The diffusion of gossip, the substitution of a new product, and the population growth of a species in an ecosystem with a bounded carrying capacity may all be described by the logistic equation dy/dt = ky(1 – y), where solutions are y = 1/1 + exp –k(t – t0). Logistic analysis also sheds light on the growth of knowledge in physics.1 The history is well documented.2 Major events can be assigned to twelve subfields: classical gravity, classical mechanics, optics and wave physics, thermodynamics, electrodynamics, atomic physics, special relativity, nuclear physics, general relativity, quantum mechanics, high-energy particle physics, and string physics.3
Grouping these discoveries into six chronological stages from the years 1500 to 2000 covering twelve subfields and analyzing their median and characteristic time of development, I found that each stage demonstrated a logistic-like development, as measured by the number of important discoveries.
|Classical gravity||1655 (1649)||92 (88)||19||1667 (1674)||145 (143)|
|Classical mechanics||1667 (1667)||163 (169)||51|
|Optics/Waves||1679 (1707)||180 (173)||32|
|Thermodynamics||1808 (1811)||123 (121)||63||1821 (1825)||117 (116)|
|Electrodynamics||1833 (1839)||111 (111)||69|
|Atomic physics||1896 (1892)||54 (52)||49||1897 (1895)||50.5 (49.5)|
|Special relativity||1898 (1898)||47 (47)||15|
|General relativity||1916 (1916)||14 (11)||24||1925 (1924)||19 (19)|
|Nuclear physics||1932 (1929)||28 (34)||57|
|Quantum mechanics||1927 (1928)||14 (13)||79|
|High energy physics||1962 (1959)||38 (34)||139||1962 (1959)||38 (34)|
Summary of subfield and stage median years, characteristic times, and number of events. The numbers in parentheses are the results of an alternative approach using a least-squares logistic fit analysis.4
As a field develops, there is an initial decrease in both the characteristic time of a stage and the time between its midpoints. This was true from stage 1 to stage 4. The fifth stage, covering high energy physics, had a larger characteristic time. The time between stages 4 and 5 was about the same as that between stages 3 and 4; the time between stages does not decrease. This change in the characteristic duration and time between stages might indicate the logistic nature of the field’s discovery.
The assumption that each stage contributes the same amount towards the development of fundamental physics makes it possible to plot the overall completion fraction curve with five completed stages and one uncompleted stage. Since the stage centered at 1925 has the smallest characteristic time, I chose this point as the center of the transition. Symmetry would then imply that there are three stages beyond the midpoint to match the three that precede it. These would be high energy physics, string physics, and another, as-yet-unidentified stage. Only the logistic width parameter, α, is needed to fit the curve. If α = 1/77, 212 years are required for the transition to go from 20% to 80% completion.
Within physics, logistic development is apparent both in its subfields and the field as a whole. This suggests that a new subfield, such as string physics, is likely to be 50% complete in 2030, and 80% complete in 2090. There are currently many approaches to the integration of quantum field theory and the general theory of relativity. Some approaches will be abandoned, others modified, and perhaps some of them will turn out to represent different views of the same underlying theory. The development curve would then be symmetric around the midpoint, which we currently identify as the 1920s. String physics, if that is what it turns out to be, is only the second stage after the midpoint inflection in the field as a whole, suggesting that there is yet another stage in the development of fundamental physics to come. If the symmetry holds, the 20%, 50%, and 80% completion times for this last stage would be 2100, 2180, and 2260 respectively.
What this analysis shows is that progress in the overall field of fundamental physics is a function of time. No claim is being made about the mechanism generating this pattern of development. The seven phases seem to be tied to increasingly smaller scales. Isaac Newton’s theory connected spatial scales from 1 to 108 m; thermodynamics and James Clerk Maxwell’s electromagnetic theory connected radio waves from 1 to 10–8 m; and nuclear physics connects phenomena from 10–8 to 10–15 m. In 1979, Bernard Carr and Martin Rees attempted to estimate the sizes of important scales, based on fundamental constants. The size scales from planets to humans to molecules is roughly a multiple of the quarter root of the ratio of the strength of gravity compared to electromagnetism (αG/α)‑1/4. This is about 10–9.
Each of the five transitions allowed discovery of phenomena that facilitated technological developments and the mathematical instruments needed to tackle smaller scales. Each phase represents a reduction in the spatial scale by about eight orders of magnitude. A logistic pattern would indicate that there will be two more phases beyond the current fifth phase. The reason for having seven nested transitions within a larger transition is not known, although it may be related to the initial step of understanding a fraction of the full problem.5 Too small an initial fraction leads to incomplete problem scope and definition. Too large an initial fraction leads to complications between the development of basic understanding and higher-level derivations. This nested set of logistics transitions is similar to a recent analysis of general evolution.6
- David LePoire, “Application of Logistic Analysis to the History of Physics,” Technological Forecasting and Social Change 72 (2005): 471–79. ↩
- Lloyd Motz and Jefferson Weaver, The Story of Physics (New York: Avon Books, 1992). ↩
- There might be other ways to characterize discoveries, such as, for example, experimental, observational, and theoretical. But this particular mix of approaches has been important throughout the history of scientific progress. ↩
- David LePoire, “Application of Logistic Analysis to the History of Physics,” Technological Forecasting and Social Change 72 (2005): 475. ↩
- David LePoire, “Potential Nested Accelerating Returns Logistic Growth,” in Big History, Evolution: From Big Bang to Nanorobots (Volgograd, Russia: Uchitel Publishing, 2015), 46–60. ↩
- David LePoire, “Interpreting ‘Big History’ as Complex Adaptive System Dynamics with Nested Logistic Transitions in Energy Flow and Organization,” Emergence: Complexity and Organization. March 31, 2015. ↩