摄动理论

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本文描述作为一般数学方法的摄动理论。应用于量子力学的摄动理论，参看摄动理论 (量子力学)。

摄动理论由用于寻找无法精确求解的问题的近似解的数学方法组成，这些方法从相关问题的精确解开始入手。摄动理论适用于可以通过加入一个微扰项到一个可以精确求解的问题上而表述的问题。

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 $ε$ ), like the following:

$A=A_0 + \epsilon A_1 + \epsilon^2 A_2 + \cdots$

In this example, $A 0$ would be the known solution to the exactly solvable initial problem and $A_1,A_2,\ldots$ represent the "higher orders" which are found iteratively by some systematic procedure. For small $ε$ 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.

[编辑] Simple example

Consider the following equation for the unknown variable $x$ :

x = 1 + ε x 5

For the initial problem with $ε = 0$ , the solution is $x 0 = 1$ . For small $ε$ the lowest order approximation may be found by inserting the ansatz

$x=x_0+\epsilon x_1 (+\ldots)$

into the equation and demanding the equation to be fulfilled up to terms that involve powers of $ε$ higher than the first. This yields $x 1 = 1$ . In the same way, the higher orders may be found. However, even in this simple example it may be observed that for (arbitrarily) small $ε > 0$ there are four other solutions to the equation (with very large magnitude). The reason we don't find these solutions in the above perturbation method is because these solutions diverge when $\epsilon\rightarrow 0$ while the ansatz assumes regular behavior in this limit.

The four additional solutions can be found using the methods of singular perturbation theory. In this case this works as follows. Since the four solutions diverge at $ε = 0$ , it makes sense to rescale $x$ . We put

x = y ε - ν

such that in terms of $y$ the solutions stay finite. This means that we need to choose the exponent $ν$ to match the rate at which the solutions diverge. In terms of $y$ the equation reads:

ε - ν y = 1 + ε 1 - 5ν y 5

The 'right' value for $ν$ is obtained when the exponent of $ε$ in the prefactor of the term proportional to $y$ is equal to the exponent of $ε$ in the prefactor of the term proportional to $y 5$ , i.e. when $ν = 1 / 4$ . This is called 'significant degeneration'. If we choose $ν$ larger then the four solutions will collapse to zero in terms of $y$ and they will become degenerate with the solution we found above. If we choose $ν$ smaller then the four solutions will still diverge to infinity.

Putting $ν = 1 / 4$ in the above equation yields:

y = ε 1 / 4 + y 5

This equation can be solved using ordinary perturbation theory in the same way as regular expansion for $x$ was obtained. Since the expansion parameter is now $ε 1 / 4$ we put:

$y=y_0 + \epsilon^{1/4}y_1 + \epsilon^{1/2}y_2 \ldots$

There are 5 solutions for $y 0$ : 0, 1, -1, i and -i. We must disregard the solution $y = 0$ . The case $y = 0$ corresponds to the original regular solution which appears to be at zero for $ε = 0$ , because in the limit $\epsilon\rightarrow 0$ we are rescaling by an infinite amount. The next term is $y 1 = - 1 / 4$ . In terms of $x$ the four solutions are thus given as:

$x = \epsilon^{-1/4}\left[y_0 - 1/4\epsilon^{1/4} +\ldots\right]$

Both regular and singular perturbation theory are frequently used in physics and engineering. Regular 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). A well known example from physics where regular perturbation theory fails is in fluid dynamics when one treats the viscosity as a small parameter. Close to a boundary, the fluid velocity goes to zero, even for very small viscosity (the no-slip condition). For zero viscosity, it is not possible to impose this boundary condition and a regular perturbative expansion amounts to an expansion about an unrealistic physical solution. Singular perturbation theory can, however, be applied here and this amounts to 'zooming in' at the boundaries (boundary layer theory, solvable using the method of matched asymptotic expansions).

Perturbation theory can fail when the system can 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). In some cases this failure manifests itself by divergent behavior of the perturbation series. Such divergent series can sometimes be resummed using techniques such as Borel resummation.

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-Poincaré technique, the method of harmonic balancing, and the method of multiple time scales.