Perturbation Theory Part I

So what is perturbation theory, and why should you care? Perturbation theory comprises a whole host of mathematical techniques that can be applied to many classes of problems to find approximate solutions. Typically problems that can be solved through perturbation theory are close to a problem with a closed form, analytic solution but with some additional small term, like say for example: \[ \left\{ \begin{array}{l l} u'' + u + \epsilon u^3 = 0\\ u(0) = a \\ u'(0) = 0 \\ \end{array} \right.\] This is a non linear ODE with no analytic solution. However what if $\epsilon$ takes a small value, say $\epsilon \ll 1$, then maybe we can make some progress. It seems like an intuitive guess to assume that when $\epsilon$ is sufficiently small the solution to this ODE isn't going to be influenced very much by this term. This allows us to think about the problem in a different way, as a simple analytic problem that we can solve, but with some additional 'perturbation' due to the small non linear term. So in the case of this example we expect that the solution to this equation should be similar to the solution to \[ \left\{ \begin{array}{l l} u'' + u = 0\\ u(0) = a \\ u'(0) = 0 \\ \end{array} \right.\] which is of course, $u(x) = a \cos x$.

So perturbation theory gives us solutions to difficult problems in terms of a solution to an easy problem with some additional correction terms. This approach is very fruitful in subjects like physics where any real, non idealised, system is likely to have irregularities that lead to solving non-linear differential equations. Pertubative solutions also carry some advantages over numerical methods as we do not need to fix numerical values to any parameters, and perturbative solutions offer more physical insight into a given problem.

Perturbation theory Ideas and Theorems

So now that we can see the uses and motivations behind perturbation theory we need to introduce the framework in which it sits. But first to talk about perturbation theory we first need to introduce some new notation.

Let $f(x)$ and $g(x)$ be two functions defined around the point $x=x_0$, then
$f = \mathcal{o}(g)$ as $x \to x_0$ if $\lim_{x \to x_0} f / g = 0$
$f = \mathcal{O}(g)$ as $x \to x_0$ if $\lim_{x \to x_0} f / g = \text{const} \ne 0$
$f \sim g$ as $x \to x_0$ if $\lim_{x \to x_0} f / g = 1$

Regular perturbation

There are two important ideas we need to do perturbation theory, first the perturbation series, which we shall use as an approximation tool and the fundamental theorem of perturbation theory which allows us to determine the functions in a perturbation series.

First the perturbation series. Perturbation theory borrows heavily from power series, we want to expand our difficult problem in terms of a small parameter $\epsilon$ around $\epsilon =0$. Naturally there are caveats with this, however for sufficiently 'nice' functions this should be fine. \[ f(x;\epsilon) = f_0(x) + \epsilon f_1(x) + \epsilon^2 f_2(x) + \cdots \] Where $\{ f_i(x) \}$ is a function to be determined.

Second the fundamental theorem of perturbation theory, which states, If an asymptotic expansion satisfies \[ A_0 + A_1 \epsilon + \cdots + A_N \epsilon^N \equiv \mathcal{O}(\epsilon^{N+1}) \] for all sufficiently small $\epsilon$ and all the coefficients,$A_i$, are independent of $\epsilon$ then \[ A_0 = A_1 = \cdots = A_N = 0 \] This is a very important result as it allows us to solve a perturbation series separately for each order of $\epsilon$ and thus determine all of the functions $\{ f_i(x) \}$ in out perturbations series.

An algorithmic approach

With the important ideas of perturbation theory in hand we turn out attention towards how to solve a general perturbation problem. The following approach will work for most perturbation problems.

1. Set $\epsilon = 0$ and solve the resulting linear, unperturbed, equation
2. Allow $\epsilon$ to be non zero and formulate the equation in terms of a perturbation series, namely let $f(x)$ be specified by the power series \[ f = f_0 + \epsilon f_1 + \epsilon^2 f_2 + \cdots \] and substituting this into all the $f(x)$ terms in the original equation.
3. Now collect terms of equal order in $\epsilon$. Solve the resulting systems of equations upto a suitable order in $\epsilon$, here we have made use of the fundamental theorem of perturbation theory. The approximate solution to the original equation is found by placing the result of the linear system into the perturbation series.

Some examples

Polynomials

Lots of talk and no real examples doesn't help understanding, so let's choose a really simple example, one that we could solve exactly. So consider the quadratic equation $x^2 - 4x + 3 + \epsilon = 0, \ \epsilon \ll 1 $ The solution to this can be found trivially via the quadratic formula and is \[ x = 2 \pm \sqrt{1-\epsilon} \] But what if we didn't know that, let's instead apply perturbation theory to this problem and see what we get. Let's work through the problem using the algorithm I suggested originally.

1. First let's identify the unperturbed problem. Of course we just set $\epsilon = 0$ and see what we're left with \[ x^2 - 4x + 3 = 0 \] And let's identify the solution to this problem, using the quadratic formula we find \[ x = 2 \pm 1 \] We've identified and solved the unperturbed problem so let's proceed to the next stage.
2. Let's assume we can write $x$ in a perturbation series as \[x = x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots \] Next we substituting this into our original equation to give us \[ (x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots)^2 - 4(x_0 + \epsilon x_1 + \epsilon^2 x_2 + \cdots) + 3 + \epsilon = 0 \] So we now have the full problem written in terms of a perturbation series, so let's proceed to the final step.
3. Now we solve the perturbation series by collecting terms of equal order in $\epsilon$ and solving the resulting system of equations. So collecting terms of equal orders \[ \left\{ \begin{array}{l l} \mathcal{O}(1) &: {x_0}^2 - 4 x_0 + 3 &=& 0 \\ \mathcal{O}(\epsilon) &: 2x_0 x_1 - 4x_1 + 1 &=& 0 \\ \mathcal{O}(\epsilon^2) &: {x_1}^2 + 2 x_2 x_0 - 4 x_2 &=& 0 \\ \vdots \end{array} \right.\] Solving perturbation series problems by hand can be rather tedious so we shall only proceed up to second order in $\epsilon$. The solution to this system of equations can be computed iteratively very simply and we find that \[ x_0 = 2 \pm 1 , \quad x_1 = \pm 1/2, \quad x_2 = \pm 1/8 \] So letting $x_{\pm}$ denote the large and smaller root respectively we find our approximate solution, \[ \left\{ \begin{array}{l l} x_+ = 3 - \epsilon/2 - \epsilon^2/8 + \cdots \\ x_- = 1 + \epsilon/2 + \epsilon^2/8 + \cdots \\ \end{array} \right.\] Perturbation success!

The next question we should be asking is how good is this approximate result compared to the actual solution. That of course is why I picked such a simple example as we can directly compare our perturbation solution to the analytic one. The analytic solution from before is \[ x = 2 \pm \sqrt{1-\epsilon} \] To compare this to our approximate we write $\sqrt{1-\epsilon}$ as a taylor series in $\epsilon$. This is \[ \sqrt{1-\epsilon} = 1 - \epsilon/2 - \epsilon^2/8 - \epsilon^3/16 + \mathcal{O}(\epsilon^4) \] Putting this expansion into the analytic solution we find \[ x = 2 \pm 1 \mp \epsilon/2 \mp \epsilon^2/8 \mp \epsilon^3/16 + \mathcal{O}(\epsilon^4) \] Comparing the solutions, they match up to order $\mathcal{O}(\epsilon^3)$ which was as far as we chose to continue our perturbation series. In this case the perturbation series actually converges to the correct solution with an infinite number of terms, however our truncated series provides a very good approximation to the actual result.

Differential equations

As I implied earlier the ideas in perturbation theory have very broad areas of application. We can use them further to find approximate solutions to differential equations. Consider the boundary value problem \[ \left\{ \begin{array}{l l} u'(x) = \epsilon u^2, \quad x>0 \\ u(0) = 1 \end{array} \right.\] This is a very simple example but demonstrates the ideas involved in applying regular perturbation theory to differential equations. Again let us use the algorithm that I originally proposed to solve this ODE using perturbation theory.

1. Let's set $\epsilon = 0$ and Identify the unperturbed problem \[ \left\{ \begin{array}{l l} u'(x) = 0 \\ u(0) = 1 \end{array} \right.\] The solution to the unperturbed problem is trivial and is. \[ u(x) = 1 \] Now we move on to identify the perturbed problem.
2. We assume that we can write $u(x)$ as a perturbation series \[ u(x) = u_0 + \epsilon u_1 + \epsilon^2 u_2 + \cdots \] We now substitute this into our original equation, including the boundary conditions and find \[ \left\{ \begin{array}{l l} (u_0 + \epsilon u_1 + \epsilon^2 u_2 + \cdots)' = \epsilon (u_0 + \epsilon u_1 + \epsilon^2 u_2 + \cdots)^2 \\ u_0(0) + \epsilon u_1(0) + \epsilon^2 u_2(0) + \cdots = 1 \end{array} \right.\] It's important to note there that the boundary conditions depend on the order of $\epsilon$ that we are interested in. In this case only the zeroth order boundary condition is inhomogeneous, $u_0(0)=1$, however all the higher order terms are homogeneous, having $u_n(0)=0$ for $n \ne 0$. Now we've formulated the problem in terms of a perturbation series let's proceed to the final step.
3. Now we collect all the terms of equal order in $\epsilon$ and form a system of equations that we need to solve. The system is \[ \left\{ \begin{array}{l l} \mathcal{O}(1) &: {u_0}' &=& 0 \\ \mathcal{O}(\epsilon) &: {u_1}' - {u_0}^2 &=& 0 \\ \mathcal{O}(\epsilon^2) &: {u_2}' - {u_1}^2 &=& 0 \\ \vdots \end{array} \right.\] and has the corresponding system of boundary conditions \[ \left\{ \begin{array}{l l} \mathcal{O}(1) &: u_0(0) = 1 \\ \mathcal{O}(\epsilon) &: u_1(0) = 0 \\ \mathcal{O}(\epsilon^2) &: u_2(0) = 0 \\ \vdots \end{array} \right.\] As with our previous example it is not difficult to solve this system of equations, as before we proceed iteratively and find that the solutions to the system are \[ u_0 = 1, \quad u_1 = x, \quad u_2 = x^2 \] Substituting these values into the perturbation series we find the approximate solution to the ODE is given by \[ u(x) = 1 + \epsilon x + \epsilon^2 x^2 + \cdots \] Perturbation success!

Again we ask ourselves, how good is this approximation compared to the actual solution and as this is a differential equation and $u$ is a function in $x$ is there a limit on the values of $x$ so that our approximate solution remains good. First we need to determine the exact solution to the problem, this can be done in a number of ways as the result is \[ u(x) = \dfrac{1}{1-\epsilon x} \] Examining the behaviour of these solutions over a range of values provides a number of insights. First the analytic solution has a singularity when $\epsilon x = 1$, we cannot find this behaviour in our perturbative solution. However is this a reasonable expectation, over what range do we expect our expansion to be valid?

It turns out in this case that our perturbation series is identical to the power series in $\epsilon x$ of $1/1-\epsilon x$, and its convergence depends upon $\epsilon x < 1$. So with the small number of terms we have in our approximate solution the only range we expect our solution to be good is one where $\epsilon x \ll 1$. Solving this for $x$ leads to the range over which our expansion is good, namely, $ 0< x \ll 1/\epsilon $.

But what about the more general cases when we can't find a power series to check when out solution is valid? Well it's often problem specific however another way to see why our approximation fails is to look at the perturbation series. The second term in the series is $\epsilon x$ and this is of order $\mathcal{O}(\epsilon)$ so all is good, but as $x \sim 1/\epsilon$ our first order perturbation is now of zeroth order, $\mathcal{O}(1)$. Thus the way in which we separated out the perturbation terms in orders of $\epsilon$ is no longer correct and thus our solution fails.