Equation 133

Thermodynamic potential (economics)
Annotated image, page from Lev Landau and Evgeny Lifschitz’ 1959 Course of Theoretical Physics (Ѻ), from Smith’s 2015 blog “Economic Potentials: How to Define an Economy”, wherein he attempts, as she says, to “construct the thermodynamic potential of an economy by elaborate analogy”, wherein eq. 15.7 is a variant of Willard Gibbs' 1876 equation 133, which Edwin Wilson (1938) told Paul Samuelson could be used to formulate the equilibrium of an economy. [3]
In equations, equation 133 (CR:8) is equation number 133 of Willard Gibbs’ 1876 700-equation numbered On the Equilibrium of Heterogeneous Substances, subsection "Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations", defined as follows:

 U - TS + PV - M_1 m_1 - M_2 m_2 ... - M_n m_n \,

where U is energy, T temperature, S entropy, P pressure, V volume, mn the quantities of substances in a "fluid enclosed in a rigid envelop which is nonconducting to heat and impermeable to all the components of the fluid", and Mn the potentials (check) of the nth substance.

Mathematical economics
In 1938, American polymath Edwin Wilson, the last protege of Willard Gibbs, who was also Paul Samuelson' economics PhD adviser, wrote the following to Samuelson, in commentary on one of Samuelson's papers, wherein, in the context of his mathematical economics course, suggested that he use Gibbs equation 133 to formulate a new foundational version of economics; specifically: [2]

“Moreover, general as the treatment is I think that there is the possibility that it is not so general in some respects as Willard Gibbs would have desired. [In] discussing equilibrium and displacements from one position of equilibrium to another position [Gibbs] laid great stress on the fact that one had to remain within the limits of stability. Now if one wishes to postulate the derivatives including the second derivatives in an absolutely definite quadratic form one doesn’t need to talk about the limits of stability because the definiteness of the quadratic form means that one has stability. I wonder whether you can’t make it clearer or can’t come nearer following the general line of ideas [that] Gibbs has given in his Equilibrium of Heterogeneous Substances, equation 133.”

The very impressive mention of "equation 133", from Gibbs' subsection "Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations", is the following:

 U - TS + PV - M_1 m_1 - M_2 m_2 ... - M_n m_n \,

Wilson, in other words, is suggested that Samuelson use the Gibbs fundamental equation to formulate a theory of economic stability. Samuelson, however, not being intellectually capable (i.e. he lacked education in pure physics, chemistry, and engineering, of the Gibbs mentality) of doing what Wilson suggested; accordingly, nine years later, in 1947, Samuelson, taking Wilson's advice, in part, as best he could, used some of this logic, via "mathematical isomorphisms", as Samuelson called them, in outline (e.g. Le Chatelier's principle), to pen his magnum opus Foundations of Economic Analysis, which invariably put economics into a new form of a more rigorous, semi-physical science conceptualized or analogized, mathematics-based science; in short, he used Gibbs' models of minima and maxima, not thermodynamically, but generally, and argued that such could be found amid various economic variables.

References
1. 1. (a) Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances" (§: "Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations", pgs. 156-62; eq. 133, pg. 157), Transactions of the Connecticut Academy, III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.
(b) Gibbs, Willard. (1906). The Scientific Papers of J. Willard Gibbs, Volume One (§: "Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations", pgs. 156-62; eq. 133, pg. 157(§: "Internal Stability of Homogeneous Fluids as indicated by Fundamental Equations", pgs. 100-05; eq. 133, pg. 100). Longmans, Green, and Co.
2. (a) Wilson, Edwin. (1938). “Letter to Paul Samuelson”, Dec 30.
(b) Weintraub, E. Roy. (1991). Stabilizing Dynamics: Constructing Economic Knowledge (pdf) (equation 133, pgs. 61-62). Cambridge University Press.
3. Smith, Jason. (2015). “Economic Potentials: How to Define an Economy” (Ѻ), Information Transfer Economics, BlogSpot, Apr 25.

Further reading

● Anon. (2015). “More on Samuelson and the Damage he has Done to Economic Theory” (Ѻ), LawofMarkets.com, Sep 12

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