Social Quantum Field Simulation Framework:

 Mathematical Formulas

Key Equations of Motion

The fields evolve via coupled Klein-Gordon-like equations:

Phi Field Evolution: ∂²φ/∂t² = ∇²φ - m_φ²φ - λφ³ + 2gφψ

Psi Field Evolution: ∂²ψ/∂t² = ∇²ψ - m_ψ²ψ - ηψ³ + gφ²

Local Energy Density

ℋ = ½π² + ½(∇φ)² + ½m_φ²φ² + (λ/4)φ⁴

Extended Energy Density (Including Both Fields)

ℋ = ½π_φ² + ½π_ψ² + ½(∇φ)² + ½(∇ψ)² + ½m_φ²φ² + ½m_ψ²ψ² + (λ/4)φ⁴ + (η/4)ψ⁴ - gφ²ψ

Lyapunov Exponent Calculation

λ_L = (1/(T·dt))·ln(dT/d₀)

Where:

• φ, ψ: Scalar fields representing social variables
• π_φ, π_ψ: Field momentum (rate of change)
• m_φ, m_ψ: Mass parameters (resistance to change)
• λ, η: Self-interaction strengths
• g: Coupling strength between fields
• ∇²: Laplacian operator (spatial diffusion)
• ∇: Gradient operator (spatial variation)

Python Code

 GitHub Link : 

Social Quantum Field Simulation Framework.py

Explanation of the Program

This program integrates all components of the Social Quantum Field Simulation Framework into a single script, with a clear social interpretation embedded in the variable names, labels, and comments. Here’s a breakdown of the key features:

1. Simulation:
   • Evolves two scalar fields (phi and psi) using the Leapfrog method, solving coupled Klein-Gordon-like equations with nonlinear and coupling terms.
   • Social Interpretation:
     • phi: Represents opinion strength (e.g., political ideology, -1 for liberal, +1 for conservative).
     • psi: Represents economic sentiment (e.g., -1 for pessimistic, +1 for optimistic).
     • The grid represents social regions (e.g., geographic areas or network nodes).
     • Nonlinear terms (lam * phi**3, eta * psi**3) model self-reinforcement (e.g., echo chambers).
     • Coupling terms (2 * g * phi * psi, g * phi**2) model interdependencies (e.g., economic sentiment affecting opinions).
2. Dynamical Symmetry Breaking:
   • The nonlinear potential allows phi or psi to settle into non-zero states, visible in the field evolution and VEV plots.
   • Social Interpretation: Represents polarization or consensus formation, where neutral opinions shift to a dominant stance (e.g., widespread conservatism).
3. Vacuum Expectation Value (VEV):
   • Computes the spatial average of phi and psi (vev_phi, vev_psi) to track stable societal states.
   • Social Interpretation: Non-zero VEVs indicate societal norms, such as a prevailing political lean or economic outlook.
4. Chaotic Phase:
   • Visualized in the phase space plot, where scattered trajectories suggest chaotic dynamics.
   • Social Interpretation: Reflects unpredictable social trends, such as viral movements or sudden economic shifts, driven by nonlinear interactions.
5. Entanglement Approximation:
   • Uses the correlation coefficient of local energy density between adjacent windows to quantify spatial correlations.
   • Social Interpretation: Measures social connectivity, with high correlations indicating unified behavior (e.g., aligned opinions) and low or fluctuating correlations suggesting fragmentation or chaos.
6. Local Energy Density:
   • Computes the energy density for phi, focusing on opinion dynamics.
   • Social Interpretation:
     • Kinetic term (0.5 * pi**2): Social volatility (rate of opinion change).
     • Gradient term (0.5 * grad_phi**2): Tension between neighboring regions.
     • Potential terms: Stability or cost of maintaining opinions.
7. Spatial Correlation Profile:
   • Computes correlations between windows separated by varying distances to show the range of social influence.
   • Social Interpretation: Indicates how far opinions or sentiment propagate, informing communication strategies.
8. Visualization:
   • A six-panel plot shows:
     • Opinion Dynamics: Spatial profiles of phi, showing polarization or clustering.
     • Economic Sentiment Dynamics: Profiles of psi, showing economic trends.
     • Societal Norms: VEVs of phi and psi, tracking dominant states.
     • Social Connectivity: Average correlation over time, reflecting coherence or chaos.
     • Activity Stability: Total energy, ensuring simulation reliability.
     • Spatial Correlation Profile: Range of social influence.
   • A separate phase space plot visualizes chaotic dynamics.
   • Social Interpretation: Labels use social terms (e.g., “Opinion Strength,” “Social Regions”) to make the plots intuitive for social analysis.

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Social Interpretation Summary

The program models a society where:

• Opinions (phi) and economic sentiment (psi) evolve across regions due to:
  • Self-reinforcement: Strong opinions amplify themselves (nonlinear terms).
  • Interdependencies: Economic sentiment influences opinions, and vice versa (coupling terms).
  • Diffusion: Social variables spread between regions (Laplacian).
• Polarization occurs when opinions shift from neutral to extreme states (symmetry breaking).
• Societal norms emerge as stable average opinions or sentiments (VEVs).
• Unpredictable trends arise from nonlinear interactions, visible as chaotic phase space trajectories.
• Social connectivity is quantified by correlations, showing how cohesive or fragmented society is.

The visualizations provide insights for policymakers or researchers:

• Opinion Dynamics: Identify polarized regions for targeted interventions.
• Societal Norms: Monitor shifts in average opinions or sentiment to anticipate societal trends.
• Social Connectivity: Assess whether society is unified or fragmented, guiding communication strategies.
• Chaotic Dynamics: Detect vulnerability to small perturbations, informing stabilizing measures.

社會量子場模擬框架解析

框架基本概念

這個框架將社會視為一個動態系統，通過對標量場的模擬來研究社會現象，例如意見形成、經濟行為或文化變遷。核心理念是:

1. 標量場 (φ 和 ψ):
   • φ: 可代表政治立場強度（-1為左傾，+1為右傾）
   • ψ: 可代表經濟信心（-1為悲觀，+1為樂觀）
   • 網格上每個點代表一個社會單位（社區或個人）
2. 運動方程:
   • 質量項 (m_φ²φ): 表示社會變量的慣性或抗變性
   • 非線性項 (λφ³): 模擬自我強化效應，如回音室效應
   • 耦合項 (2gφψ): 反映社會變量間的相互依賴
   • 拉普拉斯算子: 代表通過互動產生的空間擴散
3. 對稱性破缺:
   • 對應社會極化或共識形成
   • 初始中立意見(φ≈0)可能分裂為對立派系(φ→±1)
4. 真空期望值 (VEV):
   • φ或ψ的空間平均值代表主流社會狀態
   • 非零VEV反映穩定的社會配置，如極化的政治格局
5. 混沌相:
   • 代表不可預測的社會動態
   • 小事件可能通過非線性互動放大，導致廣泛影響
6. 糾纏近似:
   • 測量相鄰社會單位間的連通性或影響傳播
   • 高相關性表明強烈統一性，低相關性表明分裂

應用場景示例

政治極化:

• φ: 政治觀點（負值=自由派，正值=保守派）
• ψ: 對政府機構的信任度
• 非線性項使極端意見自我強化，形成回音室
• 耦合項表示對機構的信任如何調節政治觀點
• 對稱性破缺體現為中立社會分裂為對立陣營

經濟繁榮與蕭條:

• φ: 經濟情緒（負值=悲觀，正值=樂觀）
• ψ: 消費支出（負值=儲蓄，正值=消費）
• 非線性項反映過度消費如何強化樂觀情緒但最終導致崩潰
• 強烈的經濟情緒驅動消費行為
• 小型市場衝擊可能導致不可預測的消費模式

增強框架的建議

1. 映射實際社會數據:
   • 使用真實民調或經濟指標校準模型參數
   • 調整參數以匹配觀察到的社會動態
2. 納入ψ場的能量密度計算:
   • 擴展能量密度函數以包含兩個場的完整相互作用
3. 定量混沌:
   • 添加李雅普諾夫指數計算來明確測量混沌行為
   • 正指數確認混沌相，表明社會不可預測性
4. 空間相關性剖面:
   • 計算不同距離窗口間的相關性，繪製社會影響的"範圍"

這種框架的優勢在於能夠模擬複雜的社會互動和集體行為，有助於理解極化、共識形成和社會連通性等現象的動態過程。

Python Code

Enhanced Social Quantum Field Simulation Framework.py