cognitive/knowledge_base/mathematics/path_integral_verification.md

type	id	created	modified	tags	aliases
verification	path_integral_verification_001	2024-02-05	2024-02-05	verification path-integrals technical-accuracy documentation	path-verification documentation-verification

Path Integral Documentation Verification

Core Mathematical Foundations

Measure Theory Verification

1. Functional Integration

   • Add link to radon_nikodym_theorem for measure relationships
   • Verify lebesgue_integration connection to path measures
   • Include cylinder_measure construction for path spaces

2. Probability Measures

   • Link to gaussian_measures for Wiener process
   • Add kolmogorov_extension for infinite dimensions
   • Include weak_convergence of measures

Functional Analysis Enhancement

1. Topology

   class FunctionalTopology:
       """Enhanced topological structure for path spaces."""
       def __init__(self):
           self.norm_types = {
               'uniform': self._uniform_norm,
               'sobolev': self._sobolev_norm,
               'holder': self._holder_norm
           }
   

   Links to:

   • topological_vector_spaces
   • frechet_spaces
   • nuclear_spaces

2. Operators

   • Add unbounded_operators treatment
   • Include spectral_theory for path operators
   • Link to fredholm_theory for integral operators

Statistical Physics Connections

Partition Function Accuracy

1. Normalization

   def verify_partition_normalization(Z: Callable,
                                    beta_range: np.ndarray) -> bool:
       """Verify partition function normalization."""
       for beta in beta_range:
           if not np.isclose(
               integrate_states(lambda x: exp(-beta * H(x)) / Z(beta)),
               1.0,
               rtol=1e-5
           ):
               return False
       return True
   

   Links to:

   • normalization_conditions
   • thermodynamic_consistency
   • gibbs_measures

2. Physical Properties

   • Add fluctuation_dissipation relations
   • Include onsager_relations
   • Link to kubo_formulas

Active Inference Implementation

Free Energy Computation

1. Numerical Accuracy

   class FreeEnergyValidator:
       """Validate free energy computations."""
   
       def verify_bound(self,
                       free_energy: Callable,
                       log_evidence: Callable) -> bool:
           """Verify variational free energy bounds log evidence."""
           samples = self.generate_test_samples()
           F = free_energy(samples)
           L = log_evidence(samples)
           return np.all(F >= L)  # Check bound holds
   

   Links to:

   • jensen_inequality
   • information_geometry
   • variational_principles

2. Gradient Accuracy

   • Add natural_gradient verification
   • Include fisher_information computation
   • Link to wasserstein_metrics

Path Space Methods

1. Discretization

   class PathDiscretizationValidator:
       """Validate path discretization methods."""
   
       def verify_convergence_order(self,
                                  scheme: Callable,
                                  exact_solution: Callable,
                                  dt_range: np.ndarray) -> int:
           """Verify numerical convergence order."""
           errors = []
           for dt in dt_range:
               numerical = scheme(dt)
               error = np.max(np.abs(numerical - exact_solution))
               errors.append(error)
   
           # Compute convergence order
           orders = np.log(errors[:-1]/errors[1:]) / \
                   np.log(dt_range[:-1]/dt_range[1:])
           return np.mean(orders)
   

   Links to:
   • convergence_analysis
   • stability_theory
   • error_propagation

Implementation Verification

Numerical Methods

1. Integration Schemes

   class IntegrationVerifier:
       """Verify numerical integration methods."""
   
       def verify_symplectic(self,
                           integrator: Callable,
                           hamiltonian: Callable,
                           duration: float) -> bool:
           """Verify symplectic property preservation."""
           initial_state = self.random_state()
           trajectory = integrator(hamiltonian, initial_state, duration)
   
           return self.check_symplectic_form_conservation(trajectory)
   

   Links to:

   • geometric_integration
   • symplectic_methods
   • energy_conservation

2. Sampling Methods

   • Add mcmc_diagnostics
   • Include effective_sample_size
   • Link to convergence_diagnostics

Performance Optimization

1. Computational Efficiency

   class PerformanceValidator:
       """Validate computational performance."""
   
       def benchmark_scaling(self,
                           method: Callable,
                           problem_sizes: List[int]) -> Dict[str, float]:
           """Analyze computational scaling."""
           times = []
           memory = []
           for size in problem_sizes:
               t, m = self.measure_performance(method, size)
               times.append(t)
               memory.append(m)
   
           return {
               'time_complexity': self.fit_scaling_law(times),
               'space_complexity': self.fit_scaling_law(memory)
           }
   

   Links to:
   • complexity_analysis
   • optimization_techniques
   • parallel_efficiency

Documentation Updates

Theory Documents

1. Path Integral Theory

   • Add stochastic_analysis section
   • Enhance field_theory connections
   • Include quantum_mechanics analogies

2. Implementation Guide

   • Add error_estimation methods
   • Include adaptive_methods
   • Link to performance_optimization

Synthesis Document

1. Bridge Concepts

   • Add discretization_theory
   • Include numerical_analysis
   • Link to computational_physics

2. Validation Methods

   • Add unit_testing framework
   • Include integration_testing
   • Link to continuous_integration

References

• hairer_2006 - Geometric Numerical Integration
• glimm_1987 - Quantum Physics: A Functional Integral Point of View
• pavliotis_2014 - Stochastic Processes and Applications
• graham_2006 - Computational Methods for Physics