1.4. How to do better on exams (and discover new physical laws)[[Student version, December 8, 2002]] 15
- Often when we forge such a link we find that powerful theoretical tools to solve one
problem have already been created in the context of another. An example will be
the mathematical solution of the helix-coil transition model in Chapter 9. - Similarly, we get to carry over powerful existingexperimentaltechniques as well. For
example, the realization that DNA and proteins were molecules led Max Perutz,
Linus Pauling, Maurice Wilkins and others to study their structure using X-ray
diffraction, a technique invented to find the structure of simple, nonliving crystals
like quartz. - Finally, perceiving a link between two circles of ideas can lead us toask new questions
which later prove to be key. For example, even after Watson and Crick’s discovery
that the DNA molecule was a very long sentence written in an alphabet with four
letters (see Chapter 3), attention did not focus at once on the importance of finding
the dictionary, or code, relating sequences of those letters to the 20-letter alphabet of
amino acids constituting proteins. Thinking about the problem as one in information
transfer led George Gamow, a physicist interested in biology, to write an influential
paper in 1954 asking this question and suggesting that answering it might not be so
difficult as it at first seemed.
It may seem that we need no longer content ourselves with simple models. Can’t massive
computers now follow the fine details of any process? Yes and no. Many low-level processes can in
fact now be followed in molecular detail. But in practice, our ability to get a detailed picture of even
simple systems is surprisingly limited, in part by the rapid increase of computational complexity
when we study large numbers of particles. Surprisingly, though, many physical systems have simple
“emergent properties” not visible in the complex dynamics of their individual molecules. The
simple equations we’ll study seek to encapsulate these properties, and often manage to capture the
important features of the whole complex system. Examples in this book will include the powerful
property of hydrodynamic scale invariance to be explored in Chapter 5, the mean-field behavior
of ions in Chapter 7, and the simple elasticity theory of complex macromolecules in Chapter 9.
The need to exploit such simplicity and regularity in the collective behavior of many similar actors
becomes even more acute when we begin to study even larger systems than the ones discussed in
this book.
1.4 How to do better on exams (and discover new physical laws)
Equation 1.2 and the discussion below it made use of some simple ideas involving units. Students
often see units, and the associated ideas ofdimensional analysis,presented with a brush-your-teeth
attitude. This is regrettable. Dimensional analysis is more than just hygiene. It’s ashortcut to
insight.
1.4.1 Dimensions and units
Every physical quantity has abstractdimensionsthat tell uswhat kind of thingit is. Every kind
of dimension can be measured using a variety of differentunits.The choice of units is arbitrary.