Positional entropy

In thermodynamics, positional entropy is []

Overview
In 1904, James Jeans, in his The Dynamical Theory of Gases (§194), was referring to “positional entropy”, based on what seems to be, his own peculiar derivation, based on his concept of “positional coordinates” of molecules in a gas body, framed around Max Planck's 1901 entropy formula S = k log W. [1] He began citing this term the following year:

“We may perhaps summarize the question by saying that the ‘positional entropy’ of the system is a maximum when the system is in the normal and anti-normal piling.”
— James Jeans (1905), “The Kinematics of a Granular Medium in Normal Piling” [2]

In 1920, Jeans, in a latter edition of his The Dynamical Theory of Gases (pgs. 178-180), gives his own peculiar derivation of a gas system that yields an entropy equation, entropy symbolized by phi φ, consisting of two parts:

Jeans eq 2

one, the first part of the above equation, entropy depending on the velocities of the molecules, the other, the second part above, on the positional coordinates of the molecules. [3]

Entropy and life debate
On 21 Apr 1934, Jeans, amid the Jeans, Donnan, Guggenheim debate, referred to the “normal” formula for positional entropy (an invention which seems to be his own), as follows: [4]

“I am anxious to treat Prof. Donnan's views with all courtesy, but think his last letter, written in conjunction with Prof. Guggenheim, is entirely invalidated, like his previous letter, by a technical error in thermodynamics. The ordinary formula for the positional entropy of a large number of particles is:

positional entropy = k ∫∫∫ ν log ν dxdydz

where v is the number of particles per unit volume. Thus, moving N particles from a place of density ν to one of higher density ν' decreases the entropy by

kN (log ν’ – log ν)

Surely Profs. Donnan and Guggenheim have over­looked the factor N. Owing to its presence, moving a single molecule does not, as they contend, have the same effect as moving a truckload of N molecules, but only 1/Nth of this effect. The same error, I think, invalidates their second paragraph.”

(add)

Quotes
The following are related quotes:

“Typical views in the 1930's on this subject were expressed in an exchange of letters to the journal Nature between F. G. Donnan, a professor of physical and inorganic chemistry, and the physicist James Jeans. Donnan [1934] challenged the view expressed by Jeans in his book The New Background of Science that organisms must have some means for evading the second law of thermodynamics. Jeans [1934] replied that a person steering a large steamship adds little entropy in the activities of steering, but the ‘positional entropy’ of the cargo is changed considerably. This is due to moving the cargo from a location where it has a high density υ1 to a location where it has a low density υ2, thereby changing the positional entropy by ΔS = kBN[ln υ1 – ln υ2]. In this case, the intervention of an intelligent being is incidental to the entropy calculation. In spite of the entry of the physicist E. A. Guggenheim on Donnan's side, and the exchange of several more letters, this controversy died without being resolved and without, apparently, leaving any further trace. Jeans' positional entropy is only one example of the types of entropy proposed during this period.”
— Paul McEvoy (2002), Classical Theory (pg. 175)

References
1. (a) Jeans, James. (1904). The Dynamical Theory of Gases (§194). Publisher.
(b) Jeans, James. (1905). “The Kinematics of a Granular Medium in Normal Piling” (Ѻ), Proceedings of the London Mathematical Society (pg. 136), Jan 12.
(c) Jeans, James. (1920). The Dynamical Theory of Gases (entropy, pgs. 78, 178, 180; of radiation, pgs. 367, 397; positional, 15+ pgs). Publisher.
2. Jeans, James. (1905). “The Kinematics of a Granular Medium in Normal Piling” (Ѻ), Proceedings of the London Mathematical Society (pg. 136), Jan 12.
3. Jeans, James. (1920). The Dynamical Theory of Gases (entropy, pgs. 78, 178, 180; of radiation, pgs. 367, 397; positional, 15+ pgs). Publisher.
4. Jeans, James. (1934). “Letter | Activities of Life and the Second Law of Thermodynamics” (abs), Nature, Apr 21, 133:612.

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