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The Art of Approximation

Writer's picture: Mishkat BhattacharyaMishkat Bhattacharya

Theoretical physics consists of building mathematical models of physical systems. All such models are approximate to varying degrees, since they cannot take everything into account. Still, there is some difference between finding an approximate and an exact solution of the (necessarily approximate) model. In this post, I will discuss both these outcomes.


Approximate Solutions of Approximate Models


By a solution of a model we mean a solution of the relevant mathematical equation(s). That could imply the solution of an algebraic, transcendental, differential or linear algebraic (i.e. matrix) equation - there may be some others that I am missing but I think you get the idea. Approximate solutions generate varying types of feelings in me:


Beauty and terror


To me it is undoubtedly a part of the beauty of physics that by making suitable approximations one can quickly find useful solutions to many problems. Of course, not only beauty but terror is involved, because our solutions live and die by their approximations.


So it is very important to be aware of the approximations one is making, and not taken them for granted and examine them carefully when our solution cannot explain the experimental data, or does not make physical sense.


Fun


A fun type in this category is the famous 'Fermi problem', e.g.


i) How many piano tuners are there in Chicago?

ii) How much would you charge to clean every window in Seattle?

iii) How many times does the average person's hear beat in a lifetime? (Answer: About a billion times)


Quite satisfactory answers can be found to these questions by making reasonable assumptions. (Wonder what ChatGPT makes of these questions?)


Surprise


In my own experience, the physical universe is surprisingly amenable to approximations. For example, I once had a postdoc start off on building a model, and he kept coming to me with these complicated expressions, so I kept telling him to make all kinds of approximations to simplify them.


At the end of eight months, we had built a working model, which however had so many approximations that I thought not only would no one believe our predictions, we would not believe them ourselves! (I used to brag that our model had every approximation known to physics.)


Amazingly, when the postdoc compared our predictions to experiment, the two showed rather close agreement. This was so unexpected - and in a way profound - that when he came to my office to tell me this, he was shaking. I have heard similar stories from other colleagues as well.


Inevitability


Of course, for sufficiently complicated models, there is no recourse except to feed the model to a computer. The computer typically solves the model numerically (but see below). This produces an approximate solution, limited by the numerical precision, etc. But very often the solutions are highly satisfactory as well as useful.


The deep end of computational solution to physics problems includes forecasting weather, simulating stellar dynamics, computing the properties of materials, etc. Forms of computing like AI have even started suggesting which experiments should be done in physics.


Exact Solutions of Approximate Models


Simple models


An exact solution to the model can be found if it possesses enough symmetry. This can happen if the model is simple (it involves few variables and constraints). Introductory physics is based on such 'simple' models: mass on a spring, mass on a slope, stone at the end of a string, person in an elevator, etc. Based on the valuable physical intuition and the protocol for mathematical solution provided by these example, we can tackle more complex problems.


Complicated models


For more complicated models, the challenge is to find enough symmetry in it, so it can be solved. This often necessitates the identification of constants of motion, operations that leave the system unchanged, etc. There is no guarantee that any model can be solved exactly, neither a general systematic method - creative thinking is the only prescription.


Satisfaction


But it is extremely satisfying - the word 'ecstasy' is not inappropriate here - when such a challenge can be met. The first cause of satisfaction is that a more general kind of knowledge about the model - not restricted to the parameter values used in any numerical solution - now becomes available.


The second cause is aesthetic. This because symmetry - which is the source of the solvability - is associated with our sense of beauty. I have had the pleasure of making such a discovery three times (...in 25 years). In one of those cases, a symbolic calculation software (Mathematica) was responsible for the insight (so computers can provide exact analytic solutions too, in addition to approximate numerical ones).


The need for exact solutions


Exact solutions are not just vanity items, which can be made irrelevant by the power of numerical computation. They can sometimes be - in a sense - irreplaceable, able to account for important physical phenomena which approximate techniques miss. An example of this I like to keep 'in my pocket' is that of the BCS theory of superconductivity:


Feynman had had spectacular success with the approximate technique of perturbation theory (his famous diagrams are basically terms in a perturbation series) when it came to quantum electrodynamics (the study of the interactions of light and matter).


But when he used the same tool to try and solve the mystery of superconductivity, he failed. After Bardeen, Cooper and Schrieffer solved the problem exactly, it became clear why it could not be solved approximately using terms from a series à la Feynman. This was because the solution contained an essential singularity - which meant it could not be represented by a series!

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