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On the Emergent "Quantum" Theory in Complex Adaptive Systems

Tristan Hubsch, Djordje Minic, Konstantin Nikolic, Sinisa Pajevic

TL;DR

The paper develops an emergent mock quantum framework for complex adaptive systems by recasting classical dynamics in a wavefunction-like form and showing how environmental coupling can cancel a state-dependent quantum potential $V_Q$, yielding a system-specific Schrödinger equation with a non-universal mock Planck constant $\hbar_{\text{mock}}$. It applies the construction to the Lotka–Volterra model, deriving a Hamiltonian $H = a(e^Q - Q) + d(e^P - P)$, a harmonic approximation with discrete energy levels $E_n$, and explicit forms for the mock quantum potential and vacuum structures, while also incorporating stochastic dynamics through a state-dependent mock quantum statistical field theory. The work links the emergence of mock quantum dynamics to ergodic non-equilibrium statistics and the principle of maximal variety, and it develops a hydrodynamic (Madelung) formulation that suggests universal signatures across quantum-like and classical transitions, including turbulence-inspired regimes. The results point to potential experimental tests in fluid analogs and synthetic biological systems, where environment-driven cancellation and state-dependent dynamics could realize stable, adaptable quantum-like behavior in macroscopic systems.

Abstract

We explore the concept of emergent quantum-like theory in complex adaptive systems, and examine in particular the concrete example of such an emergent (or "mock") quantum theory in the Lotka-Volterra system. In general, we investigate the possibility of implementing the mathematical formalism of quantum mechanics on classical systems, and what would be the conditions for using such an approach. We start from a standard description of a classical system via Hamilton-Jacobi (HJ) equation and reduce it to an effective Schrödinger-type equation, with a (mock) Planck constant $\mockbar$, which is system-dependent. The condition for this is that the so-called quantum potential VQ, which is state-dependent, is cancelled out by some additional term in the HJ equation. We consider this additional term to provide for the coupling of the classical system under consideration to the "environment." We assume that a classical system could cancel out the VQ term (at least approximately) by fine tuning to the environment. This might provide a mechanism for establishing a stable, stationary states in (complex) adaptive systems, such as biological systems. In this context we emphasize the state dependent nature of the mock quantum dynamics and we also introduce the new concept of the mock quantum, state dependent, statistical field theory. We also discuss some universal features of the quantum-to-classical as well as the mock-quantum-to-classical transition found in the turbulent phase of the hydrodynamic formulation of our proposal. In this way we reframe the concept of decoherence into the concept of "quantum turbulence," i.e. that the transition between quantum and classical could be defined in analogy to the transition from laminar to turbulent flow in hydrodynamics.

On the Emergent "Quantum" Theory in Complex Adaptive Systems

TL;DR

The paper develops an emergent mock quantum framework for complex adaptive systems by recasting classical dynamics in a wavefunction-like form and showing how environmental coupling can cancel a state-dependent quantum potential , yielding a system-specific Schrödinger equation with a non-universal mock Planck constant . It applies the construction to the Lotka–Volterra model, deriving a Hamiltonian , a harmonic approximation with discrete energy levels , and explicit forms for the mock quantum potential and vacuum structures, while also incorporating stochastic dynamics through a state-dependent mock quantum statistical field theory. The work links the emergence of mock quantum dynamics to ergodic non-equilibrium statistics and the principle of maximal variety, and it develops a hydrodynamic (Madelung) formulation that suggests universal signatures across quantum-like and classical transitions, including turbulence-inspired regimes. The results point to potential experimental tests in fluid analogs and synthetic biological systems, where environment-driven cancellation and state-dependent dynamics could realize stable, adaptable quantum-like behavior in macroscopic systems.

Abstract

We explore the concept of emergent quantum-like theory in complex adaptive systems, and examine in particular the concrete example of such an emergent (or "mock") quantum theory in the Lotka-Volterra system. In general, we investigate the possibility of implementing the mathematical formalism of quantum mechanics on classical systems, and what would be the conditions for using such an approach. We start from a standard description of a classical system via Hamilton-Jacobi (HJ) equation and reduce it to an effective Schrödinger-type equation, with a (mock) Planck constant , which is system-dependent. The condition for this is that the so-called quantum potential VQ, which is state-dependent, is cancelled out by some additional term in the HJ equation. We consider this additional term to provide for the coupling of the classical system under consideration to the "environment." We assume that a classical system could cancel out the VQ term (at least approximately) by fine tuning to the environment. This might provide a mechanism for establishing a stable, stationary states in (complex) adaptive systems, such as biological systems. In this context we emphasize the state dependent nature of the mock quantum dynamics and we also introduce the new concept of the mock quantum, state dependent, statistical field theory. We also discuss some universal features of the quantum-to-classical as well as the mock-quantum-to-classical transition found in the turbulent phase of the hydrodynamic formulation of our proposal. In this way we reframe the concept of decoherence into the concept of "quantum turbulence," i.e. that the transition between quantum and classical could be defined in analogy to the transition from laminar to turbulent flow in hydrodynamics.
Paper Structure (15 sections, 104 equations, 3 figures)

This paper contains 15 sections, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Emergent Quantum, or "Mock Quantum" Theory
  3. Quantum, Mock Quantum and a Comparison with de-Broglie-Bohm
  4. Emergence of mock quantum description in classical systems
  5. The Lotka-Volterra system
  6. The Mock Quantum Potential and the Lotka-Volterra System
  7. Quadratic approximation
  8. Vacuum of the full MQ LV theory
  9. Stochastic Dynamics and Mock Quantum Statistical Field Theory
  10. Why $V_Q$?: complexity, variety and mock quantum theory
  11. Emergent Hydrodynamics: the (Mock) Quantum-Classical Transition
  12. Conclusion
  13. Hydrodynamic quantum-like behavior
  14. Approaching Mock Quantum Theory
  15. Quantum hydrodynamics in a nutshell

Figures (3)

  • Figure 1: The "quantum potential," $V_Q(\psi_n)$ as defined in \ref{['VQ']}, plotted for a few lowest-energy states
  • Figure 2: The "quantum potential," $V_Q(\Psi)$ as defined in \ref{['VQ']}, plotted for a few linear combinations of lowest-energy states, illustrated here for $\Psi=\psi_0{+}\psi_3$, $\Psi=\psi_0{+}\psi_3{+}\psi_4$ and $\Psi=\psi_0{+}\psi_3{+}\psi_4{+}\psi_7$, in terms of harmonic eigenstates, $\psi_n$.
  • Figure 3: Quantum-like behavior of a millimeter size droplet. (A) Creation of a 'Walker': (1st, left panel) Bath with silicon oil is subject to vertical oscillations, and a drop falls on the surface. (2nd, central panel) A drop has bounced of the fluid surface and created a transient wave. (3rd, right panel) Walker: the drop continues bouncing but also moves horizontally because of the wave, which is a combination of the Faraday wave (due to bath vibrations) and a transient wave created by the impact of the droplet. Experiments are performed in near critical regime for Faraday waves when the surface is flat in absence of the drop. (B) Double-slit experiment: (left panel) A walker is moving in the direction of a double slit. (Right panel) The histogram of the deflection angle ($\alpha$) of a walker, which is consistent with the double-slit QM experiment --- reconstructed from fluidpilot. A wavelike histogram is emerging, with three peaks (red lines), approximately in agreement with a double-slit interference pattern for a monochromatic wave of the wavelength $\lambda_F$ of the Faraday wave.