cognitive/knowledge_base/mathematics/non_equilibrium_thermodynamics.md

title	type	status	created	complexity	processing_priority	tags	semantic_relations
Non-Equilibrium Thermodynamics	concept	stable	2024-03-15	advanced	1	mathematics physics thermodynamics complexity	type links foundation_for free_energy_principle active_inference self_organization type links implements thermodynamics statistical_physics stochastic_processes type links relates information_theory dynamical_systems complex_systems

Non-Equilibrium Thermodynamics

Overview

Non-Equilibrium Thermodynamics extends classical thermodynamics to systems far from equilibrium, providing a framework for understanding self-organization, dissipative structures, and the emergence of order in biological and cognitive systems.

Mathematical Foundation

Entropy Production

Local Balance

\frac{\partial s}{\partial t} + \nabla \cdot J_s = \sigma

where:

• s is entropy density
• J_s is entropy flux
• \sigma is entropy production rate

Onsager Relations

J_i = \sum_j L_{ij}X_j

where:

• J_i are fluxes
• X_j are forces
• L_{ij} are Onsager coefficients

Implementation

Non-Equilibrium System

class NonEquilibriumSystem:
    def __init__(self,
                 state_dim: int,
                 force_matrix: np.ndarray,
                 diffusion_matrix: np.ndarray):
        """Initialize non-equilibrium system.
        
        Args:
            state_dim: State dimension
            force_matrix: Deterministic forces
            diffusion_matrix: Noise coupling
        """
        self.dim = state_dim
        self.F = force_matrix
        self.D = diffusion_matrix
        
        # Initialize state
        self.state = np.zeros(state_dim)
        
        # Initialize thermodynamic quantities
        self.entropy = 0.0
        self.entropy_production = 0.0
        self.entropy_flux = 0.0
    
    def update_state(self,
                    dt: float = 0.01) -> None:
        """Update system state.
        
        Args:
            dt: Time step
        """
        # Deterministic update
        drift = self.F @ self.state
        
        # Stochastic update
        noise = np.random.randn(self.dim)
        diffusion = self.D @ noise
        
        # Update state
        self.state += dt * drift + np.sqrt(dt) * diffusion
        
        # Update thermodynamic quantities
        self.update_thermodynamics(dt)
    
    def update_thermodynamics(self,
                            dt: float) -> None:
        """Update thermodynamic quantities.
        
        Args:
            dt: Time step
        """
        # Compute entropy production
        forces = self.F @ self.state
        fluxes = self.D @ self.D.T @ forces
        self.entropy_production = np.sum(forces * fluxes) * dt
        
        # Compute entropy flux
        self.entropy_flux = self.compute_entropy_flux(dt)
        
        # Update entropy
        self.entropy += self.entropy_production - self.entropy_flux
    
    def compute_entropy_flux(self,
                           dt: float) -> float:
        """Compute entropy flux.
        
        Args:
            dt: Time step
            
        Returns:
            flux: Entropy flux
        """
        # Implementation depends on system specifics
        pass

Fluctuation Analysis

class FluctuationAnalyzer:
    def __init__(self,
                 system: NonEquilibriumSystem):
        """Initialize fluctuation analyzer.
        
        Args:
            system: Non-equilibrium system
        """
        self.system = system
        self.trajectories = []
        
    def sample_trajectories(self,
                          n_samples: int,
                          duration: float,
                          dt: float = 0.01) -> np.ndarray:
        """Sample system trajectories.
        
        Args:
            n_samples: Number of trajectories
            duration: Trajectory duration
            dt: Time step
            
        Returns:
            trajectories: Sampled trajectories
        """
        n_steps = int(duration / dt)
        trajectories = np.zeros((n_samples, n_steps, self.system.dim))
        
        for i in range(n_samples):
            # Reset system
            self.system.state = np.zeros(self.system.dim)
            
            # Generate trajectory
            for t in range(n_steps):
                self.system.update_state(dt)
                trajectories[i,t] = self.system.state.copy()
        
        self.trajectories = trajectories
        return trajectories
    
    def compute_fluctuation_theorem(self,
                                  time_window: int) -> Dict[str, np.ndarray]:
        """Compute fluctuation theorem statistics.
        
        Args:
            time_window: Analysis window size
            
        Returns:
            stats: Fluctuation statistics
        """
        # Compute entropy production
        s_prod = np.zeros((len(self.trajectories), len(self.trajectories[0])-time_window))
        
        for i, traj in enumerate(self.trajectories):
            for t in range(len(traj)-time_window):
                window = traj[t:t+time_window]
                s_prod[i,t] = self.compute_window_entropy_production(window)
        
        # Compute probability ratio
        p_forward = np.histogram(s_prod.flatten(), bins=50, density=True)[0]
        p_backward = np.histogram(-s_prod.flatten(), bins=50, density=True)[0]
        
        return {
            'entropy_production': s_prod,
            'p_forward': p_forward,
            'p_backward': p_backward
        }

Dissipative Structure Analysis

class DissipativeStructure:
    def __init__(self,
                 spatial_dim: Tuple[int, int],
                 diffusion_coeff: float,
                 reaction_rates: np.ndarray):
        """Initialize dissipative structure model.
        
        Args:
            spatial_dim: Spatial dimensions
            diffusion_coeff: Diffusion coefficient
            reaction_rates: Reaction rate constants
        """
        self.dim = spatial_dim
        self.D = diffusion_coeff
        self.k = reaction_rates
        
        # Initialize fields
        self.concentration = np.random.rand(*spatial_dim)
        self.chemical_potential = np.zeros(spatial_dim)
        
    def update(self,
              dt: float = 0.01) -> None:
        """Update structure state.
        
        Args:
            dt: Time step
        """
        # Compute diffusion
        laplacian = self.compute_laplacian()
        diffusion = self.D * laplacian
        
        # Compute reactions
        reactions = self.compute_reactions()
        
        # Update concentration
        self.concentration += dt * (diffusion + reactions)
        
        # Update chemical potential
        self.update_chemical_potential()
    
    def compute_laplacian(self) -> np.ndarray:
        """Compute Laplacian operator.
        
        Returns:
            laplacian: Laplacian of concentration field
        """
        # Finite difference approximation
        laplacian = np.zeros_like(self.concentration)
        
        # Interior points
        laplacian[1:-1,1:-1] = (
            self.concentration[:-2,1:-1] +
            self.concentration[2:,1:-1] +
            self.concentration[1:-1,:-2] +
            self.concentration[1:-1,2:] -
            4 * self.concentration[1:-1,1:-1]
        )
        
        return laplacian
    
    def compute_reactions(self) -> np.ndarray:
        """Compute reaction terms.
        
        Returns:
            reactions: Reaction contribution
        """
        # Implementation depends on specific reaction network
        pass

Applications

Biological Systems

Cell Biology

• Membrane transport
• Metabolic networks
• Signal transduction
• Cell division

Development

• Pattern formation
• Morphogenesis
• Tissue organization
• Growth dynamics

Cognitive Systems

Neural Dynamics

• Action potentials
• Synaptic plasticity
• Network formation
• Information processing

Active Inference

• Free energy minimization
• Belief updating
• Action selection
• Learning dynamics

Best Practices

Modeling

1. Identify relevant scales
2. Define boundary conditions
3. Specify constraints
4. Include fluctuations

Analysis

1. Track energy flows
2. Monitor entropy production
3. Validate steady states
4. Check conservation laws

Implementation

1. Stable integration
2. Noise handling
3. Boundary treatment
4. Conservation checks

Common Issues

Technical Challenges

1. Multiple time scales
2. Numerical instability
3. Boundary effects
4. Conservation violations

Solutions

1. Multi-scale methods
2. Implicit schemes
3. Buffer regions
4. Constraint projection

Related Documentation

• thermodynamics
• statistical_physics
• free_energy_principle
• complex_systems