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Perturbation Theory and Ideality

| On 03, Mar 2008

By Joseph Marotta

Perturbation theory is a common technique that appears in both classical and quantum physics, where it is often used to approximate the solution to a nonlinear problem. Nonlinear problems are often difficult to solve directly, but are extremely useful in day-to-day engineering and physics situations. When using perturbation theory, the approximate solution takes the form of a dominant linear solution, with a series of small “perturbations” added to it. In essence, this involves taking a solution to a more-readily solvable problem and adding several smaller terms to it that approximate the unknown, more-difficult parts of the current problem. Both the dominant solution and the smaller additive terms take the forms of mathematical functions in space, time, energy, or any other set of parameters. The solution looks like a series of mathematical functions, all added together.

The “Ideal” in Engineering and Physics

Effectively, using the ideal final result (IFR) to solve an engineering problem takes the same form as using a perturbation theory to solve a physical problem. Both begin with an “ideal” solution and from there make small additions to the solution, to adapt the solution to the constraints of the problem. In the case of perturbation theory, the “ideal” is a problem that has already been solved (the simple harmonic oscillator in most cases). The IFR presents the solution as the ideal system, typically a problem that “solves itself.”

Using the IFR

Constraints on the IFR are often technical or physical contradictions that prevent realization of the IFR; they often can be solved by using the 40 inventive principles or 76 standard solutions of the Theory of Inventive Problem Solving (TRIZ). Solutions to these contradictions can be done as many times as is necessary to achieve a solution, but is it possible to determine how close the solution is to the IFR in a quantifiable way? Perturbation theory may provide the answer.

Using Perturbation Theory

In perturbation theory, it is required that the deviations from the initial solution be small.Specifically, perturbations are modulated by a multiplicative parameter greater than 0, but less than 1 (this parameter is typically represented by the Greek letter e). A solution typically derived by perturbation theory looks like the following:



where the small parameter e rapidly decreasing as extra terms are added (as 0<e<1)

Solving a problem in this fashion ensures that only small corrections to the initial solution can be made.

Working backward from the IFR adds a small number of mitigations to the technical and physical difficulties (if not impossibilities) of the IFR. Figure 1 illustrates a typical solution working backward from the IFR; this process can be repeated as many times as is necessary.

Figure 1: Working Backward From the IFR

An Ideal Result forElectrical Power Generation

As an example, consider electrical power generation. An ideal final result for generating power could be free, unlimited energy. The second law of thermodynamics says, however, that having a source of unlimited energy is physically impossible. So the ideal result must be modified to something like as much energy as possible for free. But infrastructure is needed to both collect and deliver power, so having a large source of energy will not be free. The solution can now change to as much energy as possible for as low a cost as possible. Various methods of implementing this solution can now be considered.

Minor Corrections to the IFR

Therefore, the specific solution should look like a perturbation theory solution to a problem, an IFR with minor corrections. As more steps are taken, the challenges should become progressively trivial, so that the specific solution should include only relatively minor modifications to the IFR after a certain point. Where this point is reached is different for every situation, but after every modification is proposed, an analysis of resources required to implement the solution should be executed.

If it appears that subsequent modifications of the IFR require equal amounts of resources as the previous modifications, either a different perspective must be taken on the problem or a different IFR must be considered. In other words, if modifying the IFR appears to stay the same or increase with respect to resources and effort required, the level of inventiveness must be raised. This is again analogous to perturbation theory, in which the theory begins to break down if it does not appear that the solution to the problem is converging on a single function. Carrying out the resource and effort analysis at each stage, then, could indicate the level of inventiveness required for a solution and help guide future efforts toward a solution.

Extending the Electrical Power Generation Solution

Returning to the electrical power generation example, one possible implementation of this solution could be solar power, which imposes an initial cost for the infrastructure and will continue to operate as long as a solar collector (or the sun itself) is in working order. The next step is to analyze how much of the limiting resources are required to implement this solution. Solar power output is limited by both the efficacy of the material (its energy conversion efficiency) and its surface area; the limiting resources are the substance and the space required (assuming the solar collectors are placed in an appropriately sunny area).

Beginning with the substance, the expense required in developing a more efficient solar cell must be analyzed. In terms of resource cost, how difficult is it to develop a more efficient solar cell and mass-produce it? One potential solution involves doping current solar cells with carbon nano-tubes – thereby increasing not only the energy conversion efficiency, but also the effective surface area of the cell. Although producing carbon nano-tubes is cheap in terms of amount and cost of the substance involved, mass-producing them effectively is not. New technology must be developed to both incorporate carbon nano-tubes and to effectively mass-produce them. Solutions to both of these problems will require significant amounts of time and probably new ideas and trends in technology. In short, this solution requires many resources before it can be implemented in an expeditious manner. Either this particular implementation of solar cells must be abandoned or a higher level of inventiveness must be considered.

Looking to use of space as a resource with existing solar cell technology provides a relatively simple and cheap solution – place more cells in a greater area. Examining places where that resource is not as valuable (less arable land, more inaccessible areas, places where fewer people live), more space for solar energy collectors can be found. Further, this is a solution that will not require many other resources to implement. The solution appears to converge on simply increasing the surface area occupied by solar cells. Unfortunately, this resource (space) is limited and this process of solution will not be forever available. In the absence of new technologies, however, simply using more space is the method that appears to converge on the modified IFR (as much energy as possible for as low a cost as possible) the quickest.

Conclusion

The similarities between using the IFR to find a solution to a specific problem and perturbation theory in physics are striking. This similarity can be used to think about the difficulty inherent in a specific problem. Then ideas about how to overcome a particular problem can be generated and these ideas can be evaluated for cost in both resources and effort for a particular solution. Using this analogy, resource costs for specific engineering problems can be saved before a specific solution is implemented.