Quantum Primordial Soup

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WHAT IS IT?

This is a model of quantum dissipative structures' emergence in a quantum chaotic medium. The model is part of the quantum complex systems science and risk mathematics research project, under development by the author at ISCSP - Technical University of Lisbon.

The model combines quantum statistical mechanics and chaos theory, and it leads to the emergence of a fluctuating complex millieu. After a few steps of the simulation, there emerge organized structures akin to unicellular organisms that have a pulsating body (pulsating either in dark grey and light grey RGB scheme (see the description below for the details), a colored nucleous, and a border (usually pulsating inversely to the body in the RGB scheme), these structures survive for a while feeding upon the environment and assimilating other organisms whenever their borders touch.

Complex quantum structures' individuation takes place in this model with a struggle for growth territory, for survival against dissolution and against assimilation from other structures.

This life-like quantum game emergence is not preprogrammed in the model, we did not pre-program the large cellular structures, they just happen to emerge.

The model has been used both for research in risk science as well as an example used in the author's MsC Classes on Decision Making Techniques, for a basic understanding of the problem of risk in complex systems science.

The model implements an example of a computational structure resulting from the article:

- Gon_ves (2012) "Quantum Chaos and Quantum Computing Structures" arXiv:1208.2610v1 [nlin.CD], http://arxiv.org/abs/1208.2610

Each patch contains a three-entry quantum register machine, each entry is assigned a color, and each entry is in a coherent state. The quantum computation implements a local unitary transition at each patch from coherent state to coherent state (by the general scheme explained in arXiv:1208.2610v1 [nlin.CD])) according to a nonlinear map rule.

The coherent state sequence is such that each coherent state leads to an occupation number quantum average that coincides with a definite energy eigenstate (even though the quantum state is NOT in a definite energy state, but in a superposition of energy eigenstates as per the coherent state notion).

Each patch is colored in accordance with an RGB scheme represenation of its mean quantum energy.

THE MODEL (FORMAL EXPLANATION):

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Each patch contains a bosonic quantum register machine, with three entries corresponding to three colors Red, Green and Blue, in a coherent state for each color:

| |psi(j,t)> = |CRed(j,t),CGreen(j,t),CBlue(j,t)>

At each round, a local quantum logical gate of the product coherent state category Coh^3 (see the article for details) is implemented such that a coherent state transition takes place:

| psi(j,t-1)> -> |psi(j,t)>

The relation between the coherent states is such that:

| |CRed(j,t),CGreen(j,t),CBlue(j,t)> = |F(CRed(j,t-1)),F(CGreen(j,t-1)), F(CBlue(j,t-1))>

Where F is a complex-valued nonlinear map that transforms the phase of each coherent state (in the polar coordinate representation of a complex number) as follows:

| PhiRed(j,t) = (1 - epsilon - delta) * Mred(j,t-1) + epsilon * mean-field(PhiRed) + mean-field(h)

| PhiGreen(j,t) = (1 - epsilon - delta) * Mred(j,t-1) + epsilon * mean-field(PhiRed) + mean-field(h)

| PhiBlue(j,t) = (1 - epsilon - delta) * Mred(j,t-1) + epsilon * mean-field(PhiRed) + mean-field(h)

With the updates:

| Mred(j,t) = b * PhiRed(j,t-1) * (1 - PhiRed(j,t-1))

| Mgreen(j,t) = b * PhiGreen(j,t-1) * (1 - PhiGreen(j,t-1))

| Mblue(j,t) = b * PhiBlue(j,t-1) * (1 - PhiBlue(j,t-1))

| h(j,t) = [Mred(j,t) + Mgreen(j,t) + Mblue(j,t)] / 3

and where the mean-field(PhiColor) is the mean of the coherent state phases of the patches in radius r of the j'th patch for the corresponding "Color", the same holding for mean-field(h).

In turn, the amplitude of each coherent state for a patch is transformed by the complex-valued nonlinear map as:

| AmplitudeRed(j,t) = sqrt{int(PhiRed(j,t) * 255)}

| AmplitudeGreen(j,t) = sqrt{int(PhiGreen(j,t) * 255)}

| AmplitudeBlue(j,t) = sqrt{int(PhiBlue(j,t) * 255)}

Thus, the expected energy for each register at each patch is given by:

| = (int(PhiRed(j,t) * 255) + 1 / 2) * h-bar * omega

| = (int(PhiGreen(j,t) * 255) + 1 / 2) * h-bar * omega

| = (int(PhiBlue(j,t) * 255) + 1 / 2) * h-bar * omega

Each patch is colored in RGB scheme by the occupation numbers, therefore each patch Red, Green Blue color in the model is in proportion to the combination of the expected energy eigenstates for each quantum register.

HOW TO USE IT

The user can control the logistic map's parameters, as well as the couplings (epsilon and delta) and the radius r for the local mean field coupling.

The slider no-plot-transient allows the user to control a the moment upon which the mean color value for the entire system of patches is plotted.

THINGS TO NOTICE

One should address the model in terms of the spatial and temporal patterns that emerge, the different behaviors of the model for different couplings, try to look and identify emerging individuated structures and look at how they interact with the environment and with each other.

You can also watch the video below which shows several examples of relevant points to notice in the model.

http://www.youtube.com/watch?v=zc61oy5R9hQ

CREDITS AND REFERENCES

- Gon_ves, (2012) "Quantum Chaos and Quantum Computing Structures" arXiv:1208.2610v1 [nlin.CD], http://arxiv.org/abs/1208.2610

- Gon_ves, (2012) "Risk Governance - A Framework for Risk Science-Based Decision Support Systems" http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2085482

Comments and Questions

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Click to Run Model

patches-own 
[
 PhiRed_t-1 PhiRed_t
 PhiGreen_t-1 PhiGreen_t
 PhiBlue_t-1 PhiBlue_t
 M0 M1 M2 h
 c2-red ; |c-red|^2 corresponds to the expected number of red quanta in the register
 c2-green ; |c-green|^2 corresponds to the expected number of green quanta in the register
 c2-blue ; |c-blue|^2 corresponds to the expected number of blue quanta in the register
 indicator
]

globals [ m-red m-green m-blue ]

to setup
  ca
  ask patches 
  [
    set PhiRed_t random-float 1.000
    set PhiGreen_t random-float 1.000
    set PhiBlue_t random-float 1.000
  ]
  ask one-of patches [ set indicator 1 ]
end 

to go
  no-display
  tick
  ask patches [ previous-state-phase ]
  ask patches [ n-map ]
  ask patches [ update-quantum-phases ]
  ask patches [ set-color ]
  set m-red mean[c2-red] of patches
  set m-green mean [c2-green] of patches
  set m-blue mean [c2-blue] of patches
  if ticks >= no-plot-transient [ do-plot ]
  display
end 

to previous-state-phase
  set PhiRed_t-1 PhiRed_t
  set PhiGreen_t-1 PhiGreen_t
  set PhiBlue_t-1 PhiBlue_t
end 

to n-map
  set M0 PhiRed_t-1 * b * (1 - PhiRed_t-1)
  set M1 PhiGreen_t-1 * b * (1 - PhiGreen_t-1)
  set M2 PhiBlue_t-1 * b * (1 - PhiBlue_t-1)
  set h ( M0 + M1 + M2 ) / 3
end 

to update-quantum-phases
  
  set PhiRed_t (1 - epsilon - delta) * M0 + epsilon * mean [ M0 ] of patches in-radius r + delta * mean [ h ] of patches in-radius r
  set PhiGreen_t (1 - epsilon - delta) * M1 + epsilon * mean [ M1 ] of patches in-radius r + delta * mean [ h ] of patches in-radius r
  set PhiBlue_t (1 - epsilon - delta) * M2 + epsilon * mean [ M2 ] of patches in-radius r + delta * mean [ h ] of patches in-radius r
end 

to set-color
  set c2-red int (PhiRed_t * 255)
  set c2-green int (PhiGreen_t * 255)
  set c2-blue int (PhiBlue_t * 255)
  set pcolor (list c2-red c2-green c2-blue )
end 

to do-plot
  set-current-plot "Mean Red"
  plot m-red
  set-current-plot "Mean Green"
  plot m-green
  set-current-plot "Mean Blue"
  plot m-blue
end

There is only one version of this model, created over 12 years ago by Carlos Pedro S. Gonçalves.

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