free_energy_principle - Obsidian Publish

# Free Energy Principle ## Overview The Free Energy Principle (FEP) is a unifying theory that explains how biological systems maintain their organization in the face of environmental fluctuations. It proposes that all self-organizing adaptive systems minimize "variational free energy"—a statistical quantity that bounds the surprise (or self-information) associated with sensory signals. Through this minimization, organisms implicitly resist the natural tendency toward disorder described by the second law of thermodynamics. Mathematically, the FEP states that biological systems minimize a functional $F[q]$ that represents the upper bound on surprise: ```math F[q] = \mathbb{E}_q[\ln q(s) - \ln p(s,o)] ``` where: - $q(s)$ is a recognition density over internal states $s$ - $p(s,o)$ is a generative model relating internal states $s$ to observations $o$ The principle has broad applications across cognitive neuroscience, artificial intelligence, and biological self-organization. ```mermaid graph TD subgraph "Free Energy Principle" A[Variational Free Energy] --> B[Perception] A --> C[Learning] A --> D[Action] B --> E[State Estimation] C --> F[Model Parameter Updates] D --> G[Active Inference] E --> H[Reduction of Perceptual Prediction Error] F --> I[Reduction of Model Complexity] G --> J[Reduction of Expected Surprise] end style A fill:#f9d,stroke:#333 style B fill:#adf,stroke:#333 style C fill:#adf,stroke:#333 style D fill:#adf,stroke:#333 ``` ## Historical Context and Development ### Origins and Evolution The Free Energy Principle emerged from the intersection of thermodynamics, information theory, and neuroscience, developed primarily by Karl Friston in the early 2000s: ```mermaid timeline title Evolution of the Free Energy Principle section Conceptual Foundations 1950s-60s : Cybernetics : Ashby's homeostasis, self-organizing systems 1970s-80s : Information Theory : Shannon's entropy, Jaynes' maximum entropy principle section Neuroscientific Roots 1990s : Predictive Coding : Rao & Ballard's hierarchical predictive coding 1995-2000 : Bayesian Brain : Helmholtz's unconscious inference revival section Formal Development 2003-2005 : First FEP Papers : Friston's initial mathematical formulation 2006-2010 : Connection to Variational Bayes : Formal link to machine learning section Extensions 2010-2015 : Active Inference : Extension to action and behavior 2016-2020 : Physics Connections : Statistical physics, non-equilibrium steady states section Current Directions 2020-Present : Markov Blankets : Formal boundary conditions for systems 2021-Present : New Mathematics : Stochastic differential equations, information geometry ``` ### Key Contributors 1. **Karl Friston**: Principal architect who formalized the Free Energy Principle 1. **Geoffrey Hinton**: Developed related concepts in Helmholtz Machines 1. **Anil Seth**: Extended FEP to interoceptive processing and consciousness 1. **Maxwell Ramstead**: Applied FEP to social systems and cultural dynamics 1. **Christopher Buckley**: Connected FEP to control theory 1. **Thomas Parr**: Developed hierarchical implementations 1. **Lancelot Da Costa**: Advanced mathematical formulations using stochastic calculus ### Intellectual Lineage The Free Energy Principle integrates several intellectual traditions: - **Helmholtz's Unconscious Inference**: The idea that perception is a form of unconscious statistical inference - **Ashby's Cybernetics**: Concepts of homeostasis and self-organization - **Jaynes' Maximum Entropy Principle**: Using entropy maximization subject to constraints - **Feynman's Path Integrals**: Statistical mechanics approaches to understanding systems - **Pearl's Causality**: Causal modeling through graphical models ## Mathematical Foundation ### Variational Free Energy Variational free energy is formally defined as: ```math F[q] = \mathbb{E}_q[\ln q(s) - \ln p(s,o)] ``` This can be decomposed in several ways: ```math \begin{aligned} F[q] &= \mathbb{E}_q[\ln q(s) - \ln p(s|o) - \ln p(o)] \\ &= D_{\mathrm{KL}}[q(s) \| p(s|o)] - \ln p(o) \\ &= \underbrace{\mathbb{E}_q[-\ln p(o|s)]}_{\text{accuracy}} + \underbrace{D_{\mathrm{KL}}[q(s) \| p(s)]}_{\text{complexity}} \end{aligned} ``` where: - $D_{\mathrm{KL}}$ is the Kullback-Leibler divergence - $p(o)$ is the marginal likelihood or evidence - $p(s|o)$ is the true posterior distribution ### Decomposition The FEP can be decomposed into different components representing accuracy and complexity: ```math \begin{aligned} \text{Accuracy} &= \mathbb{E}_q[-\ln p(o|s)] \\ \text{Complexity} &= D_{\mathrm{KL}}[q(s) \| p(s)] \end{aligned} ``` This decomposition reveals a fundamental trade-off: minimizing free energy requires balancing model fit (accuracy) against model complexity. ### Alternative Forms Free energy can also be expressed in terms of energy and entropy: ```math F[q] = \mathbb{E}_q[E(s,o)] - H[q(s)] ``` where: - $E(s,o) = -\ln p(s,o)$ is the energy function - $H[q(s)] = -\mathbb{E}_q[\ln q(s)]$ is the entropy of the recognition density ## Information Geometry Perspective ### Geometry of Free Energy The Free Energy Principle can be understood through the lens of information geometry, where variational inference corresponds to optimization on a statistical manifold: ```math \begin{aligned} \delta F[q] &= \int \frac{\delta F}{\delta q(s)} \delta q(s) ds \\ &= \int \left(\ln q(s) - \ln p(s,o) + 1 \right) \delta q(s) ds \end{aligned} ``` For parametric distributions $q_\theta(s)$ with parameters $\theta$, we can write: ```math \frac{\partial F}{\partial \theta_i} = \mathbb{E}_q\left[\frac{\partial \ln q_\theta(s)}{\partial \theta_i}\left(\ln q_\theta(s) - \ln p(s,o)\right)\right] ``` ### Fisher Information and Natural Gradients Optimization in information geometry often uses the Fisher information matrix: ```math \mathcal{G}_{ij} = \mathbb{E}_q\left[\frac{\partial \ln q_\theta(s)}{\partial \theta_i}\frac{\partial \ln q_\theta(s)}{\partial \theta_j}\right] ``` Leading to natural gradient descent: ```math \Delta \theta = -\eta \mathcal{G}^{-1}\nabla_\theta F ``` where $\eta$ is a learning rate. This approach respects the Riemannian geometry of the probability simplex and can accelerate convergence. ### Connections to Optimal Transport Recent work connects the FEP to optimal transport theory and Wasserstein gradients: ```math \partial_t q_t = \nabla \cdot (q_t \nabla \delta F/\delta q) ``` This formulation reveals free energy minimization as a gradient flow on the space of probability measures. ## Connections to Statistical Physics ### Thermodynamic Interpretation The Free Energy Principle has deep connections to statistical physics and thermodynamics: ```math \begin{aligned} F_{\text{thermo}} &= U - TS \\ F_{\text{variational}} &= \mathbb{E}_q[E] - H[q] \end{aligned} ``` Where: - $F_{\text{thermo}}$ is thermodynamic free energy - $U$ is internal energy - $T$ is temperature - $S$ is entropy - $F_{\text{variational}}$ is variational free energy - $E$ is an energy function - $H[q]$ is entropy of $q$ ### Non-equilibrium Steady States A more recent formulation connects the FEP to non-equilibrium steady states (NESS): ```math \nabla \cdot (q(x) \nabla E(x)) = 0 ``` Where $q(x)$ is a probability density function over states $x$ and $E(x)$ is a potential function. In the context of the FEP, biological systems are conceptualized as maintaining a characteristic probability distribution over their states. ### Principle of Least Action The FEP can also be connected to the principle of least action via path integrals: ```math q^*(x(t)) = \frac{1}{Z} \exp\left(-\int_0^T \mathcal{L}(x(t), \dot{x}(t)) dt\right) ``` Where $\mathcal{L}$ is a Lagrangian function and $Z$ is a normalizing constant. This formulation provides a bridge between the FEP and theoretical physics. ## Advanced Mathematical Formulations ### Stochastic Differential Equations The FEP can be formulated using stochastic differential equations: ```math \begin{aligned} dx &= (f(x) + Q\partial_x\ln p(x))dt + \sqrt{2Q}dW \\ dp &= (-\partial_xH)dt + \sqrt{2R}dW \end{aligned} ``` where: - $f(x)$ is the drift function - $Q$ and $R$ are diffusion matrices - $dW$ is a Wiener process - $H$ is the Hamiltonian ### Path Integral Formulation The path integral perspective connects FEP to quantum mechanics: ```math \begin{aligned} p(x(T)|x(0)) &= \int \mathcal{D}[x]e^{-S[x]} \\ S[x] &= \int_0^T dt\left(\frac{1}{2}|\dot{x} - f(x)|^2 + V(x)\right) \end{aligned} ``` where: - $\mathcal{D}[x]$ is the path integral measure - $S[x]$ is the action functional - $V(x)$ is the potential function ### Information Geometric Extensions Using the language of information geometry: ```math \begin{aligned} g_{ij}(\theta) &= \mathbb{E}_{p(x|\theta)}\left[\frac{\partial \ln p(x|\theta)}{\partial \theta^i}\frac{\partial \ln p(x|\theta)}{\partial \theta^j}\right] \\ \Gamma_{ijk}^{(α)} &= \mathbb{E}\left[\frac{\partial^2 \ln p}{\partial \theta^i\partial \theta^j}\frac{\partial \ln p}{\partial \theta^k}\right] \\ \nabla_\theta F &= g^{ij}(\theta)\frac{\partial F}{\partial \theta^j} \end{aligned} ``` ## Implementation Framework ### Variational Methods ```python class FreeEnergyModel: def __init__(self, state_dim: int, obs_dim: int): """Initialize free energy model. Args: state_dim: State dimension obs_dim: Observation dimension """ self.state_dim = state_dim self.obs_dim = obs_dim # Initialize distributions self.q = VariationalDistribution(state_dim) self.p = GenerativeModel(state_dim, obs_dim) def compute_free_energy(self, obs: torch.Tensor) -> torch.Tensor: """Compute variational free energy. Args: obs: Observations Returns: F: Free energy value """ # Get variational parameters mu, log_var = self.q.get_parameters() # Compute KL divergence kl_div = -0.5 * torch.sum( 1 + log_var - mu.pow(2) - log_var.exp() ) # Compute expected log likelihood expected_llh = self.p.expected_log_likelihood( obs, mu, log_var ) return kl_div - expected_llh def update(self, obs: torch.Tensor, lr: float = 0.01) -> None: """Update model parameters. Args: obs: Observations lr: Learning rate """ # Compute free energy F = self.compute_free_energy(obs) # Compute gradients F.backward() # Update parameters with torch.no_grad(): for param in self.parameters(): param -= lr * param.grad param.grad.zero_() ``` ### Markov Blanket Implementation ```python class MarkovBlanket: def __init__(self, internal_dim: int, blanket_dim: int, external_dim: int): """Initialize Markov blanket. Args: internal_dim: Internal state dimension blanket_dim: Blanket state dimension external_dim: External state dimension """ self.internal_dim = internal_dim self.blanket_dim = blanket_dim self.external_dim = external_dim # Initialize states self.internal = torch.zeros(internal_dim) self.blanket = torch.zeros(blanket_dim) self.external = torch.zeros(external_dim) def update_internal(self, blanket_state: torch.Tensor) -> None: """Update internal states given blanket.""" self.internal = self._compute_internal_update( self.internal, blanket_state ) def update_blanket(self, internal_state: torch.Tensor, external_state: torch.Tensor) -> None: """Update blanket states.""" self.blanket = self._compute_blanket_update( internal_state, self.blanket, external_state ) def _compute_internal_update(self, internal: torch.Tensor, blanket: torch.Tensor) -> torch.Tensor: """Compute internal state update.""" # Implement specific update rule return internal def _compute_blanket_update(self, internal: torch.Tensor, blanket: torch.Tensor, external: torch.Tensor) -> torch.Tensor: """Compute blanket state update.""" # Implement specific update rule return blanket ``` ### MCMC Sampling ```python class MCMCSampler: def __init__(self, target_distribution: Callable, proposal_distribution: Callable): """Initialize MCMC sampler. Args: target_distribution: Target distribution proposal_distribution: Proposal distribution """ self.target = target_distribution self.proposal = proposal_distribution def sample(self, initial_state: torch.Tensor, n_samples: int) -> torch.Tensor: """Generate samples using MCMC. Args: initial_state: Initial state n_samples: Number of samples Returns: samples: Generated samples """ current_state = initial_state samples = [current_state] for _ in range(n_samples): # Propose new state proposal = self.proposal(current_state) # Compute acceptance ratio ratio = self.target(proposal) / self.target(current_state) # Accept/reject if torch.rand(1) < ratio: current_state = proposal samples.append(current_state) return torch.stack(samples) ``` ## Advanced Applications ### 1. Biological Systems ```mermaid graph TD subgraph "Biological Self-Organization" A[Free Energy Minimization] --> B[Homeostasis] A --> C[Autopoiesis] A --> D[Adaptive Behavior] B --> E[Metabolic Regulation] C --> F[Self-maintenance] D --> G[Learning and Evolution] end style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style D fill:#bfb,stroke:#333 ``` ### 2. Cognitive Systems ```mermaid graph LR subgraph "Cognitive Architecture" A[Sensory Input] --> B[Prediction Error] B --> C[Belief Update] C --> D[Action Selection] D --> E[Environmental Change] E --> A C --> F[Model Update] F --> B end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style D fill:#bfb,stroke:#333 ``` ### 3. Artificial Systems ```mermaid graph TD subgraph "AI Applications" A[Free Energy Principle] --> B[Neural Networks] A --> C[Robotics] A --> D[Reinforcement Learning] B --> E[Deep Learning] C --> F[Control Systems] D --> G[Decision Making] end style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style D fill:#bfb,stroke:#333 ``` ## Future Directions ### Emerging Research Areas 1. **Quantum Extensions** - Quantum Free Energy - Quantum Information Geometry - Quantum Active Inference 1. **Complex Systems** - Collective Behavior - Network Dynamics - Emergence and Criticality 1. **Machine Learning** - Deep Free Energy Models - Variational Autoencoders - Generative Models ### Open Problems 1. **Theoretical Challenges** - Exact vs. Approximate Inference - Non-equilibrium Dynamics - Scale Separation 1. **Practical Challenges** - Computational Tractability - Model Selection - Real-world Applications ## Criticisms and Limitations ### Technical Critiques 1. **Mathematical Tractability**: Exact free energy minimization is often intractable in complex models. 1. **Linearity Assumptions**: Many implementations rely on linear or weakly nonlinear approximations. 1. **Discrete vs. Continuous Time**: Reconciling discrete and continuous time formulations remains challenging. 1. **Identifiability Issues**: Different generative models can yield identical behavioral predictions. ### Conceptual Critiques 1. **Circularity Concerns**: Critics argue the theory risks being unfalsifiable by explaining everything. 1. **Missing Mechanistic Details**: The abstract nature can obscure specific neural mechanisms. 1. **Teleological Interpretations**: Confusion between "as if" optimization and goal-directed behavior. 1. **Scale Problems**: Applying the principle across vastly different scales (from cells to societies) raises questions. ### Empirical Challenges 1. **Experimental Design**: Testing FEP directly is challenging due to its mathematical abstraction. 1. **Parameter Identification**: Identifying model parameters from empirical data can be difficult. 1. **Competing Explanations**: Alternative theories sometimes provide simpler explanations for the same phenomena. ## Recent Extensions and Developments ### Non-equilibrium Steady States Recent formulations cast the FEP in terms of non-equilibrium steady states using stochastic differential equations: ```math dx = f(x)dt + \sigma(x)dW ``` where $x$ represents the system state, $f(x)$ is a drift function, $\sigma(x)$ is a diffusion coefficient, and $dW$ is a Wiener process. ### Particular Physics The "particular physics" formulation grounds the FEP in specific physical principles: ```math \begin{aligned} \dot{p}(x,t) &= -\nabla \cdot (f(x)p(x,t)) + \frac{1}{2}\nabla^2 \cdot (D(x)p(x,t)) \\ &= -\nabla \cdot (f(x)p(x,t) - \nabla \cdot (D(x)p(x,t))) \end{aligned} ``` where $p(x,t)$ is a probability density function, $f(x)$ is a flow field, and $D(x)$ is a diffusion tensor. ### Symmetry Breaking Recent work connects the FEP to symmetry breaking in physics: ```math \mathcal{L}(g \cdot x) = \mathcal{L}(x) \quad \forall g \in G ``` where $\mathcal{L}$ is a Lagrangian, $g$ is a group element, and $G$ is a group of transformations. The emergence of Markov blankets is viewed as a form of symmetry breaking. ### Dynamic Causal Modeling Extensions to Dynamic Causal Modeling (DCM) allow empirical testing of FEP predictions: ```math \dot{x} = f(x,u,\theta) + w ``` where $u$ are inputs, $\theta$ are parameters, and $w$ is noise. DCM enables system identification from time series data. ## Philosophical Implications ### Epistemology The FEP has profound implications for epistemology: 1. **Knowledge as Model**: Suggests knowledge is embodied in probabilistic models. 1. **Bayesian Brain**: Positions the brain as a Bayesian inference machine. 1. **Umwelt**: Each organism constructs its own model-based reality. ### Metaphysics Several metaphysical consequences follow from the FEP: 1. **Mind-Body Problem**: Offers a naturalistic account of mind as inference. 1. **Causality**: Reframes causation in terms of generative models. 1. **Emergence**: Provides a mathematical account of how higher-order properties emerge. ### Ethics and Aesthetics The FEP has been extended to consider: 1. **Value**: Framing value in terms of expected free energy minimization. 1. **Aesthetics**: Suggesting beauty might relate to successful prediction. 1. **Social Dynamics**: Modeling social cohesion as shared free energy minimization. ## Theoretical Results ### Fisher Information and Cramér-Rao Bound The FEP connects to fundamental results in information theory: ```math \text{Var}(\hat{\theta}) \geq \frac{1}{\mathcal{I}(\theta)} ``` where $\mathcal{I}(\theta)$ is the Fisher information. This relates to the precision with which parameters can be estimated. ### Markov Blanket Conditions For a system with states partitioned into internal states $\mu$ and external states $\eta$, the Markov blanket $b$ must satisfy: ```math p(\mu | \eta, b) = p(\mu | b) ``` This condition ensures that internal states are conditionally independent of external states, given the blanket states. ### Path Integral Formulation The path of least action in state space can be formulated as: ```math q^*(x_{0:T}) = \frac{1}{Z}p(x_0)\exp\left(-\int_0^T \mathcal{L}(x, \dot{x})dt\right) ``` where $\mathcal{L}(x, \dot{x})$ is a Lagrangian and $Z$ is a normalizing constant. ### Marginal Stability Systems at non-equilibrium steady states exhibit marginal stability: ```math \lambda_{\max}(J) \approx 0 ``` where $\lambda_{\max}(J)$ is the maximum eigenvalue of the Jacobian matrix of the dynamics. This relates to critical slowing near bifurcation points. ## Best Practices ### Model Design 1. Choose appropriate state space dimensionality 1. Structure hierarchical dependencies 1. Design informative priors 1. Balance model complexity 1. Test with simulated data ### Implementation 1. Handle numerical stability issues 1. Choose appropriate approximation methods 1. Validate with synthetic data 1. Benchmark against standard methods 1. Implement efficient matrix operations ```python def compute_fisher_metric(model, q_samples, n_samples=1000): """Compute Fisher information metric for a model. Args: model: Probabilistic model q_samples: Parameter samples n_samples: Number of samples Returns: fisher_metric: Fisher information matrix """ n_params = q_samples.shape[1] fisher_metric = np.zeros((n_params, n_params)) # Compute score function for each sample scores = np.zeros((n_samples, n_params)) for i in range(n_samples): params = q_samples[i] # Compute gradients of log probability for j in range(n_params): # Finite difference approximation eps = 1e-5 params_plus = params.copy() params_plus[j] += eps params_minus = params.copy() params_minus[j] -= eps log_p_plus = model.log_prob(params_plus) log_p_minus = model.log_prob(params_minus) scores[i, j] = (log_p_plus - log_p_minus) / (2 * eps) # Compute outer product of scores for i in range(n_samples): outer_product = np.outer(scores[i], scores[i]) fisher_metric += outer_product fisher_metric /= n_samples return fisher_metric ``` ### Validation 1. Check convergence of optimization 1. Analyze sensitivity to hyperparameters 1. Compare with alternative models 1. Test edge cases 1. Evaluate generalization performance ## Related Documentation - [[active_inference]] - [[variational_inference]] - [[bayesian_brain_hypothesis]] - [[markov_blanket]] - [[predictive_coding]] - [[information_theory]] - [[statistical_physics]] - [[self_organization]] - [[autopoiesis]] ## Quantum Extensions ### Quantum Free Energy Principle The quantum extension of FEP uses quantum probability theory: ```math \begin{aligned} \text{Quantum State:} \quad |\psi\rangle &= \sum_s \sqrt{q(s)}|s\rangle \\ \text{Density Matrix:} \quad \rho &= |\psi\rangle\langle\psi| \\ \text{Quantum Free Energy:} \quad F_Q &= \text{Tr}(\rho H) + \text{Tr}(\rho \ln \rho) \end{aligned} ``` ### Quantum Information Geometry ```math \begin{aligned} \text{Quantum Fisher Information:} \quad g_{ij}^Q &= \text{Re}(\text{Tr}(\rho L_i L_j)) \\ \text{SLD Operators:} \quad L_i &= 2\frac{\partial \rho}{\partial \theta^i} \\ \text{Quantum Relative Entropy:} \quad S(\rho\|\sigma) &= \text{Tr}(\rho(\ln \rho - \ln \sigma)) \end{aligned} ``` ### Implementation Framework ```python class QuantumFreeEnergy: def __init__(self, hilbert_dim: int, hamiltonian: np.ndarray): """Initialize quantum free energy system. Args: hilbert_dim: Dimension of Hilbert space hamiltonian: System Hamiltonian """ self.dim = hilbert_dim self.H = hamiltonian def compute_quantum_free_energy(self, state_vector: np.ndarray) -> float: """Compute quantum free energy. Args: state_vector: Quantum state vector Returns: F_Q: Quantum free energy """ # Compute density matrix rho = np.outer(state_vector, state_vector.conj()) # Energy term energy = np.trace(rho @ self.H) # von Neumann entropy entropy = -np.trace(rho @ np.log(rho)) return energy + entropy ``` ## Advanced Mathematical Structures ### Category Theory Framework ```mermaid graph TD subgraph "Categorical FEP" A[State Space Category] --> B[Observable Category] B --> C[Measurement Category] C --> D[Inference Category] D --> A E[Free Energy Functor] --> F[Natural Transformation] F --> G[Adjoint Functors] end style A fill:#f9f,stroke:#333 style E fill:#bbf,stroke:#333 style G fill:#bfb,stroke:#333 ``` ### Differential Forms and Symplectic Structure ```math \begin{aligned} \text{Symplectic Form:} \quad \omega &= \sum_i dp_i \wedge dq_i \\ \text{Hamiltonian Vector Field:} \quad X_H &= \omega^{-1}(dH) \\ \text{Liouville Form:} \quad \theta &= \sum_i p_i dq_i \end{aligned} ``` ### Lie Group Actions ```math \begin{aligned} \text{Group Action:} \quad \Phi: G \times M &\to M \\ \text{Momentum Map:} \quad J: T^*M &\to \mathfrak{g}^* \\ \text{Coadjoint Orbit:} \quad \mathcal{O}_\mu &= \{Ad^*_g\mu : g \in G\} \end{aligned} ``` ## Advanced Implementation Frameworks ### Deep Free Energy Networks ```python class DeepFreeEnergyNetwork(nn.Module): def __init__(self, state_dim: int, obs_dim: int, latent_dim: int): """Initialize deep free energy network.""" super().__init__() # Recognition network self.recognition = nn.Sequential( nn.Linear(obs_dim, 256), nn.ReLU(), nn.Linear(256, 2 * latent_dim) # Mean and log variance ) # Generative network self.generative = nn.Sequential( nn.Linear(latent_dim, 256), nn.ReLU(), nn.Linear(256, obs_dim) ) # State transition network self.dynamics = nn.Sequential( nn.Linear(latent_dim, 256), nn.ReLU(), nn.Linear(256, latent_dim) ) def encode(self, x: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]: """Encode observations to latent space.""" h = self.recognition(x) mu, log_var = torch.chunk(h, 2, dim=-1) return mu, log_var def decode(self, z: torch.Tensor) -> torch.Tensor: """Decode latent states to observations.""" return self.generative(z) def transition(self, z: torch.Tensor) -> torch.Tensor: """Predict next latent state.""" return self.dynamics(z) def compute_free_energy(self, x: torch.Tensor, z: torch.Tensor, mu: torch.Tensor, log_var: torch.Tensor) -> torch.Tensor: """Compute variational free energy.""" # Reconstruction term x_recon = self.decode(z) recon_loss = F.mse_loss(x_recon, x, reduction='none') # KL divergence term kl_div = -0.5 * torch.sum( 1 + log_var - mu.pow(2) - log_var.exp(), dim=1 ) return recon_loss.sum(dim=1) + kl_div ``` ### Markov Blanket Analysis ```python class MarkovBlanketAnalyzer: def __init__(self, n_variables: int, threshold: float = 0.1): """Initialize Markov blanket analyzer.""" self.n_vars = n_variables self.threshold = threshold def identify_blanket(self, data: np.ndarray) -> Dict[str, np.ndarray]: """Identify Markov blanket structure. Args: data: Time series data (samples × variables) Returns: partitions: Dictionary of partitioned variables """ # Compute correlation matrix corr = np.corrcoef(data.T) # Threshold to get adjacency matrix adj = (np.abs(corr) > self.threshold).astype(int) # Find strongly connected components components = self._find_components(adj) # Identify internal, blanket, and external states internal = self._identify_internal(components) blanket = self._identify_blanket(adj, internal) external = self._identify_external(internal, blanket) return { 'internal': internal, 'blanket': blanket, 'external': external } ``` ## Advanced Applications ### 1. Biological Systems ```mermaid graph TD subgraph "Biological Free Energy" A[Cellular Dynamics] --> B[Metabolic Networks] B --> C[Gene Regulation] C --> D[Protein Synthesis] D --> E[Cellular State] E --> A end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style E fill:#bfb,stroke:#333 ``` ### 2. Neural Systems ```mermaid graph LR subgraph "Neural Free Energy" A[Sensory Input] --> B[Prediction Errors] B --> C[State Estimation] C --> D[Parameter Updates] D --> E[Synaptic Weights] E --> F[Neural Activity] F --> A end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style E fill:#bfb,stroke:#333 ``` ### 3. Social Systems ```mermaid graph TD subgraph "Social Free Energy" A[Individual Beliefs] --> B[Social Interactions] B --> C[Cultural Norms] C --> D[Collective Behavior] D --> E[Environmental Change] E --> A end style A fill:#f9f,stroke:#333 style C fill:#bbf,stroke:#333 style D fill:#bfb,stroke:#333 ``` ## Computational Complexity Analysis ### Time Complexity 1. **Belief Updates**: $O(d^3)$ for $d$-dimensional state space 1. **Free Energy Computation**: $O(d^2)$ for matrix operations 1. **Markov Blanket Identification**: $O(n^2)$ for $n$ variables ### Space Complexity 1. **State Representation**: $O(d)$ for state vectors 1. **Covariance Matrices**: $O(d^2)$ for precision matrices 1. **Model Parameters**: $O(d^2)$ for transition matrices ### Optimization Methods 1. **Gradient Descent**: $O(kd^2)$ for $k$ iterations 1. **Natural Gradient**: $O(kd^3)$ with Fisher information 1. **Variational EM**: $O(kd^3)$ for full covariance ## Best Practices ### Model Design 1. Choose appropriate state space dimensionality 1. Design informative priors 1. Structure hierarchical dependencies 1. Balance model complexity 1. Implement efficient numerics ### Implementation 1. Use stable matrix operations 1. Handle numerical precision 1. Implement parallel computation 1. Monitor convergence 1. Validate results ### Validation 1. Test with synthetic data 1. Compare with baselines 1. Cross-validate results 1. Analyze sensitivity 1. Benchmark performance ## Future Research Directions ### Theoretical Developments 1. **Quantum Extensions**: Quantum probability theory 1. **Category Theory**: Functorial relationships 1. **Information Geometry**: Wasserstein metrics ### Practical Advances 1. **Scalable Algorithms**: Large-scale systems 1. **Neural Implementation**: Brain-inspired architectures 1. **Real-world Applications**: Complex systems ### Open Problems 1. **Non-equilibrium Dynamics**: Far-from-equilibrium systems 1. **Scale Separation**: Multiple time scales 1. **Emergence**: Collective phenomena ## Markov Blanket Theory ### Formal Definition ```math \begin{aligned} & \text{Markov Blanket Partition:} \\ & \mathcal{S} = \{η, s, a, μ\} \\ & \text{where:} \\ & η: \text{external states} \\ & s: \text{sensory states} \\ & a: \text{active states} \\ & μ: \text{internal states} \end{aligned} ``` ### Conditional Independence ```mermaid graph TD subgraph "Markov Blanket Structure" E[External States η] --> S[Sensory States s] S --> I[Internal States μ] I --> A[Active States a] A --> E S --> A end style E fill:#f9f,stroke:#333 style S fill:#bbf,stroke:#333 style I fill:#bfb,stroke:#333 style A fill:#fbb,stroke:#333 ``` ### Statistical Properties ```math \begin{aligned} & \text{Conditional Independence:} \\ & P(η|s,a,μ) = P(η|s,a) \\ & P(μ|s,a,η) = P(μ|s,a) \\ & \text{Flow:} \\ & \dot{x} = f(x) + ω \\ & \text{where } x = [η,s,a,μ]^T \end{aligned} ``` ## Technical Derivations ### Free Energy Decomposition ```math \begin{aligned} F &= \mathbb{E}_q[\log q(s) - \log p(o,s)] \\ &= \mathbb{E}_q[\log q(s) - \log p(s|o) - \log p(o)] \\ &= D_{KL}[q(s)||p(s|o)] - \log p(o) \\ &= \underbrace{D_{KL}[q(s)||p(s|o)]}_{\text{accuracy}} + \underbrace{\mathbb{E}_q[\log q(s)]}_{\text{complexity}} \end{aligned} ``` ### Variational Principle ```mermaid graph TD subgraph "Free Energy Minimization" F[Free Energy] --> A[Accuracy Term] F --> C[Complexity Term] A --> P[Prediction Error] C --> R[Model Complexity] P --> M[Model Update] R --> M M --> F end style F fill:#f9f,stroke:#333 style M fill:#bfb,stroke:#333 ``` ### Path Integral Formulation ```math \begin{aligned} & \text{Action:} \\ & S[q] = \int_0^T \mathcal{L}(q,\dot{q})dt \\ & \text{Lagrangian:} \\ & \mathcal{L}(q,\dot{q}) = F[q] + \frac{1}{2}\dot{q}^T\Gamma\dot{q} \\ & \text{Hamilton's Equations:} \\ & \dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q} \end{aligned} ``` ## Mathematical Foundations ### Information Geometry ```math \begin{aligned} & \text{Fisher Metric:} \\ & g_{ij}(θ) = \mathbb{E}_p\left[\frac{\partial \log p}{\partial θ^i}\frac{\partial \log p}{\partial θ^j}\right] \\ & \text{Natural Gradient:} \\ & \dot{θ} = -g^{ij}\frac{\partial F}{\partial θ^j} \\ & \text{Geodesic Flow:} \\ & \ddot{θ}^k + \Gamma^k_{ij}\dot{θ}^i\dot{θ}^j = 0 \end{aligned} ``` ### Differential Geometry ```mermaid graph TD subgraph "Geometric Structure" M[Statistical Manifold] --> T[Tangent Bundle] M --> C[Connection] T --> G[Geodesics] C --> P[Parallel Transport] G --> F[Free Energy Flow] P --> F end style M fill:#f9f,stroke:#333 style F fill:#bfb,stroke:#333 ``` ### Stochastic Processes ```math \begin{aligned} & \text{Langevin Dynamics:} \\ & dx = f(x)dt + σdW \\ & \text{Fokker-Planck Equation:} \\ & \frac{\partial p}{\partial t} = -\nabla \cdot (fp) + \frac{1}{2}\nabla^2(Dp) \\ & \text{Path Integral:} \\ & p(x_T|x_0) = \int \mathcal{D}[x]e^{-S[x]} \end{aligned} ``` ```python def compute_fisher_metric(model, q_samples, n_samples=1000): """Compute Fisher information metric for a model. Args: model: Probabilistic model q_samples: Parameter samples n_samples: Number of samples Returns: fisher_metric: Fisher information matrix """ n_params = q_samples.shape[1] fisher_metric = np.zeros((n_params, n_params)) # Compute score function for each sample scores = np.zeros((n_samples, n_params)) for i in range(n_samples): params = q_samples[i] # Compute gradients of log probability for j in range(n_params): # Finite difference approximation eps = 1e-5 params_plus = params.copy() params_plus[j] += eps params_minus = params.copy() params_minus[j] -= eps log_p_plus = model.log_prob(params_plus) log_p_minus = model.log_prob(params_minus) scores[i, j] = (log_p_plus - log_p_minus) / (2 * eps) # Compute outer product of scores for i in range(n_samples): outer_product = np.outer(scores[i], scores[i]) fisher_metric += outer_product fisher_metric /= n_samples return fisher_metric ``` ### Validation 1. Check convergence of optimization 1. Analyze sensitivity to hyperparameters 1. Compare with alternative models 1. Test edge cases 1. Evaluate generalization performance ## Non-Equilibrium Statistical Mechanics Extensions ### Theoretical Framework **Definition** (Non-Equilibrium Steady State): A system state where currents flow but macroscopic properties remain constant: $\langle\dot{X}\rangle = 0, \quad \text{but} \quad \langle J_i \rangle \neq 0$ where $J_i$ are probability currents and $X$ represents macroscopic observables. **Theorem** (Fluctuation-Dissipation Relations): For systems near equilibrium, the response to perturbations is related to spontaneous fluctuations: $\chi(\omega) = \frac{1}{k_B T} \int_0^\infty \langle\delta X(t)\delta X(0)\rangle e^{-i\omega t} dt$ ```python class NonEquilibriumFreeEnergy: """Non-equilibrium extensions of the free energy principle.""" def __init__(self, temperature: float, dissipation_rate: float, external_driving: Callable[[float], np.ndarray]): """Initialize non-equilibrium system. Args: temperature: System temperature (energy scale) dissipation_rate: Rate of energy dissipation external_driving: Time-dependent external forces """ self.kT = temperature self.gamma = dissipation_rate self.external_force = external_driving def compute_entropy_production_rate(self, state: np.ndarray, probability_current: np.ndarray) -> float: """Compute entropy production rate for non-equilibrium system. The entropy production rate quantifies irreversibility: dS/dt = ∫ J(x) · ∇ ln P(x) dx ≥ 0 Args: state: Current system state probability_current: Probability current J(x) Returns: Entropy production rate """ # Compute probability gradient prob_gradient = self._compute_probability_gradient(state) # Entropy production as current-force product entropy_production = np.sum(probability_current * prob_gradient / self.kT) return max(0, entropy_production) # Non-negative by second law def compute_nonequilibrium_free_energy(self, beliefs: np.ndarray, time: float) -> Dict[str, float]: """Compute free energy including non-equilibrium contributions. F_neq = F_eq + F_driving + F_dissipation where: - F_eq is equilibrium free energy - F_driving accounts for external driving - F_dissipation accounts for dissipative processes Args: beliefs: Current belief distribution time: Current time Returns: Complete non-equilibrium free energy decomposition """ # Equilibrium component F_eq = self._compute_equilibrium_free_energy(beliefs) # Driving contribution external_work = self._compute_driving_contribution(beliefs, time) # Dissipation contribution dissipation_cost = self._compute_dissipation_cost(beliefs) # Non-equilibrium correction terms correlation_correction = self._compute_correlation_correction(beliefs) total_free_energy = F_eq + external_work + dissipation_cost + correlation_correction return { 'total_nonequilibrium_free_energy': total_free_energy, 'equilibrium_component': F_eq, 'driving_contribution': external_work, 'dissipation_cost': dissipation_cost, 'correlation_correction': correlation_correction, 'entropy_production_rate': self.compute_entropy_production_rate( beliefs, self._compute_probability_current(beliefs)) } def _compute_probability_gradient(self, state: np.ndarray) -> np.ndarray: """Compute gradient of log probability.""" # Finite difference approximation h = 1e-6 gradient = np.zeros_like(state) for i in range(len(state)): state_plus = state.copy() state_minus = state.copy() state_plus[i] += h state_minus[i] -= h log_prob_plus = np.log(self._probability_density(state_plus) + 1e-15) log_prob_minus = np.log(self._probability_density(state_minus) + 1e-15) gradient[i] = (log_prob_plus - log_prob_minus) / (2 * h) return gradient def _compute_probability_current(self, beliefs: np.ndarray) -> np.ndarray: """Compute probability current in belief space.""" # Simplified current computation # Full implementation would require solving continuity equation return np.gradient(beliefs) def _probability_density(self, state: np.ndarray) -> float: """Compute probability density at given state.""" # Simplified Gaussian density return np.exp(-0.5 * np.sum(state**2) / self.kT) def compute_fluctuation_dissipation_relation(self, time_series: np.ndarray, perturbation_response: np.ndarray, frequencies: np.ndarray) -> Dict[str, np.ndarray]: """Verify fluctuation-dissipation relation. Tests whether: χ(ω) = (1/kT) ∫ ⟨δX(t)δX(0)⟩ e^(-iωt) dt Args: time_series: Time series of system fluctuations perturbation_response: Response to small perturbations frequencies: Frequency range for analysis Returns: Fluctuation-dissipation analysis """ # Compute autocorrelation function autocorr = self._compute_autocorrelation(time_series) # Fourier transform to get susceptibility susceptibility_theory = np.fft.fft(autocorr) / self.kT # Compare with measured response susceptibility_measured = np.fft.fft(perturbation_response) # Compute violation measure fdr_violation = np.abs(susceptibility_theory - susceptibility_measured) return { 'theoretical_susceptibility': susceptibility_theory, 'measured_susceptibility': susceptibility_measured, 'fdr_violation': fdr_violation, 'violation_magnitude': np.mean(fdr_violation), 'frequencies': frequencies } def _compute_autocorrelation(self, time_series: np.ndarray) -> np.ndarray: """Compute autocorrelation function.""" n = len(time_series) autocorr = np.correlate(time_series, time_series, mode='full') return autocorr[n-1:] # Take positive lags only ### Quantum Generalizations **Definition** (Quantum Free Energy): For quantum systems with density matrix $\rho$: $F_Q = \text{Tr}[\rho H] - T S_Q[\rho]$ where $S_Q[\rho] = -\text{Tr}[\rho \ln \rho]$ is the von Neumann entropy. ```python class QuantumFreeEnergyPrinciple: """Quantum mechanical extensions of the free energy principle.""" def __init__(self, hilbert_space_dim: int, hamiltonian: np.ndarray, temperature: float = 1.0, hbar: float = 1.0): """Initialize quantum free energy system. Args: hilbert_space_dim: Dimension of Hilbert space hamiltonian: System Hamiltonian matrix temperature: Temperature parameter hbar: Reduced Planck constant (set to 1 in natural units) """ self.dim = hilbert_space_dim self.H = hamiltonian self.kT = temperature self.hbar = hbar # Validate Hamiltonian if not self._is_hermitian(hamiltonian): raise ValueError("Hamiltonian must be Hermitian") def compute_quantum_free_energy(self, density_matrix: np.ndarray) -> Dict[str, float]: """Compute quantum free energy F = ⟨H⟩ - T S_Q. Args: density_matrix: Quantum state density matrix ρ Returns: Quantum free energy components """ # Validate density matrix if not self._is_valid_density_matrix(density_matrix): raise ValueError("Invalid density matrix") # Compute internal energy internal_energy = np.real(np.trace(density_matrix @ self.H)) # Compute von Neumann entropy eigenvals = np.linalg.eigvals(density_matrix) eigenvals = eigenvals[eigenvals > 1e-15] # Remove zero eigenvalues von_neumann_entropy = -np.sum(eigenvals * np.log(eigenvals)) # Free energy free_energy = internal_energy - self.kT * von_neumann_entropy return { 'quantum_free_energy': free_energy, 'internal_energy': internal_energy, 'von_neumann_entropy': von_neumann_entropy, 'thermal_energy': self.kT * von_neumann_entropy } def quantum_variational_principle(self, trial_density_matrix: np.ndarray, target_hamiltonian: np.ndarray) -> Dict[str, Any]: """Apply variational principle to quantum free energy. Minimizes F[ρ] = Tr[ρH] - T S[ρ] over density matrices ρ. Args: trial_density_matrix: Trial quantum state target_hamiltonian: Target Hamiltonian Returns: Variational optimization results """ def objective(rho_vec: np.ndarray) -> float: """Objective function for optimization.""" # Reshape vector to matrix rho = rho_vec.reshape(self.dim, self.dim) # Ensure Hermiticity and trace normalization rho = 0.5 * (rho + rho.conj().T) rho = rho / np.trace(rho) # Compute free energy result = self.compute_quantum_free_energy(rho) return result['quantum_free_energy'] # Initial guess - thermal state beta = 1.0 / self.kT thermal_state = scipy.linalg.expm(-beta * target_hamiltonian) thermal_state = thermal_state / np.trace(thermal_state) # Optimization from scipy.optimize import minimize result = minimize( objective, thermal_state.flatten(), method='L-BFGS-B' ) optimal_rho = result.x.reshape(self.dim, self.dim) optimal_rho = 0.5 * (optimal_rho + optimal_rho.conj().T) optimal_rho = optimal_rho / np.trace(optimal_rho) return { 'optimal_density_matrix': optimal_rho, 'optimal_free_energy': result.fun, 'convergence_info': result, 'thermal_state_comparison': self.compute_quantum_free_energy(thermal_state) } def quantum_master_equation(self, initial_state: np.ndarray, lindblad_operators: List[np.ndarray], time_span: Tuple[float, float], num_points: int = 100) -> Dict[str, np.ndarray]: """Solve quantum master equation for open system dynamics. dρ/dt = -i/ℏ [H,ρ] + ∑_k (L_k ρ L_k† - ½{L_k†L_k, ρ}) Args: initial_state: Initial density matrix lindblad_operators: List of Lindblad jump operators time_span: Time interval for evolution num_points: Number of time points Returns: Time evolution of quantum state and free energy """ def lindblad_equation(t: float, rho_vec: np.ndarray) -> np.ndarray: """Lindblad master equation in vectorized form.""" rho = rho_vec.reshape(self.dim, self.dim) # Hamiltonian evolution drho_dt = -1j/self.hbar * (self.H @ rho - rho @ self.H) # Lindblad dissipation terms for L in lindblad_operators: drho_dt += (L @ rho @ L.conj().T - 0.5 * (L.conj().T @ L @ rho + rho @ L.conj().T @ L)) return drho_dt.flatten() # Time evolution times = np.linspace(time_span[0], time_span[1], num_points) from scipy.integrate import solve_ivp solution = solve_ivp( lindblad_equation, time_span, initial_state.flatten(), t_eval=times, method='RK45' ) # Extract density matrices and compute free energies density_matrices = [] free_energies = [] entropies = [] for i in range(len(times)): rho = solution.y[:, i].reshape(self.dim, self.dim) density_matrices.append(rho) fe_result = self.compute_quantum_free_energy(rho) free_energies.append(fe_result['quantum_free_energy']) entropies.append(fe_result['von_neumann_entropy']) return { 'times': times, 'density_matrices': density_matrices, 'free_energies': np.array(free_energies), 'entropies': np.array(entropies), 'solution': solution } def quantum_information_geometry(self, density_matrix: np.ndarray) -> Dict[str, np.ndarray]: """Compute quantum information geometric quantities. Args: density_matrix: Quantum state density matrix Returns: Information geometric measures """ # Quantum Fisher information matrix quantum_fisher = self._compute_quantum_fisher_information(density_matrix) # Bures metric bures_metric = 0.25 * quantum_fisher # Quantum relative entropy (quantum KL divergence) thermal_state = self._thermal_state() quantum_relative_entropy = self._quantum_relative_entropy( density_matrix, thermal_state) return { 'quantum_fisher_information': quantum_fisher, 'bures_metric': bures_metric, 'quantum_relative_entropy': quantum_relative_entropy, 'eigenvalues': np.linalg.eigvals(density_matrix), 'purity': np.real(np.trace(density_matrix @ density_matrix)) } def _is_hermitian(self, matrix: np.ndarray, tol: float = 1e-10) -> bool: """Check if matrix is Hermitian.""" return np.allclose(matrix, matrix.conj().T, atol=tol) def _is_valid_density_matrix(self, rho: np.ndarray, tol: float = 1e-10) -> bool: """Validate density matrix properties.""" # Check Hermiticity if not self._is_hermitian(rho, tol): return False # Check trace normalization if not np.allclose(np.trace(rho), 1.0, atol=tol): return False # Check positive semidefiniteness eigenvals = np.linalg.eigvals(rho) if np.any(eigenvals < -tol): return False return True def _thermal_state(self) -> np.ndarray: """Compute thermal equilibrium state.""" beta = 1.0 / self.kT thermal = scipy.linalg.expm(-beta * self.H) return thermal / np.trace(thermal) def _compute_quantum_fisher_information(self, rho: np.ndarray) -> np.ndarray: """Compute quantum Fisher information matrix.""" # Simplified computation for demonstration # Full implementation requires more sophisticated methods eigenvals, eigenvecs = np.linalg.eig(rho) # Remove zero eigenvalues mask = eigenvals > 1e-15 eigenvals = eigenvals[mask] eigenvecs = eigenvecs[:, mask] # Compute Fisher information (simplified) fisher = np.zeros((self.dim, self.dim)) for i, lam in enumerate(eigenvals): if lam > 1e-15: fisher += (1.0 / lam) * np.outer(eigenvecs[:, i], eigenvecs[:, i].conj()) return fisher def _quantum_relative_entropy(self, rho: np.ndarray, sigma: np.ndarray) -> float: """Compute quantum relative entropy S(ρ‖σ).""" # S(ρ‖σ) = Tr[ρ(ln ρ - ln σ)] # Eigendecomposition eigenvals_rho, eigenvecs_rho = np.linalg.eig(rho) eigenvals_sigma, eigenvecs_sigma = np.linalg.eig(sigma) # Remove zero eigenvalues eigenvals_rho = eigenvals_rho[eigenvals_rho > 1e-15] eigenvals_sigma = eigenvals_sigma[eigenvals_sigma > 1e-15] # Compute matrix logarithms ln_rho = eigenvecs_rho @ np.diag(np.log(eigenvals_rho)) @ eigenvecs_rho.conj().T ln_sigma = eigenvecs_sigma @ np.diag(np.log(eigenvals_sigma)) @ eigenvecs_sigma.conj().T return np.real(np.trace(rho @ (ln_rho - ln_sigma))) ### Stochastic Thermodynamics Integration class StochasticThermodynamics: """Stochastic thermodynamics framework for free energy principle.""" def __init__(self, temperature: float, protocol: Callable[[float], np.ndarray]): """Initialize stochastic thermodynamics analysis. Args: temperature: System temperature protocol: Time-dependent protocol λ(t) """ self.kT = temperature self.protocol = protocol def compute_stochastic_work(self, trajectory: np.ndarray, times: np.ndarray) -> Dict[str, float]: """Compute work along stochastic trajectory. W = ∫ (∂H/∂λ)(λ(t)) dλ/dt dt Args: trajectory: Stochastic trajectory x(t) times: Time points Returns: Work statistics """ work_increments = [] for i in range(len(times) - 1): dt = times[i+1] - times[i] # Protocol change lambda_t = self.protocol(times[i]) lambda_next = self.protocol(times[i+1]) dlambda_dt = (lambda_next - lambda_t) / dt # Work increment (simplified) dW = self._force_on_protocol(trajectory[i], lambda_t) * dlambda_dt * dt work_increments.append(dW) total_work = np.sum(work_increments) work_variance = np.var(work_increments) * len(work_increments) return { 'total_work': total_work, 'work_variance': work_variance, 'work_increments': np.array(work_increments) } def jarzynski_equality_verification(self, work_samples: np.ndarray) -> Dict[str, float]: """Verify Jarzynski equality: ⟨e^(-βW)⟩ = e^(-βΔF). Args: work_samples: Work values from multiple realizations Returns: Jarzynski equality analysis """ beta = 1.0 / self.kT # Compute exponential average exp_work = np.exp(-beta * work_samples) jarzynski_estimator = np.mean(exp_work) # Free energy difference estimate delta_F_estimate = -self.kT * np.log(jarzynski_estimator) # Theoretical free energy difference (would be computed separately) delta_F_theory = 0.0 # Placeholder # Verification metric verification_error = abs(delta_F_estimate - delta_F_theory) return { 'jarzynski_estimator': jarzynski_estimator, 'delta_F_estimate': delta_F_estimate, 'delta_F_theory': delta_F_theory, 'verification_error': verification_error, 'work_statistics': { 'mean_work': np.mean(work_samples), 'work_std': np.std(work_samples), 'min_work': np.min(work_samples), 'max_work': np.max(work_samples) } } def _force_on_protocol(self, state: np.ndarray, lambda_val: float) -> float: """Compute force exerted on protocol parameter.""" # Simplified force computation return np.sum(state * lambda_val) # Example usage and validation def validate_advanced_extensions(): """Validate advanced theoretical extensions.""" # Non-equilibrium system neq_system = NonEquilibriumFreeEnergy( temperature=1.0, dissipation_rate=0.1, external_driving=lambda t: np.array([np.sin(t)]) ) beliefs = np.array([0.3, 0.4, 0.3]) neq_result = neq_system.compute_nonequilibrium_free_energy(beliefs, 1.0) print("Non-equilibrium free energy:", neq_result['total_nonequilibrium_free_energy']) # Quantum system dim = 3 H = np.array([[1.0, 0.1, 0], [0.1, 2.0, 0.1], [0, 0.1, 3.0]]) quantum_system = QuantumFreeEnergyPrinciple( hilbert_space_dim=dim, hamiltonian=H, temperature=1.0 ) # Random density matrix rho = np.random.random((dim, dim)) + 1j * np.random.random((dim, dim)) rho = 0.5 * (rho + rho.conj().T) rho = rho / np.trace(rho) quantum_result = quantum_system.compute_quantum_free_energy(rho) print("Quantum free energy:", quantum_result['quantum_free_energy']) if __name__ == "__main__": validate_advanced_extensions() ```text ## Related Documentation - [[active_inference]] - [[variational_inference]] - [[bayesian_brain_hypothesis]] - [[markov_blanket]] - [[predictive_coding]] - [[information_theory]] - [[statistical_physics]] - [[self_organization]] - [[autopoiesis]]