tuning of the software/hardware structure to solve problem (as the problems often can be very large, in processing power need or in memory requests).

Divisions

It is possible to find a corresponding computational branch for every major field in physics, for example computational mechanics and computational electrodynamics. Computational mechanics consists of computational fluid dynamics (CFD), computational solid mechanics and computational contact mechanics.

One subfield at the overlap between CFD and electromagnetic modelling is computational magnetohydrodynamics. The quantum many-body problem leads directly to the large and rapidly growing field of computational chemistry.

A field related to computational condensed matter is computational statistical mechanics, which deals with the simulation of models and theories (such as percolation and spin models) that are difficult to solve otherwise. Computational statistical physics makes heavy use of Monte Carlo-like methods. More broadly, (particularly through the use of agent based modeling) it also concerns itself with (and can be used in) the social sciences, network theory, and mathematical models for the spread of disease (most notably, the SIR Model) and the spread of forest fires.

On the more obscure side, numerical relativity is a (relatively) new field interested in finding numeric solutions to the field equations of general (and special) relativity, while computational particle physics deals with problems motivated by particle physics.

Likewise Computational astrophysics is the application of these techniques and methods to astrophysical problems and phenomena.

Applications

Due to the broad class of problems computational physics deals, it is an essential component of modern research in different areas of physics, namely: accelerator physics, astrophysics, fluid mechanics (computational fluid dynamics), lattice field theory/lattice gauge theory (especially lattice quantum chromodynamics), plasma physics (see plasma modeling), simulating physical systems (using e.g. molecular dynamics), protein structure prediction, solid state physics, soft condensed matter physics etc.

Computational solid state physics, for example, uses density functional theory to calculate properties of solids, a method similar to that used by chemists to study molecules. Other quantities of interest in solid state physics, such as the electronic band structure, magnetic properties and charge densities can be calculated by this and several methods, including the Luttinger-Kohn/k.p method and ab-initio methods.

References

^ ^a ^b Thijssen, Joseph (2007). Computational Physics. Cambridge University Press. ISBN 0521833469.
^ ^a ^b Vesely, Franz J. (2001). Computational Physics-An Introduction. Springer Science+Business Media.
^ ^a ^b ^c ^d ^e Potter, David (1973). Computational Physics. John Wiley & Sons.
^ Ercolessi, Furio (1997). "A molecular dynamics primer" (PDF). University of Udine.
^ Landau, Rubin H.; Páez, Manuel J.; Bordeianu, Cristian C. (2015). Computational Physics: Problem Solving with Python. John Wiley & Sons.
^ Landau, Rubin H.; Paez, Jose; Bordeianu, Cristian C. (2011). A survey of computational physics: introductory computational science. Princeton University Press.
^ "How Long Do Numerical Chaotic Solutions Remain Valid?". doi:10.1103/PhysRevLett.79.59. {{cite journal}}: Cite journal requires |journal= (help)

External links

[ThijssenBook-1] Thijssen, Joseph (2007). Computational Physics. Cambridge University Press. ISBN 0521833469.

[CompPhysIntro-2] Vesely, Franz J. (2001). Computational Physics-An Introduction. Springer Science+Business Media.

[CompPhysPotter-3] Potter, David (1973). Computational Physics. John Wiley & Sons.

[ercolessi-4] Ercolessi, Furio (1997). "A molecular dynamics primer" (PDF). University of Udine.

[ComPhysPy-5] Landau, Rubin H.; Páez, Manuel J.; Bordeianu, Cristian C. (2015). Computational Physics: Problem Solving with Python. John Wiley & Sons.

[SurveyCompSci-6] Landau, Rubin H.; Paez, Jose; Bordeianu, Cristian C. (2011). A survey of computational physics: introductory computational science. Princeton University Press.

[Sauer1997-7] "How Long Do Numerical Chaotic Solutions Remain Valid?". doi:10.1103/PhysRevLett.79.59. {{cite journal}}: Cite journal requires |journal= (help)

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