Order of Magnitude Physics a Textbook with Applications to the Retinal Rod(1998) [pdf]

Order of Magnitude Physics A Textbook with Applications to the Retinal Rod and to the Density of Prime Numbers Thesis by Sanjoy Mahajan In partial fulfillment of the requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1998 (Submitted 19 May 1998) ii c©1998 Sanjoy Mahajan All rights reserved iii Acknowledgments I have had wonderful teachers. John Hopfield suggested that a textbook could be part of a thesis, and always shared the joy that he finds in physics. Sterl Phinney (my advisor) and Peter Goldreich taught me order-of-magnitude physics. Carver Mead shared his great scientific insight and his gentle personal wisdom. Steve Frautschi encouraged me to lead a section of Physics 1a, and generously gave me guidance on this unusual thesis project. David Middlebrook taught me that to calculate is human, but to approximate is di- vine. Henry Lester, and the Computation and Neural Systems program, introduced me to biophysics. Lyn Dupr´e copyedited the parts of this dissertation with no mistakes, and, in the process, trained my ear. Erik Winfree showed me what clear thinking means. Joshua Zucker guided my experiments in teaching. Margaret Lambert taught me how to play Bach; and Don Caldwell welcomed me into the choir, although I had never sung before. Comments from Jennifer Linden, Dap Hartmann, and John Hopfield have made the textbook useful to a wider audience. Comments from Joseph Keane, Dinakar Ramakrishnan, Warren Smith, and Erik Winfree have made the primality model more rigorous. Donald Knuth wrote the TEX typesetting system. John Hobby wrote MetaPost, with which I drew most of the figures. Thousands of volunteers wrote the Linux operating system. The Hertz Foundation supported me in my last three years of graduate study. The Whitaker Foundation supported me in my first year. I also received support from Carver Mead, John Hopfield, and William Goddard, all of whom have been my advisors. Many people here have shared their friendship. I particularly want to mention Sarah Bates, Carlos Brody, Craig Jin, Elisabeth Moyer, Sam Roweis, Rahul Sarpeshkar, Micah Siegel, Steven and Athina Quake, and Erik Winfree. To my other friends, here and from the years BC (before Caltech), thank you as well.

Laura Rodriguez, den mother of the Hopfield group, kept us sane and happy. My brothers, Sunit and Ashish, and my parents, Sushma and Subhash, have given me a home. My thanks to all. iv Abstract I develop tools to amplify our mental senses: our intuition and reasoning abilities. The first five chapters—based on the Order of Magnitude Physics class taught at Caltech by Peter Goldreich and Sterl Phinney—form part of a textbook on dimensional analysis, approxi- mation, and physical reasoning. The text is a resource of intuitions, problem-solving meth- ods, and physical interpretations. By avoiding mathematical complexity, order-of-magnitude techniques increase our physical understanding, and allow us to study otherwise difficult or intractable problems. The textbook covers: (1) simple estimations, (2) dimensional analysis, (3) mechanical properties of materials, (4) thermal properties of materials, and (5) water waves. As an extended example of order-of-magnitude methods, I construct an analytic model for the flash sensitivity of a retinal rod. This model extends the flash-response model of Lamb and Pugh with an approximate model for steady-state response as a function of background light Ib. The combined model predicts that the flash sensitivity is proportional to I −1.3 b . This result roughly agrees with experimental data, which show that the flash sensitivity follows the Weber–Fechner behavior of I−1 b over an intensity range of 100. Because the model is simple, it shows clearly how each biochemical pathway determines the rod’s response. The second example is an approximate model of primality, the square-root model. Its goal is to explain features of the density of primes. In this model, which is related to the Hawkins’ random sieve, divisibility and primality are probabilistic. The model implies a recurrence for the probability that a number n is prime. The asymptotic solution to the recurrence is (log n)−1, in agreement with the prime-number theorem. The next term in the solution oscillates around (log n)−1 with a period that grows superexponentially. These oscillations are a model for oscillations in the density of actual primes first demonstrated by Littlewood, who showed that the number of primes ≤ n crosses its natural approximator, the logarithmic integral, infinitely often.

No explicit crossing is known; the best theorem, due to te Riele, says that the first crossing happens below 7 × 10370. A consequence of the square-root model is the conjecture that the first crossing is near 1027. v Contents 1 Contributions 1 1.1 Textbook 2 1.2 Original research 3 1.3 Words to the reader 4 Part 1: Textbook 2 Wetting Your Feet 6 2.1 Warmup problems 6 2.2 Scaling analyses 22 2.3 What you have learned 34 3 Dimensional Analysis 36 3.1 Newton’s law 36 3.2 Pendula 42 3.3 Drag in fluids 48 3.4 What you have learned 64 4 Materials I 66 4.1 Sizes 66 4.2 Energies 79 4.3 Elastic properties 82 4.4 Application to white dwarfs 91 4.5 What you have learned 96 5 Materials II 97 5.1 Thermal expansion 97 5.2 Phase changes 101 5.3 Specific heat 111 5.4 Thermal diffusivity of liquids and solids 117 5.5 Diffusivity and viscosity of gases 120 vi 5.6 Thermal conductivity 121 5.7 What you have learned 125 6 Waves 127 6.1 Dispersion relations 127 6.2 Deep water 131 6.3 Shallow water 157 6.4 Combining

deep- and shallow-water gravity waves 161 6.5 Combining deep- and shallow-water ripples 162 6.6 Combining all the analyses 163 6.7 What we did 163 Part 2: Applications 7 Retinal rod 165 7.1 Problem that the rod solves 165 7.2 Scope of this analysis 166 7.3 Flash response in darkness: The model of Lamb and Pugh 167 7.4 Steady-state response 171 7.5 Flash sensitivity 177 7.6 Conclusion 180 8 Density of primes 181 8.1 Approximate primality by stages 181 8.2 Convert the recurrence into a delay–differential equation 187 8.3 Solve the delay–differential equation 191 8.4 Analyze the oscillations 200 8.5 Speculate on the Riemann hypothesis 203 8.6 Go beyond the square root 203 8.7 Lower the temperature 205 8.8 Finish 206 1 1 Contributions Human hands are weak, and human arms are short. Tools—such as the wedge, the crane, and the waterwheel—strengthen our grasp and lengthen our reach. To extend our mind’s reach, we need mental tools. This dissertation is a resource of such tools. It is in two parts: a textbook on order-of-magnitude physics, and original research in biophysics and number theory that illustrates order-of-magnitude methods. In an order-of-magnitude analysis, we approximate because “an approximate answer is not only almost as useful as the exact answer, it’s more useful.” An approximate answer, because it is simple, shows us the important principles in a physical problem. Thirty thou- sand pages of detailed velocity and temperature tables might give us accurate knowledge of the winds, but we would rather know that a hurricane with 100 mph winds is approaching from sea. An approximate model, because it is compact, fits in our limited brains; once it is in there, we can refine it, and reason from it to new problems.

No exact, analytic solutions exist for many problems, such as hurricane motion or river turbulence. Even when solutions exist, they are often so complicated that they teach us little about the physical principles important in the problem. It is these principles that have general value, because we can use them to understand new pieces of the world. Often, these principles remain hidden behind a maze of mathematical manipulations; the prin- ciples determine the scaling laws—the functional form of the solution—and the complex mathematics determines a dimensionless constant that multiples the functional form. An order-of-magnitude analysis reverses the traditional approach: You forget about the dimen- sionless constant, and hope that it is close to unity. You make estimates using procedures no more complicated than high-school algebra and geometry. If your scaling law is grossly wrong, you find out quickly, and can begin looking for new physics to create a new model; 1. Contributions 2 if it is accurate, you know what physical principles to include in the next, more refined analysis, and you have added a physical picture to your mental toolbox. The textbook teaches such techniques. You learn how to identify important factors, how to combine the factors into an approximation, and how to appraise and refine an approximation. The theme woven throughout the chapters is successive approximation, be- cause all order-of-magnitude techniques are derivatives of this one, and because an order-of- magnitude analysis is itself a first approximation to a more accurate analysis. The biophysics chapter (Chapter 7) shows how to apply these methods to a problem more in-depth than a typical textbook problem. The number-theory chapter (Chapter 8) shows how to use order- of-magnitude methods to create a series of successively more accurate, yet still tractable, models of primality; and how to use more standard methods to solve these models. 1.1 Textbook The textbook is based on the Physics 103 class taught at the California Institute of Tech- nology by Peter Goldreich and Sterl Phinney. My contribution has been to make explicit the methods of approximation, and to show how to use physical reasoning to complement the methods of dimensional analysis. Many books—for example those by Bridgman [4], Pankhurst [45], and Ipsen [26]— describe the method of dimensional analysis: how to combine physical quantities into groups that have no units.

This textbook begins earlier and ends later than most texts. Chapter 2 introduces methods to estimate everyday numbers, such as city budgets or annual diaper production. If dimensional analysis is the first method of approximation, these methods are the zeroth. Chapter 3 introduces dimensional analysis. The discussion goes beyond the discussion in most textbooks: You learn how to use physical reasoning to complete dimensional-analysis arguments. With a repertoire that includes simple estimation methods, dimensional analysis, and physical reasoning, we then study the mechanical and thermal properties of materials (Chapters 4 and 5); the calculation of boiling points (Section 5.2.1) is an extended example of successive approximation, a method that the text introduces in successively more complicated problems. The final textbook chapter studies the properties 1. Contributions 3 of water waves. It analyzes one function, the dispersion relation, in all its limits. The chapter combines dimensional analysis and successive approximation with a physical model of wave propagation: the slab model. 1.2 Original research I apply this style of reasoning to a problem in biophysics, the flash sensitivity of a retinal rod; and to a problem in number theory, the distribution of primes. 1.2.1 Retinal Rod Imagine that we want to analyze the flash sensitivity of a retinal rod, to determine how the sensitivity varies as a function of background light Ib. (The flash sensitivity is the maximum change in membrane current divided by the flash strength.) To make an accurate model, we would model the biochemistry with a bundle of nonlinear differential equations that contain a slew of rate and time constants. We would throw up our hands and integrate the equations numerically; if we set up the equations correctly, our simulation would match reality. We could then get a valuable check on the completeness of our experimental knowledge. But what insight would this model give us about how the rod works? We would not know what function the various biochemical pathways perform. We could explain the response to light by saying only, “It works as the computer says.” Instead, Chapter 7 develops an analytic model of the flash sensitivity. This model compactly represents important features of rod biochemistry; it shows how negative feedback and cooperative binding compute a flash sensitivity that approximates the Weber–Fechner sensitivity of I−1 b .