Perturbation theory

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The theory of perturbation consists of mathematical methods to find approximate solutions to problems, by starting from the right solution of related problems and simpler. An important feature of this technique is the middle step that breaks the problem into "solvable" and "perturbation" parts. The theory of perturbation applies if the problem encountered can not be solved precisely, but can be formulated by adding the term "small" to the mathematical description of the problem that is completely solvable.

The theory of perturbation leads to an expression for the desired solution in terms of formal strength circuits in some "minor" parameter - known as the interrupt series - which quantifies the deviation from a completely solvable problem. The main term in this power series is the solution of a completely solvable problem, while the term further describes the deviations in the solution, due to the deviation from the initial problem. Formally, we have an approach for the full solution A , a series in the small parameter (here called ? ), as follows:

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂAÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â Â Â Â Â Â Â Â A Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 1 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â A Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 1 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 2 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â A Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 2 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle A = A_ {0} \ varepsilon ^ {1} A_ {1} \ varepsilon ^ {2} A_ {2} \ cdots} Â Â}

In this example, A ₀ will be the known solution to a truly unbreakable initial problem and A ₁ , A ₂ ,... represents the high-order term found iteratively by some systematic procedure. For ? is small, these high-level terms are becoming smaller.

Perkiraan "solusi gangguan" diperoleh dengan memotong seri, biasanya dengan menyimpan hanya dua istilah pertama, solusi awal dan koreksi perturbanan "urutan pertama"

${\ displaystyle A \ kira-kira A_ {0} \ varepsilon A_1 ~.} Â Â$

Video Perturbation theory

General description

The theory of perturbation is closely related to the method used in numerical analysis. The earliest use of what is now called the theory of perturbation is to deal with the insoluble mathematical problems of sky mechanics: for example the orbit of the Moon, which moves strikingly different from simple Keplerian ellips because of the gravity of competing Earth and the Sun.

The method of perturbation begins with a simplified form of the original problem, simple enough to be solved precisely. In celestial mechanics, these are usually Keplerian ellipses. Under non-relativistic gravity, the ellipse is true when there are only two bodies of gravity (say, Earth and the Moon) but not quite right when there are three or more objects (say, Earth, Moon, Sun, and the rest of the solar system) and not enough true when gravitational interactions are expressed using formulations of general relativity.

The problem solved, but simplified is then "disturbed" to make the condition that the disturbed solution actually meets closer to the formula in the original problem, such as including the gravitational attraction of the third body (the Sun). ). Typically, the "condition" representing reality is a formula (or several) that specifically reveals some physical laws, such as Newton's second law, the force acceleration equation,

${\ displaystyle {\ mathbf {F}} = m {\ mathbf {a}} ~.}$

In the case of example, the F style is calculated based on the number of bodies that are gravitatively relevant; acceleration a is obtained, using calculus, from the Moon path in orbit. Both have two forms: approximate values ​​for force and acceleration, resulting from simplification, and exact exact values ​​for strength and acceleration, which will require a complete answer to compute.

A slight change resulting from disruption, which may have been simplified, is used as a correction to approximate solutions. Because simplification is introduced throughout each step, correction is never perfect, and conditions fulfilled by corrected solutions are not perfect according to the equations demanded by reality. However, even just one correction cycle often provides an excellent approximation answer to what the real solution should be.

There is no requirement to stop only on one correction cycle. Partially corrected solutions can be reused as new starting points for other cycles of interference and correction. In principle, the cycle of finding a better correction can take place indefinitely. In practice, it usually stops at one or two correction cycles. The usual difficulty with this method is that progressive correction makes the new solution much more complicated, so each cycle is much more difficult to manage than the previous correction cycle. Isaac Newton was reported to have said, about the matter of the orbits of the Moon, that "It caused my head to hurt."

This general procedure is a mathematical tool widely used in advanced science and engineering: start with a simplified problem and gradually add corrections that make the formula that the corrected problem becomes closer and closer to the original formula. This is a natural extension to the mathematical functions of the "guess, check and fix" method that was first used by older civilizations to compute certain numbers, such as square roots.

Maps Perturbation theory

Example

Examples for "mathematical descriptions" are: algebraic equations, differential equations (eg equations of motion or wave equations), free energy (in statistical mechanics), radiation transfer, Hamiltonian operators (in quantum mechanics).

Examples for the types of solutions to be found are perturbatively: solutions of equations (eg, particle trajectories), statistical averages of some physical quantities (eg, average magnetization), basic energy states of quantum mechanical problems.

Examples for really solvable problems to begin with are: linear equations, including linear motion equations (harmonic oscillators, linear wave equations), statistical systems or quantum mechanics of unrelated particles (or, in general, Hamiltonian or free energy only contains the term squared in all degrees of freedom).

Examples of "perturbations" to be faced: Nonlinear contributions to equations of motion, interaction between particles, higher power terms in Hamiltonian/Free Energy.

For physical problems involving interaction between particles, the term interference series can be displayed (and manipulated) using the Feynman diagram.

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History

The theory of perturbation was first designed to solve problems that are difficult to solve in calculating planetary motion in the solar system. For example, Newton's universal law of gravity explains gravity between two astronomical bodies, but when a third body is added, the problem is, "How does each body attract one another?" Newton's equation only allows the mass of two objects to be analyzed. The gradually increasing accuracy of astronomical observations led to additional demands in solution accuracy for Newton's gravity equations, which led to several mathematicians of the 18th and 19th centuries, such as Lagrange and Laplace, to broaden and generalize the methods of perturbation theory. This well-developed method of perturbation was adopted and adapted to solve new problems arising during the development of quantum mechanics in atomic and subatomic physics of the 20th century. Paul Dirac developed the theory of perturbation in 1927 to evaluate when a particle would be emitted in a radioactive element. It was later named the Fermi golden rule.

Beginning in a planetary motion study

Since the planets are so distant from each other, and because their masses are small compared to the mass of the Sun, the force of gravity between the planets is negligible, and planetary motion is regarded, to the first approach, as a place as long as Kepler's orbit, defined by equations from the problem of two bodies, two bodies become planets and the Sun.

Because astronomical data is known for much greater accuracy, it becomes important to consider how the motion of planets around the Sun is influenced by other planets. This is the origin of the three-body problem; thus, in studying the Moon-Earth-Sun system, the mass ratio between the Moon and Earth is chosen as a small parameter. Lagrange and Laplace are the first to advance the view that constants describing planetary motions around the Sun are "disturbed", as if, by other planetary motions and vary as a function of time; hence the name "perturbation theory".

The theory of perturbation is investigated by classical scholars - Laplace, Poisson, Gauss - as a result of calculations that can be done with very high accuracy. The discovery of Neptune's planet in 1848 by Urbain Le Verrier, based on deviations in Uranus planetary motion (he sent the coordinates to Johann Gottfried Galle who managed to observe Neptune through his telescope), representing the triumph of the theory of disorder.

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The perturbation command

The standard exposure of perturbation theory is given within the order framework in which the interruption is made: first-order perturbation theory or second order distortion theory , and whether the status is interrupted degenerate, requiring singular noise . In a single case, extra care must be taken, and the theory is a bit more complicated.

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In chemistry

Many ab initio chemical methods use the theory of perturbation directly or closely related methods. The theory of implicit perturbation worked with a complete Hamiltonian from the beginning and never determined operators of such disturbances. The MÃÆ'¸ller-Plesset perturbation theory uses the distinction between Hartree-Fock Hamiltonian and the non-relativistic Hamiltonian that is appropriate as a nuisance. The zero-order energy is the amount of orbital energy. First-order energy is the energy and electron correlation Hartree-Fock included in the second or higher order. Calculations for second, third or fourth order are very common and this code is included in most ab initio quantum chemistry programs. The related but more accurate method is the combined cluster method.

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See also

• An alternative approach to the theory of perturbation
• The theory of cosmological perturbation
• Dynamic nuclear polarization
• Eigenvalue perturbation
• Homotopy perturbation method
• FEM Interval
• Lyapunov Stability
• Approximation order
• The theory of perturbation (quantum mechanics)
• Structural stability

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References

External links

• Introduction to Eric Vanden-Eijnden's regular perturbation theory (PDF)
• The Perturbation Method of Many Scale

Source of the article : Wikipedia