Emergy Synthesis 9, Proceedings of the 9th Biennial Emergy Conference (2017)

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Emergy and Econophysics

Dennis Glenn Collins

ABSTRACT This talk discusses the relationship between Emergy theory and recent attempts to bring physics theory

into economics and social science, as for example the 2006 book Econophysics and Sociophysics:

Trends and Perspectives by B.K.and A. Chakrabarti and A. Chatterjee (Eds.).

INTRODUCTION

This paper follows up on three previous directions of work by the author: 1) Toward a

Mathematical Origin of Species (2012) and From Energy to Entropy to Emergy (2003), 2) Emergy-

Symplexity (2014) and “Tropical” Generalized Gravity (2015), 3) Reliscience (with David Scienceman)

(2012). Although many chapters of the above-mentioned econophysics book are of interest, the two

chapters of most relevance for this paper are those by Juergen Mimkes (Chapter 1 and Chapter 10).

ROLE OF THERMODYNAMICS IN EMERGY

As presented in 2003 and again 2014, the mass quantity Mi in the Sakur-Tetrode entropy formula

bears a close resemblance to emergy, since it takes much energy to increase atomic mass from, say

Carbon 6 to Silicon 14. Looking at the Sakur-Tetrode formula

S(T,P, Ni) = Σ Ni Log[(2* Pi * Mi)^(1/2)*(e T)^(3/2)/(yi*P)]

where the summation Σ is over gas components i, Ni is the number of moles of component with atomic

weight Mi and yi is the mole fraction yi =Ni/ΣNi, it is apparent that the entropy is an increasing

function of Mi.

This fact gives a way of looking at maximum emergy. Suppose there is a competition between two

gases at the same temperature and pressure, but with gas 1 having value M1=1 and gas 2 having M2=10

and suppose (contrary to fact) the gas molecules are free to choose which Mi value they have.

The internal energy U for each molecule will be the same according to the formula U= (½)(Σ Ni)T.

Supposing the entropy tends to a maximum for a constant internal energy, what will happen. The problem

of maximizing entropy over the two terms can be easily solved with the FindMaximum command of

Mathematica, with the result (if correct) at T=10 and P=5, m1=1, m2 =10 that gas 1 has mole fraction

y1=.24 and gas 2 has mole fraction .76. In short, the entropy is maximized if most of the molecules

select the M2=10 option.

Program 1. Here y2=1-y1= 1-.24=.76.

T=10.0;P=5;m1=1;m2=10;

x=FindMaximum[y1*Log[(2*Pi*m1)^.5*(Exp[1]*T)^1.5/(y1*P)]+(1-

y1)*Log[(2*Pi*m2)^.5*(Exp[1]*T)^1.5/(P*(1-y1))],{y1,.7}]

Out {5.689,{y1->.240253}}

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Thinking in terms of evolution, this means that entropy reaches a higher maximum if the molecules

go for higher emergy (or mass in this case) 10 =M2 > M1= 1, whenever and however hard it may be to

do this option.

Thus there is a way to bring emergy into thermodynamics in terms of M2 having a higher ordinality

than M1.

It seems the above argument and even math can be repeated over manifold cases (for example

some immigrants risking everything to get to the higher emergy values of Western society), leading to a

maximum emergy principle while retaining maximum entropy.

More extensive programs can do calculations over a range of values and solve for emergy m2 in

terms of entropy S for fixed mole fractions, a question raised by Prof. Tennenbaum. Please see Tables 2

and 3.

Program 2 for Table 2

T=10.0;P=5;m1=1;

U= Table[{k,FindMaximum[y1*Log[(2*Pi*m1)^.5*(Exp[1]*T)^1.5/(y1*P)]+(1-

y1)*Log[(2*Pi*m2)^.5*(Exp[1]*T)^1.5/(P*(1-y1))],{y1,.1}]},{k,1,20}];MatrixForm[U]

Table 2. Entropy S and mole fraction y1 as function of m2 from 1 to 20 with m1 = 1. The value at

m2=10 returns the above y1=.24. Here y2=1-y1. As m2 increases, the mole fraction y2 increases to 1-

.18=.82.

emergy m2 entropy S mole fraction y1

1 4.95 .500

2 5.14 .414

3 5.26 .366

4 5.36 .333

5 5.43 .309

6 5.50 .289

7 5.55 .274

8 5.60 .261

9 5.64 .250

10 5.68 .240

11 5.72 .231

12 5.75 .224

13 5.79 .217

14 5.81 .210

15 5.84 .205

16 5.87 .200

17 5.89 .195

18 5.92 .1907

19 5.94 .186

20 5.96 .182

Entropy S emergy m2 with y1= .40 and y2=.60 held constant.

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Table 3. Emergy m2 as function of Entropy S with m1=1 and mole fraction y1=.40 and y2=.60 held

constant. Observe emergy m2 increases by a factor of almost 1000 as entropy increases by 2 units from

5.0 to 7.0.

5.1 1.72

5.2 2.46

5.3 3.36

5.4 4.68

5.5 6.54

5.6 9.13

5.7 12.74

5.8 17.79

5.9 24.82

6.0 34.65

6.1 48.36

6.2 67.49

6.3 94.19

6.4 131.46

6.5 183.46

6.6 256.04

6.7 357.34

6.8 498.71

6.9 696.01

7.0 971.363

Program 3. for Table 3.

T=10.0;P=5;m1=1;

V= Table[{5+.1*n, NSolve[5+.1*n=.40*Log[(2*Pi*m1)^.5*(Exp[1]*T)^1.5/(.40*P)]+(1-

.40)*Log[(2*Pi*k)^.5*(Exp[1]*T)^1.5/(P*(1-.40))],k]},{n,1,20}];MatrixForm[V]

How might the above arguments enter the formulas of thermodynamics—Well, entropy can be

made a function of Mi, which is more or less an extensive property, and also transformity Tr, which is

the corresponding intensive property.

Then S = S(T,P, Ni, Mi) and there is a standard method of Legendre transforms to deal with the

new variables Mi.

In fact originally internal energy U is a function of the extensive variables entropy S, volume V

and moles Ni and the variables are gradually changed to intensive variables by Legendre transforms:

Helmholtz A(T,V,Ni) = U – TS gets rid of S for T,

Gibbs G(T,P,Ni) = A+PV = U-TS+ PV gets rid of V for P.

Here, writing U= U(S,V,Ni, Mi) as above, one could go to say

L(T,P,Ni,Tri) = G -ΣTri*Mi= A+PV-ΣTri*Mi=U-TS+PV-ΣTri*Mi to deal with emergy in terms of

transformity.

The reader may observe that the extensive Ni's can also be replaced by the corresponding intensive

variables (Battino and Wood, p.255), called chemical potentials μi, writing something like

W(T,P,μi,Tri) = L+Σμi*Ni

= G – ΣTri*Mi+Σμi*Ni

= A+PV– ΣTri*Mi+Σμi*Ni

= U -TS+PV– Σtri*Mi+Σμi*Ni.

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Supposedly only intensive quantities remain at the last equation, where all the second quantities in

each term are solved for in terms of intensive quantities.

Another interesting question is if the Giannantoni factorization of transformity into dissipative

transformity and generative transformity could be applied to other intensive variables. For example

could temperature be factored into chaotic-causing temperature (like breaking up ice) and symmetry-

causing temperature (like smoothly-flowing water) could pressure be factored into crushing pressure

(like a car getting crushed) and symmetry-causing pressure (like uniformity inside a neutron star).

It seems necessary to study many more examples.

ADDENDUM 1

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ADDENDUM 2

The Peace Movement Paradox

Presented at SIDIM 2017 (Seminario Interuniversitario de Investigación en Ciencias Matemáticas),

UPR-Mayaguez, Ponce, Puerto Rico

ABSTRACT This talk follows up a presentation “A conjectured hypercube invariant in generalized gravity” Oct. 8,

2016 at the Indiana MAA (Math Assoc. of America) Section meeting at Purdue University, West

Lafayette, IN, which derived a gravitational potential for how maximum entropy solutions can fluctuate

around an equilibrium with points being successively pulled closer together (peace) or drifting apart

(war). Peace movements put more energy into the potential well that determines oscillations or pulsing

around maximum entropy (rminimum Fisher information) point(s) at the bottom of the well. This added

energy means greater excursions in the direction of more unity (peace) but ALSO (with pulsing) greater

excursions away from unity toward greater conflict. As examples the Stockholm Olympics of 1912 were

followed in 1914 by World War I, the Berlin Olympics of 1936 were followed by World War II in 1939

and the Sochi Olympics of 2014 were/are followed by the Crimea and Syrian conflicts. At present there

is a similar danger of the Rio de Janeiro Olympics and Columbian peace deals being followed by

increased conflict. R.J. Rummel (The Conflict Helix) tried to deal with this situation in terms of war and

peace being equilibrium points and catastrophe theory explaining transitions. Since the harmonic

oscillator spends most time (probability) at the turning points, there is some reason for this point of view;

however the theory has not produced a convincing helix whereas the authors theory does, including the

so-called “shearing” of the helix. Mathematical models involve adding so-called “damping” or “ramping

up” terms. There is also the increased threat of nuclear war with the 1962 Cuban missile crisis following

the 1960 Rome Olympics and the Pope’s “Pacem in Terris” communication 1963 trying to counter.

INTRODUCTION

“How could you say that a peace movement could cause war”--said my friend Clif Well, I might

answer, “Do you believe in Newton’s third law of motion”--For every action there is an equal and

opposite reaction. . Could there be an opposite reaction to a peace conference or Olympic competition.

There’s also what I call the “butterfly collection effect”--To improve understanding of biology, when I

was in high school, every student had to find a butterfly/insect collection, with the best grades going to

the students who could capture the most exotic species. There are a lot of high school students. Could

this homework contribute to the practical non-existence of any butterflies anymore where I lived. Do

they still require it. How else can you learn about butterflies. Of course the “butterfly collection effect”

is different from the “butterfly effect” of chaos theory.

This presentation attempts to follow up on four of the Auhor’s talks on generalized gravity:

1. Jan. 8, 2016, Emergy and Econophysics at 9th Biennial Research Conference, Univ. of

Florida, Gainesville, FL

2. March 29, 2016, “Tropical” Generalized Gravity UPR-Mayaguez Math Sciences and

Sigma Xi Colloquium, U. ofPuerto Rico, Mayaguez

3. Aug. 4, 2016, Toward generalized gravity, MAA (Math Assoc. of America) Mathfest,

The Ohio State. Univ., Columbus, OH.

4. Oct. 8, 2016, A conjectured hypercube invariant in generalized gravity, MAA (Math

Assoc of America Indiana Section Meeting, Purdue University, West Lafayette, IN.

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One topic of these talks is the optimal value to which a given set of points in a given dimension n

will be pulled together under a SCALE CHANGE “a” to

1) minimize Fisher information: n — f(x), minimize gradients (such as heat difference)

2) maximize covered sides f(x), max protection (covered sides)

3) maximize entropy f(x), 2law of thermodynamics

4) maximize emergy f(x), evolution (protected survive more)

5) solve a Newton dynamical system based on potential f(x) move according to force

Under a scale change operation the Giannantoni emergy = constant *generative transformity Tg *

dissipative transformity Td =SYM* f(x) has the same maximum as the entropy because the symmetry

measurement SYM is invariant under scale change.”a”. Of course the above theory makes finding the

correct f(x) is very important. Based on the above work (Please see Appendix A) and the FindFit

operation of Mathematica, the approximately correct f(x) for 4 points in the form of a square (n=2) is

found to be f(x) 2.15 x”2 Exp[.5*x\1.5].

It is conjectured there is a logarithmic type transformation whereby f(x) is multiplied by constant*n

for larger dimension n for a hypercube configuration. Applying this function to determine the potential

well for Newton’s second law F= M*A leads to an oscillating solution, or a helix in phase space.

The dynamical equation is x”[t]=(4.3 *x[t] - 1.6125*x[t]^2. 5)*Exp[-0.5*x[t]^1.5] since

f[x]=(4.3 x[t] - 1.6125*x[t]^2)*Exp[-0.5*x[t]^1.5].

THE MODEL

Recall x here is a scale change, so that x small corresponds to the points getting squashed together

by gravity and x large mans the points are dispersing outward IN THE SAME PATTERN. The optimal

value around x 2 corresponds to the four tiles of the WINDOWS logo or four cows putting their backs

together for protection. The positions of the four points would be then about {(0,0),(2,0),(0,2), (2,2)}

representing a buffer zone of about one unit around each point.

In comparison to the damped harmonic oscillator equation ax”+bx’ +cx =0, the above equation

will be modified to include a damping or ramping up term based on x’[t], and which represents the peace

or war movement. Here positive b means damping, so that the helix collapses to one line at the

equilibrium value of x, somewhere around x = 2. However negative b means ramping up or (Go with

the flow.) putting energy into the system, so that the helix expands into a tornado shape. Of course a

negative b on the left side of the equation corresponds to a positive b on the right side.

MODEL RESULTS

There are several cases of interest:

DO NOTHING: The system oscillates between moderate shifts toward peace (small x) or war (large x).

Were it not for the fact that “peace” (very small x) is never attained, this result might be considered

satisfactory.

GO WITH THE FLOW: Here a term 0.2*x[t] is added inside parentheses on the right side of the

equation. This term puts energy into the system and moves it higher up the potential wall. It says if there

is a war threat x’[t] >0, support the war and if there is a peace movement x’[t] <0, support peace. Since

for the given f(x) the potential constraining walls do not keep going up but taper off, eventually there is

a tornado shape that shears off (“The shearing of the helix”) representing all-out war.

ONLY SUPPORT PEACE: Here nothing (0) is done if there is a tendency to war x’>0 but the 0.2*x[t]

term is added if there is a tendency to peace x’<0. Technically the term added is 0.2*Min[x’[t],0]. This

case is the peace movement paradox because it still puts energy into the system, and creates a tornado

shape that shears off, although not as fast (soon).

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REALITY

R J. Rummel (1981, 1.2 THE REALITY OF CONFLICT) states: “Conflict and cooperation are

then complementary phases in the progress of social life. However these processes are not cyclical. If

the conditions or environment of a social relationship remain fairly constant, then the progression of

conflict and cooperation forms a helix: conflict gets shorter and less intense, cooperation more durable

and deeper, as people learn from previous conflict, expectations and interaction. As a couple surviving

the fights of their early married years and growing old together.

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However it seems Rummel was never able to get a mathematical model of the helix as done above,

partly because he ruled out a cyclical model and after all a helix is a cyclical model. Furthermore there

is a question if there is learning by mankind, as evidenced by the following case studies. It appears trying

to get people together too much can lead to a backlash, after about two years.

CASE STUDIES

1912 Stockholm Olympics May5-July27, 1912

World War I July 28, 1914-Nov. 1918 two years and 1day

Medal count: U.S. 63, Sweden 65, Britain 41, Finland 26, France 14, Germany 25, South Africa 6

Commissioning of 5 “Kaiser” class battleships Aug. 1, Aug. 1, Oct. 15, 1912 and May 14, July 31,

1913

1936 Berlin Olympics Aug.1-Aug. 16, 1936

World War II Sept. 1, 1939—Sept. 2, 1945 three years 15 days

Medal count: Germany 89, U.S. 56,Hungary 16, Italy 22,Finlandl9, Sweden 20,Japan 18,

Netherlands 17, Great Britain 14

Munich peace agreement Sept 30, 1938 Sudetenland occupation by Hitler 2 years 1.5 months

1960 Rome Olympics Aug. 25-Sept. 11, 1960

Cuban missile crisis Oct. 16-Oct. 28, 1962, two years 35 days

Medal count USSR 103, U.S. 71, Italy 36

April 1 1, 1963 Pacem in Terris by Pope John XXIII, dies June 3, 1963

Partial Test Ban Treaty, signed Aug.5, 1963 in effect Oct. 10, 1963

1964 Tokyo Olympics Oct. 10-Oct. 24, 1964

Viet Nam war, Gulf of Tonkin Aug. 2, 1964 1975 before Olympics

Medal count U.S. 36, Soviet Union 30, Japan 16, United Germany 10

U.S. troops in Viet Nam 1964 23,300; 1965 184,300; 1966 385,300; 1967 485,600

1999 Robert H. Schuller meets Grand Mufti Kuftaro of Syria in Damascus Dec. 16, 1999

2001 Sept. 11 (9-11) Bin Laden attacks on U.S. two years minus 3 months

Schuller, p. 502 “As I write these words (question p.5 1 1 March 2001) our Hour of Power is

being broadcast every week from the Crystal Cathedral to a new audience—Indonesia, the

world’s largest Muslim nation.”

U.S. Operation Enduring Freedom Oct. 7, 2001

2014 Sochi Olympics Feb.7-Feb. 23, 2014

Russia annexes Crimea March 1 8, 2014 23 days

Syria war two years

Medal count: Russia 33, Norway 26, Canada 25, U.S. 28, Netherlands 24, Germany 19

2016 Rio de Janeiro Olympics Aug. 5-Aug. 21, 2016

Medal count U.S. 121, Great Britain 67, China 70, Russia55, Germany 42

2020 Tokyo Olympics

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DISCUSSION

The transition from maximum entropy to dynamical system is, and is likely to remain, murky. The

“fluctuation theorem” (Dewar 2003) discusses motions away from maximum entropy. (Liyun Zhao et

al. 2016) studies how oscillators may self-organize into the same frequency. Caticha (2007) does an

“entropic gravity” derivation of Newtonian dynamics, although experimental results have cast doubts.

It seems quantum effects may be involve Wilczek (2012) talks about time crystals. The Author’s book

(2011) and BLOG The Quantum History Institute discuss some theory and cases.

CONCLUSION

There is still a gap between the Author’s above cyclic model and the Author’s 2011 book Conflict

in History, which does study the two-year delay. However it is hoped the above work moves things

forward.

ADDENDUM

Peace movement paradox. CASE ADDED 6-25-2017

1858 Aug. 16, 1st communication on Atlantic telegraph (1st cable failed Sept. 1858)

1860 December, South Carolina secedes from Union, 2 years four months

1861 April 12, Ft. Sumter attacked, U.S. Civil War starts.

The book Lightning Man: The Accursed Life of Samuel F.B. Morse by Kenneth Silverman (Alfred Knopf

2003) describes Samuel Morse, inventor of the telegraph, believing something like the Author’s above

theory.

p. 329 “The prospect of instant communication between continents stirred utopian fantasies of universal

brotherhood . . . Morse shared this ideology of redemption through communication and predicted an end

to war ‘in a not distant future’.”

The word “accursed” in the book’s title refers to reverses, such as the Civil War, following success and

honors, and calls to lock up Morse in New York as favoring slavery although a strong supporter of the

Union. p.393 “Brooding on the matter, he speculated that it might be the moral equivalent of the physical

law of equilibrium. Take for instance the antagonism between the opposite poles of a magnet: ‘if the

positive be strengthened, the negative is also in an equal degree strengthened and vica [sic] versa.’ . . .

p.384 Morse reasoned that such physical and moral patterns could hardly be accidental. Events that

seemed cursed, ultimately showed the workings of Providence.”

On October 24, 1861, the transcontinental telegraph was completed from San Francisco to

Newfoundland and Washington D.C. And after the end of the Civil War, in 1866, a second transatlantic

cable opened.

(Note, the above report does not reflect any support of slavery on the part of the author.)

ACKNOWLEDGMENTS

The Author thanks the SD1M Committee and Prof. Pablo Negron for hopefully allowing this talk,

previous communications with David Scienceman and Glenn Collins, and assistance from Dr. Sharlynn

Sweeney.

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REFERENCES

2007 Caticha, Ariel and Cafaro, Carlo. From information geometry to Newtonian dynamics. ArXiv:

0710.1071 vi Oct. 2007.

1972 Cousins, Norman. Book: The Improbable Triumvirate: John F. Kennedy, Pope John, Nikita

Khmschev, W.W. Norton.

2011 Collins, Dennis G. book: Conflict in History, Measuring Symmetry, Thermodynamic Modeling

and Other Work. Author House, Bloomington, IN.

2016 Collins, Dennis G. presentation: A conjectured hypercube invariant in generalized gravity. MAA

(Math Association of America) Indiana Section Meeting, Purdue University, West Lafayette, IN,

Oct. 8, 2016.

2003 Dewar, Robert. Information theory explanation of the fluctuation theorem, maximum entropy

production, and self-organized criticality in non-equilibrium stationary states. Journal of Physics

A, iopscience.iop.org.

2017 Olympics articles on Internet.

1981 Rummel, RI. Understanding Conflict and War, Vol.5 The Just Peace. Hawaii.edu/power

kills/NOTE 14.HTM from Internet 3-2-2017.

1991 Rummel, RI. Book: The Conflict Helix. Transaction Publishers, New Brunswick and London

(original 1984).

2001 Schuller, Robert H. book: My Journey, Harper San Francisco.

2012 Wilczek, Frank. Quantum Time Crystals, Physical Review Letters 109, 160401 (Oct. 19, 2012).

2016 Liyun Zhao, Jun Liu, Lan Xiang, Jin Zhou. Group Synchronization of Diffusely Coupled Harmonic

Oscillators, Kybernetica 52 No.4, pp.629-647.

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APPENDIX

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REFERENCES

Blower, David J. 2015.. Information Processing Vol. 2 The Maximum Entropy Principle, Third

Millennium Inferencing, Pensacola, FL pp.377-378.

Collins, Dennis G. 2015. Emergy/Symplexity Chapter 3 in Emergy Synthesis 8, Center for

Environmental Policy, Univ. of Florida, Gainesville, FL, pp.13-28.

Collins, Dennis G. 2011. Architecture Case Study in Transformity Factorization in Conflict in History,

Measuring Symmetry, Thermodynamic Modeling, and other Work, Author House, Bloomington, IN,

pp.28-33.

Williams, Pharos E. 2010. Physics Against the Odds phi = (k/r) Exp[-lambda subN/r], Williams

Research, Lexington, KY.