Computational physics

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Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. [1] Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science.

Numerical analysis study of algorithms that use numerical approximation for the problems of mathematical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. As an aspect of mathematics and computer science that generates, analyzes, and implements algorithms, the growth in power and the revolution in computing has raised the use of realistic mathematical models in science and engineering, and complex numerical analysis is required to provide solutions to these more involved models of the world. Ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Physics Study of the fundamental properties of matter and energy

Physics is the natural science that studies matter and its motion and behavior through space and time and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Computational science is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems. It is an area of science which spans many disciplines, but at its core it involves the development of models and simulations to understand natural systems.

It is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics, a third way that supplements theory and experiment. [2]

Experimental physics is the category of disciplines and sub-disciplines in the field of physics that are concerned with the observation of physical phenomena and experiments. Methods vary from discipline to discipline, from simple experiments and observations, such as the Cavendish experiment, to more complicated ones, such as the Large Hadron Collider.

Overview

A representation of the multidisciplinary nature of computational physics both as an overlap of physics, applied mathematics, and computer science and as a bridge among them. Computational physics diagram.svg
A representation of the multidisciplinary nature of computational physics both as an overlap of physics, applied mathematics, and computer science and as a bridge among them.

In physics, different theories based on mathematical models provide very precise predictions on how systems behave. Unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. 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 required. Computational physics is the subject that deals with these numerical approximations: the approximation of the solution is written as a finite (and typically large) number of simple mathematical operations (algorithm), and a computer is used to perform these operations and compute an approximated solution and respective error. [1]

A theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking. Depending on the context, the results might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings.

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations, and functions, but usually no limit. The set of operations and functions admitted in a closed-form expression may vary with author and context.

Algorithm an unambiguous specification of how to solve a class of problems

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, and other tasks.

Status in physics

There is a debate about the status of computation within the scientific method. [4]

Sometimes it is regarded as more akin to theoretical physics; some others regard computer simulation as "computer experiments", [4] yet still others consider it an intermediate or different branch between theoretical and experimental physics, a third way that supplements theory and experiment. While computers can be used in experiments for the measurement and recording (and storage) of data, this clearly does not constitute a computational approach.

A computer experiment or simulation experiment is an experiment used to study a computer simulation, also referred to as an in silico system. This area includes computational physics, computational chemistry, computational biology and other similar disciplines.

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.

Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, meaning there is no reasonable higher instruction to define the various possible interactions.

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. A macroscopic system typically has a size of the order of constituent particles, so it is somewhat of a problem. Solving quantum mechanical problems is generally of exponential order in the size of the system [5] and for classical N-body it is of order N-squared.

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 rendering the 'solution' useless. [6]

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:

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

Computational physics also borrows a number of ideas from computational chemistry - for example, the density functional theory used by computational solid state physicists to calculate properties of solids is basically the same as that used by chemists to calculate the properties of molecules.

Furthermore, computational physics encompasses the tuning of the software/hardware structure to solve the problems (as the problems usually 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 confluence between CFD and electromagnetic modelling is computational magnetohydrodynamics. The quantum many-body problem leads naturally to the large and rapidly growing field of computational chemistry.

Computational solid state physics is a very important division of computational physics dealing directly with material science.

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 and cellular automata) it also concerns itself with (and finds application in, through the use of its techniques) in the social sciences, network theory, and mathematical models for the propagation of disease (most notably, the SIR Model) and the spread of forest fires.

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

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

Computational biophysics is a branch of biophysics and computational biology itself, applying methods of computer science and physics to large complex biological problems.

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), nuclear engineering computer codes, protein structure prediction, weather prediction, solid state physics, soft condensed matter physics, hypervelocity impact 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.

See also

Related Research Articles

Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids. It is necessary because, apart from relatively recent results concerning the hydrogen molecular ion, the quantum many-body problem cannot be solved analytically, much less in closed form. While computational results normally complement the information obtained by chemical experiments, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials.

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems. It is also called molecular quantum mechanics.

Theoretical chemistry academic field

Theoretical chemistry is a branch of chemistry, which develops theoretical generalizations that are part of the theoretical arsenal of modern chemistry, for example, the concept of chemical bonding, chemical reaction, valence, the surface of potential energy, molecular orbitals, orbital interactions, molecule activation etc.

Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data.

Computational fluid dynamics branch of fluid mechanics that uses numerical analysis and data structures to solve and analyze problems that involve fluid flows

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests.

Molecular modelling Discovering chemical properties by physical simulations

Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials science to study molecular systems ranging from small chemical systems to large biological molecules and material assemblies. The simplest calculations can be performed by hand, but inevitably computers are required to perform molecular modelling of any reasonably sized system. The common feature of molecular modelling methods is the atomistic level description of the molecular systems. This may include treating atoms as the smallest individual unit, or explicitly modelling electrons of each atom.

The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations. including fluid mechanics, acoustics, electromagnetics, fracture mechanics, and contact mechanics.

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 of the quantum many-body problem. The diverse flavor of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem. The quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in the wave function, going beyond mean field theory and offering an exact solution of the many-body problem in some circumstances. In particular, there exist numerically exact and polynomially-scaling algorithms to exactly study static properties of boson systems without geometrical frustration. For fermions, there exist very good approximations to their static properties and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are both.

Multiphysics is defined as the coupled processes or systems involving more than one simultaneously occurring physical fields and the studies of and knowledge about these processes and systems. As an interdisciplinary study area, multiphysics spans over many science and engineering disciplines. Multiphysics is a practice built on mathematics, physics, application, and numerical analysis. The mathematics involved usually contains partial differential equations and tensor analysis. The physics refers to common types of physical processes, e.g., heat transfer (thermo-), pore water movement (hydro-), concentration field, stress and strain (mechano-), dynamics (dyno-), chemical reactions, electrostatics (electro-), and magnetostatics (magneto-).

In engineering, mathematics, physics, chemistry, bioinformatics, computational biology, meteorology and computer science, multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena.

Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.

Engineering mathematics is a branch of applied mathematics concerning mathematical methods and techniques that are typically used in engineering and industry. Along with fields like engineering physics and engineering geology, both of which may belong in the wider category engineering science, engineering mathematics is an interdisciplinary subject motivated by engineers' needs both for practical, theoretical and other considerations outwith their specialization, and to deal with constraints to be effective in their work.

Fluid–structure interaction

Fluid–structure interaction (FSI) is the interaction of some movable or deformable structure with an internal or surrounding fluid flow. Fluid–structure interactions can be stable or oscillatory. In oscillatory interactions, the strain induced in the solid structure causes it to move such that the source of strain is reduced, and the structure returns to its former state only for the process to repeat.

Computational engineering

Not to be confused with computer engineering.

Applied mathematics Application of mathematical methods to other fields

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

The Kansa method is a computer method used to solve partial differential equations. Partial differential equations are mathematical models of things like stresses in a car's body, air flow around a wing, the shock wave in front of a supersonic airplane, quantum mechanical model of an atom, ocean waves, socio-economic models, digital image processing etc. The computer takes the known quantities such as pressure, temperature, air velocity, stress, and then uses the laws of physics to figure out what the rest of the quantities should be like a puzzle being fit together. Then, for example, the stresses in various parts of a car can be determined when that car hits a bump at 70 miles per hour.

Numerical modeling (geology)

In geology, numerical modeling is a widely applied technique to tackle complex geological problems by computational simulation of geological scenarios.

Alexander Andreevich Samarskii Russian mathematician

Alexander Andreevich Samarskii was a Soviet and Russian mathematician and academician, specializing in mathematical physics, applied mathematics, numerical analysis, mathematical modeling, finite difference methods.

References

  1. 1 2 Thijssen, Jos (2007). Computational Physics. Cambridge University Press. ISBN   978-0521833462.
  2. Landau, Rubin H.; Páez, Manuel J.; Bordeianu, Cristian C. (2015). Computational Physics: Problem Solving with Python. John Wiley & Sons.
  3. Landau, Rubin H.; Paez, Jose; Bordeianu, Cristian C. (2011). A survey of computational physics: introductory computational science. Princeton University Press.
  4. 1 2 A molecular dynamics primer, Furio Ercolessi, University of Udine, Italy. Article PDF.
  5. Feynman, Richard P. (1982). "Simulating physics with computers". International Journal of Theoretical Physics. 21 (6–7). doi:10.1007/bf02650179.pdf (inactive 2018-09-05). ISSN   0020-7748.
  6. Sauer, Tim; Grebogi, Celso; Yorke, James A. (1997-07-07). "How Long Do Numerical Chaotic Solutions Remain Valid?". Physical Review Letters. 79 (1): 59–62. Bibcode:1997PhRvL..79...59S. doi:10.1103/PhysRevLett.79.59.

Further reading