• John Baez, Brandon Coya and Franciscus Rebro, Props in network theory.

Engineers have been using diagrammatic methods for circuits since time immemorial. And in the 1940s, Olson explained how to apply circuit diagrams to networks of mechanical, hydraulic, thermodynamic and chemical components:

• Harry F. Olson, *Dynamical Analogies*, Van Nostrand, New York, 1943.

By 1961, Paynter had made the analogies between these various systems mathematically precise using ‘bond graphs’:

• Henry M. Paynter, *Analysis and Design of Engineering Systems*, MIT Press, Cambridge, Massachusetts, 1961.

Here he shows a picture of a hydroelectric power plant, and the bond graph that abstractly describes it:

By 1963, Forrester was using circuit diagrams in economics:

• Jay Wright Forrester, *Industrial Dynamics*, MIT Press, Cambridge, Massachusetts, 1961.

In 1984, Odum published a beautiful and influential book on their use in biology and ecology:

• Howard T. Odum, *Ecological and General Systems: An Introduction to Systems Ecology*, Wiley, New York, 1984.

We can use props to study circuit diagrams of all these kinds! The underlying mathematics is similar in each case, so we focus on just one example: electrical circuits. For other examples, take a look at this:

• John Baez, Network theory (part 29), *Azimuth*, 23 April 2013.

http://link.springer.com/article/10.1007/BF02476608

He seems to take the view that this is always possible to construct an analogy (at least the analogy between any physical system with any other subsystem), not unique and not “special” in the sense that there is no universal implication about nature in our analogies :(

]]>Thanks for these links! This really seems like the stuff Gerard Westendorp likes. Check out his webpage on Electric circuit diagram equivalents of fields.

]]>Thanks—that looks really interesting!

]]>Thank you, for the answer. I was looking for further particulars on this topic and found this link:

• Peter Hezemans and Leo van Geffen, Analogy theory for a systems approach to physical and technical systems.

]]>These two additional papers by Kron seem to be interesting too:

• Gabriel Kron, Equivalent circuits to represent the electromagnetic field equations, *Phys. Rev.* **64** (1943), 126–128.

• G. K. Carter and Gabriel Kron, A.C. network analyzer study of the Schrödinger equation, *Phys. Rev.* **67** (1945), 44–49.

I did not know that paper/work by Kron? Does someone have access to it? I can not get it in my current position…

]]>Thanks for the excellent article! Have you seen the work of Gabriel Kron who applied similar reasoning to modeling Schrödinger’s Equation using circuits?

• Gabriel Kron, Electric circuit models of the Schrödinger equation, *Phys. Rev.* **67** (1943), 39–43.

Or did you already consider these things?

It may be the missing link for your theory, if it is not flawed somewhere.

Please have a look and tell me if you can make something out of this.

Slides from a talk:

Here is a longer paper elaborating on some of his basic ideas.

PROCESS, SYSTEM, CAUSALITY, AND QUANTUM MECHANICS

A Psychoanalysis of Animal Faith

Tom Etter

ABSTRACT

I shall argue in this paper that a central piece of modern physics does not really belong to physics at all but to elementary probability theory. Given a joint probability distribution D on a set of random variables containing x and y, define a link between x and y to be the condition x=y on D. Define the state S of a link x=y as the joint probability distribution matrix on x and y without the link. The two core laws of quantum mechanics are the Born probability rule, and the unitary dynamical law whose best known form is the Schrödinger’s equation. Von Neumann formulated these two laws in the language of Hilbert space as prob(P) = trace(PS) and S’T = TS respectively, where P is a projection, S and S’ are density matrices, and T is a unitary transformation. We’ll see that if we regard link states as density matrices, the algebraic forms of these two core laws occur as completely general theorems about links. When we extend probability theory by allowing cases to count negatively, we find that the Hilbert space framework of quantum mechanics proper emerges from the assumption that all S’s are symmetrical in rows and columns. On the other hand, Markovian systems emerge when we assume that one of every linked variable pair has a uniform probability distribution. By representing quantum and Markovian structure in this way, we see clearly both how they differ, and also how they can coexist in natural harmony with each other, as they must in quantum measurement, which we’ll examine in some detail. Looking beyond quantum mechanics, we see how both structures have their special places in a much larger continuum of formal systems that we have yet to look for in nature.

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All the best,

Marcus