warp-bubble-parameter-constraints

This repository provides a comprehensive analysis of warp bubble parameter constraints using both symbolic and numerical methods.

Features

1. analyze_constraints.py
   A Python script that:

   • Models a simplified warp bubble with Gaussian profile
   • Computes the total negative energy analytically using SymPy
   • Derives extremum conditions for bubble radius (R) and width (σ)
   • Performs nondimensional analysis using x = R/σ
   • Conducts numerical root-finding for transcendental equations
   • Provides asymptotic analysis for thin-wall and thick-wall limits
   • Exports comprehensive results as LaTeX

2. parameter_constraints.tex
   A ready‐to‐compile LaTeX document that presents:

   • The analytic form of the warp bubble model
   • Total negative energy E_neg(R,σ,A) with error functions
   • Extremum conditions and their nondimensional form
   • Asymptotic analysis for different parameter regimes
   • Physical interpretation of the absence of critical points
   • Numerical approaches for practical optimization

Key Findings

The analysis reveals that this simplified Gaussian warp bubble model has no analytical critical points because the extremum conditions reduce to transcendental equations of the form:

F(x) = exp(-2x²) + exp(-8x²)/2 = 0

where x = R/σ. Since both exponential terms are strictly positive, F(x) > 0 for all x, indicating:

• No natural "optimal" bubble size exists in this model
• Energy decreases monotonically with increasing R or σ
• Physical constraints must determine practical parameters
• More sophisticated models are needed for realistic optimization

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Prerequisites

• Python 3.7+
• SymPy - Symbolic mathematics
• NumPy - Numerical computing
• SciPy - Scientific computing (optional, for advanced numerical methods)
• mpmath - Arbitrary precision arithmetic (optional)

Install dependencies via:

pip install sympy numpy scipy mpmath

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Usage

1. Clone the repo

git clone https://github.com/arcticoder/warp-bubble-parameter-constraints.git
cd warp-bubble-parameter-constraints

2. Run the analysis script

python analyze_constraints.py

This will:

-   Perform symbolic analysis of the warp bubble model
-   Compute total negative energy and extremum conditions
-   Conduct nondimensional analysis with x = R/σ
-   Perform numerical root-finding and asymptotic analysis
-   Generate comprehensive LaTeX documentation

3. Compile the LaTeX

pdflatex parameter_constraints.tex

to produce a PDF report with complete analysis results.

Numerical Methods Implemented

The script demonstrates several approaches for handling transcendental equations:

1. Nondimensionalization - Reduces the problem to a single variable x = R/σ
2. Numerical root-finding - Uses robust solvers for transcendental equations
3. Asymptotic expansions - Provides approximate solutions for limiting cases
4. Parameter scanning - Grid-based optimization for practical applications
5. Physical interpretation - Analysis of why no critical points exist

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Files

• analyze_constraints.py
  Enhanced Python script performing symbolic computations, numerical analysis, and LaTeX export.

• parameter_constraints.tex
  Comprehensive LaTeX document with complete analysis results and physical interpretation.

• README.md
  This documentation file.

Example Output

The analysis produces output like:

Transcendental equation F(x) = exp(-2x²) + exp(-8x²)/2 = 0
where x = R/σ is the dimensionless bubble width ratio

Function values at test points:
F(0.1) = 1.441757
F(0.5) = 0.674198  
F(1.0) = 0.135503
F(2.0) = 0.000335

Observation: F(x) > 0 for all x > 0 (both exponentials are positive)
This means dE/dR ≠ 0 for any finite R, σ - no critical points exist!

This demonstrates the power of combining symbolic and numerical methods to understand the behavior of complex physical systems.

Scope, Validation & Limitations

• Scope: The materials and numeric outputs in this repository are research-stage examples and depend on implementation choices, parameter settings, and numerical tolerances.
• Validation: Reproducibility artifacts (scripts, raw outputs, seeds, and environment details) are provided in docs/ or examples/ where available; reproduce analyses with parameter sweeps and independent environments to assess robustness.
• Limitations: Results are sensitive to modeling choices and discretization. Independent verification, sensitivity analyses, and peer review are recommended before using these results for engineering or policy decisions.