Citation
Mahajan, Sanjoy Sondhi (1998) Order of Magnitude Physics: A Textbook with Applications to the Retinal Rod and the Density of Prime Numbers. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8MJZA984. https://resolver.caltech.edu/CaltechTHESIS:10302009110303489
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, approximation, and physical reasoning. The text is a resource of intuitions, problemsolving methods, and physical interpretations. By avoiding mathematical complexity, orderofmagnitude 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 orderofmagnitude methods, I construct an analytic model for the flash sensitivity of a retinal rod. This model extends the flashresponse model of Lamb and Pugh with an approximate model for steadystate response as a function of background light I_{b}. The combined model predicts that the flash sensitivity is proportional to I_{b}^{1.3}. This result roughly agrees with experimental data, which show that the flash sensitivity follows the WeberFechner behavior of I_{b}^{1} 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 squareroot 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 primenumber 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 to Riele, says that the first crossing happens below 7 x 10^{370}. A consequence of the squareroot model is the conjecture that the first crossing is near 10^{27}.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Physics ; approximation, back of the envelope  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Physics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Group:  TAPIR, Astronomy Department  
Thesis Committee: 
 
Defense Date:  19 May 1998  
NonCaltech Author Email:  sanjoy (AT) mit.edu  
Additional Information:  From author: [Thesis] eventually became a textbook, which is also freely available from MIT Press (CCBYNCSA): "The Art of Insight in Science and Engineering: Mastering Complexity" (MIT Press, 2014).  
Record Number:  CaltechTHESIS:10302009110303489  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:10302009110303489  
DOI:  10.7907/8MJZA984  
Related URLs: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  5338  
Collection:  CaltechTHESIS  
Deposited By:  Tony Diaz  
Deposited On:  17 Nov 2009 23:57  
Last Modified:  09 Oct 2024 22:01 
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