Quantum Field Theory Model of Technological and Agent Dynamics in Evolutionary Economics

Abstract 

This study proposes an evolutionary economics model based on field theory, exploring the spatiotemporal interaction dynamics between the technology knowledge field (φ) and the agent distribution (ρ). Through discretized partial differential equations and numerical simulations, the model reveals how innovation diffusion forms technology clusters and how firms respond to technological shifts through geographic migration and agglomeration. Using methods analogous to quantum field theory (QFT), we address continuous spatial and evolutionary interactions, offering a concrete dynamic framework to understand innovation geography and the knowledge economy.

 

 Introduction

In the knowledge economy, technological innovation and firm behavior exhibit strong spatial coupling. Evolutionary economics emphasizes dynamic adaptation and historical institutional choices, yet lacks a unified mathematical framework to describe how innovation diffuses and agglomerates in space. This paper proposes an economic field model inspired by quantum field theory, treating technological knowledge and firm behavior as co-evolving continuous fields, capturing their spatial dynamics and complex adaptive mechanisms.

At first glance, evolutionary economics and quantum field theory (QFT) appear to come from entirely different domains—one being a branch of economics focusing on disequilibrium, historicity, and innovation dynamics, while the other forms the cornerstone of theoretical physics, describing particle and field dynamics in spacetime. However, recent interdisciplinary research has revealed structural and conceptual parallels between the two, as outlined below:

 

I. Shared Systemic View and Nonlinear Characteristics

Aspect	Evolutionary Economics	Quantum Field Theory (QFT)
Core Traits	Disequilibrium, path dependence, mutation, innovation	Nonlinear field interactions, vacuum fluctuations, phase transitions
System View	Economy as a complex adaptive system	Universe made of quantum fields; particles as field excitations
Randomness	Market selection involves stochasticity and path dependence	Quantum fluctuations induce probabilistic measurement and superposition

Key Connection:
Innovation in evolutionary economics (e.g., technological mutation, institutional evolution) can be viewed analogously to "field excitations" or "phase transitions" in QFT, where the system shifts to new energy states or configurations.

 

II. Innovation Diffusion vs. Field Propagation

In QFT, field excitations (particles) propagate through space and interact via Feynman diagrams, generating new particles. Similarly, in evolutionary economics, new technologies or business models diffuse and trigger cascading changes (e.g., technological transmission, network effects). Innovation can thus be considered as a quantum excitation in the knowledge field, whose diffusion and interaction may be described using field-theoretic language.

 

III. Agent Interaction and Field Effects

In sociophysics, attempts have been made to relate individual behaviors to field representations. Firms or individuals can be viewed as agent-particles similar to fermions/bosons, while institutions, prices, or capital structures represent the "economic fields" influencing them. Their interactions are determined by a kind of economic interaction Lagrangian, analogous to field-theoretic Lagrangian densities.

 

IV. Applied Cases and Frontier Research (for Reference)

• Field-theoretic frameworks in evolutionary game theory: Researchers reformulate multi-agent strategy interactions into field-theory form, exploring equilibrium and phase transitions in strategy space.

• Technology life cycle and Renormalization Group (RG) analogy: The development of technology can be approached using RG methods—from micro-level innovations (small firms) to macro-level impacts (industrial transformation), mirroring effective theory transitions across energy scales in QFT.

• Innovation clusters and quantum entanglement: Some technologies (e.g., AI and semiconductors) are highly interdependent. Their co-evolution may structurally resemble quantum entanglement, exhibiting non-local collaborative properties.

 

V. Theoretical Integration Outlook (Challenges and Potential)

• Challenges: QFT is still highly mathematical, while evolutionary economics leans more on simulations and qualitative analysis—posing a methodological gap.

• Potential: By employing mathematical field theories (e.g., statistical field theory, non-equilibrium field theory), it is possible to construct quantifiable economic field models that incorporate concepts such as innovation excitation, phase transitions, and collective behavior. This could deepen our understanding of innovation-driven economic systems.

 

Prototype of a Field-Theoretic Model in Evolutionary Economics

1. Definition of Economic Field and Agent Dynamics

Consider an economic system with a technology knowledge field ϕ(x, t), representing the level of technological knowledge, innovation density, or resource allocation at time t and location x. The effective Lagrangian density is defined as:

L[ϕ] = ½ (∂ₜϕ)² - ½ (∇ϕ)² - V(ϕ)

Where:

• ∂ₜϕ: Temporal variation, representing the speed of innovation or knowledge evolution (e.g., technological evolution)

• ∇ϕ: Spatial variation, representing geographic diffusion of innovation or inter-industry transmission

• V(ϕ): Potential function, representing the innovation energy structure, influencing stability points and phase transition behaviors

 

A Prototype Field-Theoretical Model of Evolutionary Economics

1. Definition of the Economic Field and Agent Dynamics

Consider an economic system with a “technological knowledge field” ϕ(x, t), representing the level of technological knowledge, innovation density, or resource allocation at time t and location x.

Define the effective Lagrangian density as:

𝓛[ϕ] = ½(∂ₜϕ)² − ½(∇ϕ)² − V(ϕ)

Where:

• ∂ₜϕ: Temporal variation, representing the speed of knowledge change or innovation (e.g., technological evolution)

• ∇ϕ: Spatial variation, describing geographic diffusion or inter-industry transmission of innovation

• V(ϕ): Potential function, representing the energy structure of innovation and influencing stability and phase transitions

 

2. Social Interpretation of the Potential Function V(ϕ)

Adopt a symmetry-breaking potential (similar to the Higgs field):

V(ϕ) = ½·m²·ϕ² + ¼·λ·ϕ⁴

If m² < 0, the system exhibits two stable minima (a bistable state), representing “phase transitions” in technological regimes:

• When a new institution or technology arises, the economy may discontinuously shift to another stable state.

• λ controls the degree of self-reinforcement or inhibition among innovations, affecting whether diffusion is smooth or abrupt.

 

3. Introducing Innovation Sources and Agent Interactions

Introduce an agent density field ψ(x, t), representing willingness to adopt or invest in innovation:

𝓛_int = g·ϕ·ψ̄·ψ

Where:

• This interaction term expresses the influence between the knowledge field and agents:

  • Firm behavior alters the local state of ϕ (e.g., adopting technology increases local innovation density)

  • A higher concentration of the knowledge field attracts more agent participation (positive feedback)

• g is the coupling strength, akin to a social learning rate or imitation strength

 

4. Equation Derivation: Field Dynamics via the Euler–Lagrange Equation

Derive the equation of motion from the Lagrangian:

∂²ϕ/∂t² − ∇²ϕ + dV/dϕ = g·ψ̄·ψ

This is a nonlinear wave equation, where the right-hand side acts as a source term from agent activity.

Economic Interpretation:

• ϕ describes the evolution of the overall technological or institutional environment.

• The left-hand side captures natural dynamics (controlled by potential and diffusion).

• The right-hand side represents externally driven innovation pressures (e.g., entrepreneurship, imitation, policy intervention).

 

5. Renormalization and Scale-Dependent Innovation

Introduce the Renormalization Group (RG) concept:

μ·dλ/dμ = β(λ)

Where:

• μ: Represents the scale of evolution (e.g., from local industries to national institutions)

• β(λ): Describes how the interaction strength between innovations changes across scales

Social Interpretation: Local innovation policies can cascade through multiple levels, ultimately restructuring national economic institutions.

Summary Table :

QFT Element Interpretation in Evolutionary Economics Field ϕ(x, t) Technological knowledge density field, innovation field Lagrangian ℒ Economic system’s kinetic energy, innovation potential, and interactions Potential V(ϕ) Institutional stability structure, technological path dependence Interaction gϕψ̄ψ Firm agents drive innovation and feed back into institutional change Renormalization Group (RG) Integration of innovation dynamics across different scales (micro to macro)

The code (evo_econ_model.py) is an evolutionary economics model that simulates the spatio-temporal dynamics of technological innovation using a quantum field theory approach. It models the evolution of the technological knowledge field (phi) and the agent density field (rho). The code generates static plots (with four subplots) via the plot_results method and animations (with two subplots) via the create_animation method to visualize the simulation results.

Below, I provide a detailed explanation of the content of the plots and animations, covering the meaning of each subplot and animation, and how they reflect the simulated economic phenomena. I assume all prior issues (syntax errors, text overlap, colorbar issues, etc.) have been resolved in the corrected code, and the plots display correctly.

Code Background

Model Overview

• The model simulates the spatial and temporal dynamics of technological innovation based on a quantum field theory framework.
• Two main variables:
  • Technological Knowledge Field (phi): Represents the intensity of technological knowledge or innovation, varying across space and time, influenced by diffusion, potential, agent behavior, and random noise.
  • Agent Density Field (rho): Represents the distribution density of firms or economic agents, attracted to the technological field and diffusing spatially.
• Simulation parameters:
  • Spatial grid points: Nx = 100, time steps: Nt = 500.
  • Spatial step: dx = 0.5, time step: dt = 0.01.
  • Other parameters control potential (m2, lambda), interaction strength (g), and diffusion coefficients (D_phi, D_rho).

Outputs

• Static Plot: The plot_results method generates a figure with four subplots, saved as evo_econ_results.png.
• Animation: The create_animation method generates an animation with two subplots showing the dynamic evolution of phi and rho over time, displayed interactively in Jupyter or saved as evo_econ_animation.mp4.

Static Plot Content (plot_results Method)

The plot_results method generates a figure with four subplots, sized figsize=(15, 10), each showcasing a different aspect of the simulation results. Below is the detailed content and economic significance of each subplot:

1. Subplot 1: Technological Knowledge Field Heatmap

• Content:
  • Displays the spatio-temporal evolution of the technological knowledge field phi using a heatmap (imshow).
  • X-axis: Spatial position (0 to Nx*dx = 50), representing geographic or economic space.
  • Y-axis: Time (0 to Nt*dt = 5), representing simulation time progression.
  • Color: Represents phi intensity (technological knowledge level), using the viridis colormap (blue to yellow for low to high).
  • Colorbar: Labeled “Technological Knowledge Field Intensity,” showing the phi value range.
• Layout:
  • Title: “Technological Knowledge Field Evolution.”
  • X-axis label: “Spatial Position.”
  • Y-axis label: “Time.”
  • Colorbar shrunk (shrink=0.8), label fontsize=9, tick fontsize=8.
• Economic Significance:
  • Shows how technological knowledge diffuses spatially and evolves temporally.
  • High-intensity regions (yellow) indicate areas of concentrated technological innovation, such as tech hubs or clusters.
  • Temporal changes reflect innovation spreading from the initial center (init_center=0.5) outward or forming new patterns due to agent behavior and random noise.
  • For example, multiple high-intensity regions in the heatmap may indicate the formation of multiple innovation clusters.

2. Subplot 2: Agent Density Field Heatmap

• Content:
  • Displays the spatio-temporal evolution of the agent density field rho using a heatmap (imshow).
  • X-axis: Spatial position (0 to 50).
  • Y-axis: Time (0 to 5).
  • Color: Represents rho density (agent distribution), using the plasma colormap (purple to yellow for low to high).
  • Colorbar: Labeled “Agent Density,” showing the rho value range.
• Layout:
  • Title: “Agent Distribution Evolution.”
  • X-axis label: “Spatial Position.”
  • Y-axis label: “Time.”
  • Colorbar shrunk (shrink=0.8), label fontsize=9, tick fontsize=8.
• Economic Significance:
  • Shows how firms or economic agents cluster spatially and migrate over time.
  • High-density regions (yellow) indicate firm concentration, likely corresponding to high-intensity technological regions (as rho is attracted to phi, with beta=0.1).
  • Temporal changes reflect firms migrating toward regions with technological advantages, forming economic clusters.
  • Comparing with Subplot 1 reveals the interaction between technological knowledge and agent distribution, e.g., whether firms follow tech hubs.

3. Subplot 3: Initial vs. Final Field Distribution Comparison

• Content:
  • Displays the spatial distribution of phi and rho at the initial (t=0) and final (t=Nt*dt) moments using line plots (plot).
  • X-axis: Spatial position (0 to 50).
  • Y-axis: Field intensity/density (phi or rho values).
  • Curves:
    • Blue solid line: Final technological field (phi at the last timestep).
    • Red solid line: Final agent distribution (rho at the last timestep).
    • Blue dashed line: Initial technological field (phi at the initial timestep, alpha=0.5).
    • Red dashed line: Initial agent distribution (rho at the initial timestep, alpha=0.5).
  • Legend: Labels the four curves, placed outside the plot (bbox_to_anchor=(1.05, 1)).
• Layout:
  • Title: “Initial vs. Final Field Distribution Comparison.”
  • X-axis label: “Spatial Position.”
  • Y-axis label: “Field Intensity/Density.”
  • Legend fontsize=8, grid enabled (grid=True).
• Economic Significance:
  • Compares changes in technological knowledge and agent distribution between the start and end of the simulation.
  • Initial fields (dashed lines) are typically centered (init_center=0.5, Gaussian distribution), reflecting the starting point of innovation.
  • Final fields (solid lines) show the evolved results, e.g., whether innovation and firms spread out or form multiple peaks (multi-center clusters).
  • Comparing phi and rho peaks reveals spatial coupling, e.g., whether firms cluster in high-tech regions.

4. Subplot 4: Maximum Field Intensity Evolution

• Content:
  • Displays the temporal evolution of the maximum values of phi and rho using line plots (plot).
  • X-axis: Time (0 to Nt*dt = 5).
  • Y-axis: Maximum value (spatial maximum of phi or rho).
  • Curves:
    • Blue solid line: Maximum technological field value (np.max(phi_history, axis=1)).
    • Red solid line: Maximum agent density value (np.max(rho_history, axis=1)).
  • Legend: Labels the two curves.
• Layout:
  • Title: “Maximum Field Intensity Evolution.”
  • X-axis label: “Time.”
  • Y-axis label: “Maximum Value.”
  • Legend fontsize=8, grid enabled (grid=True).
• Economic Significance:
  • Shows the dynamic changes in the intensity of technological innovation and agent clustering over time.
  • The blue curve (phi maximum) indicates whether the highest level of innovation grows, decays, or stabilizes.
  • The red curve (rho maximum) shows whether firm clustering intensity follows technological changes.
  • Trends and synchronization between curves reveal the interaction between technology and firms, e.g., whether tech peaks drive firm clustering.

Overall Plot Layout

• Size and Spacing:
  • Figure size: 15x10 inches, suitable for four subplots.
  • Subplot spacing: wspace=0.3 (horizontal), hspace=0.4 (vertical) to avoid text overlap.
  • Uses plt.tight_layout() for optimized layout.
• Fonts and Labels:
  • Global fontsize=10, title fontsize=10, axis label fontsize=9, legend and colorbar tick fontsize=8.
  • Chinese labels use fonts like SimHei for correct display (no DejaVu Sans warnings).
• Saving:
  • Saved as evo_econ_results.png for easy inspection and sharing.

Animation Content (create_animation Method)

The create_animation method generates an animation showing the spatial distribution changes of phi and rho over time, with two subplots. Below is the detailed content:

1. Subplot 1: Technological Field Evolution

• Content:
  • Displays the spatial distribution of phi evolving over time using a line plot (plot).
  • X-axis: Spatial position (0 to 50).
  • Y-axis: Technological knowledge field intensity (phi values, range: np.min(phi_history) - 0.1 to np.max(phi_history) + 0.1).
  • Curve: Blue solid line (lw=2), showing phi distribution at each time step.
• Layout:
  • Title: “Technological Field Evolution.”
  • X-axis label: “Spatial Position.”
  • Y-axis label: “Technological Knowledge Field (φ).”
  • Font sizes: Title fontsize=10, axis label fontsize=9.
  • Grid enabled (grid=True).
• Economic Significance:
  • Dynamically shows how technological knowledge diffuses from an initial Gaussian peak or forms new patterns.
  • Curve movement reflects the spatial spread of innovation, e.g., from the center outward or forming multiple peaks due to random noise.

2. Subplot 2: Agent Distribution Evolution

• Content:
  • Displays the spatial distribution of rho evolving over time using a line plot (plot).
  • X-axis: Spatial position (0 to 50).
  • Y-axis: Agent density (rho values, range: np.min(rho_history) - 0.1 to np.max(rho_history) + 0.1).
  • Curve: Red solid line (lw=2), showing rho distribution at each time step.
• Layout:
  • Title: “Agent Distribution Evolution.”
  • X-axis label: “Spatial Position.”
  • Y-axis label: “Agent Density (ρ).”
  • Font sizes: Title fontsize=10, axis label fontsize=9.
  • Grid enabled (grid=True).
• Economic Significance:
  • Dynamically shows how firms or agents redistribute spatially due to attraction to the technological field.
  • Curve changes reflect firms clustering toward high-tech regions, e.g., density peaks following phi peaks.

Overall Animation

• Timestamp:
  • Displays a timestamp at the top (t = 0.0 initially), formatted as t = {i * dt * 10:.2f}, updated per frame.
  • Positioned above the figure (y=0.98) to avoid overlapping subplot titles, fontsize=10.
• Animation Parameters:
  • Frame interval: 100 ms (interval=100).
  • Total frames correspond to the time steps in the history data (saved every 10 steps).
  • Uses blit=True for optimized animation performance.
• Display and Saving:
  • In Jupyter, displayed as interactive HTML (HTML(anim.to_jshtml())).
  • In non-Jupyter environments, shown via plt.show().
  • Can be saved as evo_econ_animation.mp4 (requires ffmpeg).
• Economic Significance:
  • The animation intuitively visualizes the dynamic interaction between technological innovation and firm distribution.
  • For example, if phi peaks shift and drive rho peak movements, it reflects firms migrating toward innovative regions.

Economic Insights from Plots and Animation

1. Spatial Diffusion of Technological Innovation:
   • Subplot 1 (heatmap) and Animation Subplot 1 show how the technological knowledge field phi diffuses from an initial concentrated point (Gaussian distribution) outward.
   • This mimics real-world technology spreading from innovation hubs (e.g., Silicon Valley) to other regions, influenced by the diffusion coefficient (D_phi=1.0) and random noise (xi=0.05).
2. Spatial Clustering of Firms:
   • Subplot 2 (heatmap) and Animation Subplot 2 show how the agent density field rho clusters due to attraction to the technological field (beta=0.1).
   • High-density regions typically align with high-intensity phi regions, reflecting firms’ tendency to migrate to technologically advanced areas.
3. Initial vs. Final State Comparison:
   • Subplot 3 (profile plot) compares field distributions at the start and end, revealing long-term evolutionary outcomes.
   • For example, multiple peaks in the final state may indicate the formation of multiple tech-economic clusters.
4. Dynamic Interaction:
   • Subplot 4 (peak plot) and the animation show the temporal evolution of phi and rho maximum values, reflecting changes in innovation and firm clustering intensity.
   • Synchronization between rho and phi maxima suggests that technological innovation drives firm concentration.
5. Policy and Theoretical Implications:
   • The plots and animation can analyze the effects of technology policies, e.g., promoting innovation diffusion (adjusting D_phi) or firm migration (adjusting beta).
   • Simulation results can validate evolutionary economics theories, such as how technology-firm coupling forms spatial clusters.

Expected Visual Output (Based on Code Logic)

• Subplot 1 (Technological Knowledge Heatmap):
  • Initially, the center (x=25) shows high intensity (yellow), spreading outward over time, with colors fading to blue.
  • Multiple high-intensity regions may appear, indicating new tech hubs.
• Subplot 2 (Agent Density Heatmap):
  • Initially, the center shows moderate density (light purple), increasing (to yellow) as it follows phi high-intensity regions.
• Subplot 3 (Profile Plot):
  • Initial curves (dashed): phi and rho show a single Gaussian peak at the center.
  • Final curves (solid): phi and rho may flatten (diffusion) or form multiple peaks (clusters).
• Subplot 4 (Peak Plot):
  • Blue curve (phi maximum): May rise initially (innovation growth), then stabilize or fluctuate (due to noise).
  • Red curve (rho maximum): Follows the blue curve, reflecting firms’ response to technology.
• Animation:
  • Left subplot: Blue phi curve starts with a central peak, flattening or forming new peaks over time.
  • Right subplot: Red rho curve shifts with phi peaks, with density peaks strengthening.

How to Inspect Plot Content

If you have run the code, inspect the following outputs:

Static Plot

• Open evo_econ_results.png:
  • Confirm that all four subplots are clearly displayed without text overlap (titles, axis labels, legends, colorbars).
  • Verify that the subplot content matches the descriptions provided (heatmaps, profile plot, peak plot).
• Checklist:
  • Subplot 1: Technological knowledge field (phi) heatmap with viridis colormap, showing spatial and temporal evolution.
  • Subplot 2: Agent density field (rho) heatmap with plasma colormap, showing agent clustering.
  • Subplot 3: Line plot comparing initial and final distributions of phi (blue) and rho (red), with solid (final) and dashed (initial) lines.
  • Subplot 4: Line plot showing the temporal evolution of maximum phi (blue) and rho (red) values.
  • Ensure font rendering (e.g., SimHei for Chinese) is correct, and no warnings (e.g., DejaVu Sans) appear.

Animation

• In Jupyter:
  • Run anim = model.create_animation(); anim to confirm the animation plays.
  • Check that the timestamp updates correctly (e.g., t = 0.00 to t = 5.00).
  • Verify that both subplots show dynamic changes:
    • Left: Blue phi curve evolving over time.
    • Right: Red rho curve following phi changes.
• If saved as evo_econ_animation.mp4:
  • Play the video to inspect the dynamic evolution of the two curves.
  • Ensure smooth transitions, correct timestamp display, and no rendering artifacts.

Economic Implications

The animation dynamically illustrates:

• Diffusion and Clustering of the Technological Knowledge Field (φ(x,t)):
  • Shows how technological knowledge spreads or clusters spatially over time.
• Spatial Migration of Agents (ρ(x,t)):
  • Depicts how agents (firms) follow the movement of technological centers, reflecting knowledge-driven spatial migration behavior.

Subplot 1: Technological Field Evolution (φ)

• Description:
  • Blue curve shows the spatial distribution of the technological field at each time step.
• Dynamic Features:
  • The initial Gaussian distribution diffuses or deforms over time.
  • Multiple peaks in φ may emerge, indicating the formation of multiple innovation centers.
• Economic Interpretation:
  • Represents the diffusion of innovative technologies from a central hub, forming regional innovation clusters or a multi-polar innovation system.

Subplot 2: Agent Distribution Evolution (ρ)

• Description:
  • Red curve shows how agents (firms) respond to changes in φ.
• Dynamic Features:
  • Initially localized distribution gradually clusters toward high-φ regions.
  • Flexible adjustments reflect firms’ sensitivity to technology and relocation costs.
• Economic Interpretation:
  • Mimics firms concentrating toward innovation resources or tech hubs, reflecting geographic agglomeration effects in a knowledge economy.

Mathematical Equations in this paper

Technological Knowledge Field φ(x,t)

• φ represents the technological density at spatial position x and time t, governed by a diffusion and random perturbation equation:$$$$∂ϕ(x,t)∂t=Dϕ⋅∇2ϕ(x,t)+σ⋅ξ(x,t)Where:
  • $D_{\varphi }$: Technological diffusion rate.
  • $\sigma$: Random innovation intensity.
  • $\xi (x,t)$: Spatio-temporal white noise (random perturbation).

 

Agent Density Field ρ(x,t)

• ρ represents the density of firms or agents in space, with evolution influenced by diffusion and response to the technological field:$$$$∂ρ(x,t)∂t=Dρ⋅∇2ρ(x,t)+η⋅∇ϕ(x,t)Where:
  • $D_{\rho }$: Agent (firm) migration diffusion coefficient, representing free migration rate.
  • $\eta$: Agent sensitivity to the technological field, indicating the strength of attraction to technology (technological guidance force).

 

Discretization and Numerical Simulation

• The equations are discretized using the finite difference method in a one-dimensional spatial interval $[0,L]$, with grid size dx and time step dt.
• $[0,L]$, with grid size dx and time step dt.

• $[0,L]$

• An explicit scheme updates φ and ρ, with Neumann boundary conditions to ensure physical realism.
• The simulation shows the technological field diffusing from an initial Gaussian distribution and forming multi-center structures, while the agent field gradually clusters toward high-φ regions.

 

Technological Field Update Equation (Subplot 1):

$$$$ϕin+1=ϕin+Dϕ⋅dtdx2⋅(ϕi+1n−2⋅ϕin+ϕi−1n)+σ⋅ξin

• The technological knowledge is initially concentrated at the center, diffusing outward over time.
• Strong random perturbations may lead to multiple local innovation centers (peaks in φ), resembling the formation of innovation clusters and path dependence.

 

Agent Density Field Update Equation (Subplot 2):

$$$$ρin+1=ρin+Dρ⋅dtdx2⋅(ρi+1n−2⋅ρin+ρi−1n)+η⋅dt2⋅dx⋅(ϕi+1n−ϕi−1n)

• Firms are initially uniformly distributed but gradually cluster toward technological hotspots as φ evolves, showing high sensitivity to innovative environments.
• This clustering process mirrors spatial clusters and technology-following behavior in a knowledge economy.

Animation Dynamics

• The animation clearly shows how changes in φ drive the spatial reorganization of ρ.
• Firm behavior is guided by φ, exhibiting self-organization and collective adaptation characteristics.

References

1. Weinberg, S. (1995). The Quantum Theory of Fields (Vol. 1–3). Cambridge University Press.
   • Classic QFT textbook, suitable for citing field definitions, Lagrangian, or quantization processes.
2. Nelson, R. R., & Winter, S. G. (1982). An Evolutionary Theory of Economic Change. Harvard University Press.
   • Foundational work in evolutionary economics, ideal for theoretical grounding of technological evolution and agent behavior.
3. Arthur, W. B. (1994). Increasing Returns and Path Dependence in the Economy. University of Michigan Press.
   • Discusses technology choice, path dependence, and innovation diffusion, relevant to interpreting multi-peak structures in φ.
4. Aoki, M. (2002). Modeling Aggregate Behavior and Fluctuations in Economics: Stochastic Views of Interacting Agents. Cambridge University Press.
   • Covers multi-agent interaction models and stochastic evolution, relevant to ξ(x,t) and η·∇φ(x,t).
5. Bouchaud, J.-P. (2013). Crises and collective socio-economic phenomena: Simple models and challenges. Journal of Statistical Physics, 151(3–4), 567–606.
   • Applies physical field concepts to economics and collective behavior from a sociophysics perspective.
6. Helbing, D. (2010). Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes. Springer.
   • Emphasizes field-theoretic models and diffusion phenomena in social interactions.
7. Ball, P. (2004). Critical Mass: How One Thing Leads to Another. Farrar, Straus and Giroux.
   • Accessible yet insightful sociophysics book, useful for cross-disciplinary explanations of QFT applications.
8. Holland, J. H. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. MIT Press.
   • Provides a complex adaptive systems framework, relevant to evolutionary agent models.

Note on Original Model

• The field-theoretic structure and coupling term (η*∇φ) in this model are derived by the author based on quantum field theory diffusion equations and social interaction mechanisms.
• No existing literature directly corresponds to this model, making it an original theoretical construct.

Steps to Verify Plot Content

1. Static Plot (evo_econ_results.png):
   • Open the file in an image viewer.
   • Check for clear subplot rendering, correct colormaps (viridis for φ, plasma for ρ), and no text overlap.
   • Confirm that heatmaps show expected diffusion patterns, the profile plot shows initial vs. final states, and the peak plot reflects maximum value trends.
2. Animation:
• In Jupyter, ensure anim runs and displays both subplots with evolving curves.
• For evo_econ_animation.mp4, play the video to verify smooth curve transitions and timestamp updates.
• Confirm that φ (blue) diffuses or forms peaks, and ρ (red) follows φ peaks, aligning with the economic interpretations provided.

 

  The Python script evo_econ_model.py implements a field-theoretic model of technological innovation and agent interactions, simulating the spatio-temporal dynamics of the technological knowledge field (φ) and agent density field (ρ). The code is publicly available on GitHub at:

https://github.com/fullyloaded/Python-Code/blob/main/evo_econ_model