Perturbation theory

This article describes perturbation theory as a general mathematical method. For perturbation theory as applied to quantum mechanics, see perturbation theory (quantum mechanics).

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Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. Perturbation theory is applicable if the problem at hand can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem. Perturbation theory leads to an expression for the desired solution in terms of a power series in some "small" parameter that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution A a series in the small parameter (here called <math>\epsilon<math>), like the following:

<math> A=A_0 + \epsilon A_1 + \epsilon^2 A_2 + \cdots<math>

In this example, <math>A_0<math> would be the known solution to the exactly solvable initial problem and <math>A_1,A_2,\ldots<math> represent the "higher orders" which are found iteratively by some systematic procedure. For small <math>\epsilon<math> these higher orders become successively more unimportant.

Examples for the "mathematical description" are: an algebraic equation, a differential equation (e.g. the equations of motion in celestial mechanics or a wave equation), a free energy (in statistical mechanics), a Hamiltonian operator (in quantum mechanics).

Examples for the kind of solution to be found perturbatively: the solution of the equation (e.g. the trajectory of a particle), the statistical average of some physical quantity (e.g. average magnetization), the ground state energy of a quantum mechanical problem.

Examples for the exactly solvable problems to start with: Linear equations, including linear equations of motion (harmonic oscillator, linear wave equation), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).

Examples of "perturbations" to deal with: Nonlinear contributions to the equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/Free Energy.

For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams.

Simple example

Consider the following equation for the unknown variable <math>x<math>:

<math>x=1+\epsilon x^5<math>

For the initial problem with <math>\epsilon=0<math>, the solution is <math>x_0=1<math>. For small <math>\epsilon<math> the lowest order approximation may be found by inserting the ansatz

<math>x=x_0+\epsilon x_1 (+\ldots)<math>

into the equation and demanding the equation to be fulfilled up to terms that involve powers of <math>\epsilon<math> higher than the first. This yields <math>x_1=1<math>. In the same way, the higher orders may be found. However, even in this simple example it may be observed that for (arbitrarily) small <math>\epsilon>0<math> there are two other solutions to the equation (with very large magnitude) which cannot be found using perturbation theory.

The same problem occurs in many real applications in physics and elsewhere: Perturbation theory may only be used to find those solutions of a problem that evolve smoothly out of the initial solution when changing the parameter (that are "adiabatically connected" to the initial solution). In physics, this fails whenever the system may go to a different "phase" of matter, with a qualitatively different behaviour that cannot be understood by perturbation theory (e.g. a solid crystal melting into a liquid).

Perturbation techniques can be also used to find approximate solutions to non-linear differential equations. Examples of techniques used to find approximate solutions to these types of problems are the Lindstead-Poincare technique, the method of harmonic balancing, and the method of multiple time scales.

Caveats

There is absolutely no guarantee perturbative methods would result in a convergent solution. In fact, asymptotic series are the norm.ru:Òåîðèÿ âîçìóùåíèé ja:摂動 de:Störungstheorie