In economic thermodynamics, Julius Davidson (c.1875-c.1935) (SN:22) was an American economist noted for his 1919 article “One of the Physical Foundations of Economics” in which, using American engineer Willard Gibbs’ 1901 Elementary Principles of Statistical Mechanics as a basis, he argues that the law of diminishing returns is based on chemistry and physics, comparing human chemical reactions to basic equilibrium adjusting chemical reactions.

Law of diminishing returns
Davidson, of significance, compares the pairing of single men and single women in a fixed sized society, with varying starting numbers of single men in respect to a fixed number of single women to a type of and Le Chatelier’s principle based equilibrium-seeking chemical reaction (although, to note, he doesn’t mention French chemist Henry Le Chatelier or his 1888 chemical equilibrium principle). In introducing this scenario, Davidson describes six different possible reactions in which a thousand single woman are introduced into a society with varying starting numbers of single men (1, 500, 1000, 2000, 3000, and 4000), to give the following hypothetical reactions, each with different equilibrium state outcomes (as shown below). Under normal circumstances, evolutionary psychologists estimate that 90% of people typically marry or pair. On the following reaction scenario:

Davidson states: "If in a certain community there were a thousand women all of marriageable age, and all desiring to be married, and if in this same community there were only one man of marriageable age also desiring to be married, and if furthermore polygamy were prohibited, the probability that this one man would, within a measurable period of time, be married, would amount, for practical purposes, to certainty." In other words, it would be nearly certain (~100% probability) that in this situation the man would pair (MF). On the following reaction scenario:

Davidson states: "we should not be at all certain that all of them would be married off within a given time". In other words, a certain number, say z=495 or 95% of the single men, would be paired up in the equilibrium or end state of the reaction. In Le Chatelier's principle terms, more reactants M (males) have been introduced into the system, and the equilibrium shifts to the products side of the reaction, yielding more paired products MF (male-female pairs). On the following reaction scenario:

Davidson states: "if the number of marriageable men reached 1000, the chances of marriage for each man, though good, would be much less than before". In this case, about 90%, or z=900, of the single reactants would be paired up in the finalize end (equilibrium) state of the reaction. On the following reaction scenario:

Davidson states: "if the number of men increased further to 2000, 3000, 4000, etc., each man's chances of marriage would gradually diminish, while the chances of each woman would increase." In other words, in the M=2000 scenario, we should expect the number of pairings to be about 95% or z=950 of limiting reactant (females), because the increase in the concentration of male reactants would shift the equilibrium of the reaction to the products side. On the following reaction scenario:

Davidson states that, similar to previous, we should expect the number of pairings MF to be about z=970, i.e. about ~97% or limiting reactant (females) would have converted to pairings. On the following reaction scenario:

Davidson states, similar to the previous, we should expect the number of pairings MF to be about z=980, i.e. about ~98% or limiting reactant (females) would have converted to pairings.

Julian goes on to note that a number of factors have been neglected in his reaction model, by stating that: "here some one will protest that human factors are being neglected. Men and women have qualities which are sometimes better and sometimes worse; in one instance there may be more likelihood of attachment than in another; in one community social life gives to each sex more opportunities to meet the other than in another. All this is true and even more. Despite this, however, the element of chance still remains, and we are justified in our conclusion that as the number of either sex increases, the number of the other remaining constant, the probability that any one of the first will be married diminishes." Likewise, in the first and second reaction scenario, for example, many female-female homosexual relationships would likely form, owing to the limited number of men.

In sum, as the number of initial reactant single men M increased further, each man's chances of marriage MF would gradually diminish, while the chances of each women marrying would increase

Chemical comparison
Davidson the compares this male-female reaction to the reaction of ethanol (ethyl alcohol) and acetic acid to produce ethyl acetate and water:

 CH3CH2OH + CH3COOH ⇌ CH3COOCH2CH3 + H2O (ethanol) (acetic acid) (ethyl acetate) (water)

a reaction according to which, as he states, if the starting reactants are introduced to each other in equivalent ratio (similar to 1000-to-1000 ratio of single men to single women as starting reactant), action will cease, i.e. chemical equilibrium will be reached, when two-thirds of each reactant is transformed into ethyl acetate and water.

Davidson then states that if ethanol and acetic acid are brought into a ratio of three to one (similar to 3000-to-1000 ratio of single men to single women as starting reactants), the action will proceed until nine-tenths of the acetic acid is converted (similar to 9 out of every 10 women ending up married, in a society in which there is a 3-to-1 ratio of single men to single women at the start of the society).

Davidson goes on to discuss how this model extend to the three big factors of economics: land, labor, and capital, where increase in any given input (such as by adding an extra plow to a farmer's capital) will produce diminishing returns in the long run.

Notes
Interestingly, Davidson's comparison is similar to American astrophysicist Christopher Hirata's circa 2000 article "The Physics of Relationships", wherein he attempts to describe equilibrium shifts in a student body at a hypothetical college campus using the equilibrium constant and related chemical thermodynamic equations. [2]

Later citations of Davidson’s paper, such as made by American mathematical economist Barkley Rosser, erroneously state that Davidson mentions the “entropy law” (and or entropy) as underpinning economics; which, however, is not the case, as Davidson does not use the word entropy, nor does he allude to it, but rather only seems to discuss the law of mass action, the law of chemical equilibrium, and Le Chatelier’s principle (although he doesn’t mention this term either) in the general sense. [3]

Davidson, according to American political economist Kenneth Stokes (1994), is listed as one of the ‘heretical philosophers of social energetics’, along with Wilhelm Ostwald, Leon Winiarski, Alexander Bogdanov, Nikolai Bukharin (Historical Materialism: a System of Sociology, 1921), Eduard Sacher, Felix Auerbach, Rudolf Clausius, Patrick Geddes, Leopold Pflaunder, Georg Helm, Thomas Carver, F. Ackerman, Fred Henderson (The Economic Consequences of Power Production, 1923), Alfred Lotka, and Frederick Soddy. [4]

References
1. Davidson, Julius. (1919). “One of the Physical Foundations of Economics” (abs), Quarterly Journal of Economics, 33: 717-24.
2. (a) Hirata, Christopher M. (c. 2000). “The Physics of Relationships” (section: Fun), Tapir.Caltech.edu.
(b) Hirata, Christopher M. (2010). "The Physics of Relationships", Journal of Human Thermodynamics, 6(5): 62-76.
3. (a) Rosser, John Barkley. (1991). From catastrophe to Chaos: a General Theory of Economic Discontinuities (pg. 268). Kluwer Academics.
(b) Rosser, J. Barkley. (2011). Complex Evolutionary Dynamics in Urban-Regional and Ecological-Economic Systems: from Catastrophe to Chaos and Beyond (Julius Davidson, pgs. 194, 275). Springer.
4. Stokes, Kenneth M. (1994). Man and Biosphere: Toward a Coevolutionary Political Economy (pg. 90-91). M.E. Sharpe.