Dynamical Variational Principle

• Dynamical variational principle is a framework that defines system evolution via extremal solutions of well-defined variational problems in physics and dynamics.
• It unifies reversible and irreversible processes by incorporating holonomic and nonholonomic constraints to derive key equations like the Navier–Stokes and spectral variational formulas.
• The approach extends to complex settings—using tools like maximum caliber and Bayesian inference—to link microscopic trajectories with macroscopic observables in dynamical systems.

A dynamical variational principle is a general framework in mathematical physics and dynamical systems theory that characterizes the evolution, stationary distributions, or statistical properties of a system as (generalized) extremal solutions of well-defined variational problems. This concept extends the traditional action principles of classical and quantum mechanics, as well as equilibrium thermodynamic variational formulations, to encompass nonequilibrium, irreversible, stochastic, and operator-theoretic dynamical settings. Dynamical variational principles unify constraint enforcement, structural symmetries, and optimality in a single organizing framework across diverse physical and mathematical contexts, including irreversible fluid dynamics, transfer operators, statistical inference on trajectory spaces, and information-theoretic learning procedures.

1. Classical and Dissipative Fluid Dynamics

In continuum mechanics, the dynamical variational principle generalizes the least-action framework to encompass dissipative, irreversible, and viscoelastic phenomena. A central example is provided by Fukagawa & Fujitani’s formulation of viscous and viscoelastic fluid motion (Fukagawa et al., 2011):

• The action functional for a compressible one-component fluid is

$$S = \int dt \int d^3x~ \mathcal{L}(\rho, \mathbf{v}, s) + \text{(constraints)},$$

with Lagrangian density

$${\cal L} = \rho\left( \tfrac{1}{2} v^2 - \epsilon(\rho, s) \right),$$

where $\rho$ is mass density, $\mathbf{v}$ velocity, $s$ entropy per unit mass, and $\epsilon$ internal energy.

• For perfect fluids, holonomic constraints enforce mass and entropy conservation:

$$\rho J - \rho_{\text{init}} = 0, \qquad s - s_{\text{init}} = 0.$$

These admit Lagrange-multiplier treatment.

• For dissipative systems (viscous, viscoelastic), entropy is not conserved but instead satisfies a local balance involving stress and heat flux. The generalized nonholonomic constraint is imposed on the virtual work:

$$\int J\left[ \rho T \delta s + f_i \delta X_i \right] d^3a = 0,$$

where $f_i = \partial_j \sigma_{ij}$ is the force density from viscous stress.

• Varying the action with these constraints yields directly the Navier–Stokes equation for momentum balance:

$$\rho \left( \partial_t v_i + v_j \partial_j v_i \right) = -\partial_i p + \partial_j \sigma_{ij}.$$

For viscoelastic models, additional tensorial elastic degrees of freedom and conjugate forces enter, and the variational structure produces the corresponding Maxwell- or Oldroyd-type constitutive laws.

• The same formalism admits a control-theoretic (pre-Hamiltonian) repackaging, where the velocity field becomes an input and the state–costate structure is made explicit.

The key structural insight is that both reversible and irreversible contributions—pressure, viscous force, viscoelastic force—can be incorporated through suitable holonomic and nonholonomic constraints in one unified variational framework, obviating ad hoc Rayleigh dissipation functions or separate entropy production procedures (Fukagawa et al., 2011).

2. Dynamical Variational Principles for Transfer Operators

Antonevich–Bakhtin–Lebedev (Antonevich et al., 2008) established a general dynamical variational principle characterizing spectral properties (notably, the spectral radius) of transfer and weighted shift operators indexed on an underlying dynamical system:

• For a transfer operator $A$ on $C(X)$ over a compact space $X$ and automorphism $a:X\to X$, the so-called spectral potential for a weight $g$ is

$\phi(g)=\log r(A_g),$

where $A_g f = A(e^g f)$. This $\phi$ is convex and continuous.

• The dual variational formula expresses $\phi(g)$ as a Legendre dual over all $a$-invariant Borel measures $\mu$ using a new entropy-like invariant, the "t-entropy" $\tau_t(\mu)$:

$$\phi(g) = \max_{\mu\in M_a} \Bigl\{ \int g\,d\mu + \tau_t(\mu) \Bigr\}.$$

Here, $\tau_t(\mu)$ generalizes Kolmogorov-Sinai entropy and reduces to it in the Markovian or reversible limit.

• The proof relies on convex duality theory, approximation by partitions of unity, and construction of test functions that concentrate on empirical measure neighborhoods.
• In classical cases (Markov chain, two-sided shift), one recovers the classical entropy variational principle for pressure. In general, $\tau_t$ encodes subleading or non-KS entropy statistics.

This formulation conceptually places spectral data of non-selfadjoint operators (e.g., Ruelle-Perron–Frobenius type) within the broader landscape of dynamical variational calculus (Antonevich et al., 2008).

3. Path Entropy Maximization: Maximum Caliber

The maximum caliber ("MaxCal") principle is a time-global variational framework for inferring trajectory ensemble distributions in dynamical systems, directly paralleling maximum entropy in equilibrium statistical mechanics (Dixit et al., 2017):

• Let $p(\Gamma)$ be the probability assigned to each possible system trajectory (or path) $\Gamma$. The "Caliber" is the path entropy function

$$S[p] = -\sum_{\Gamma} p(\Gamma) \ln \frac{p(\Gamma)}{q(\Gamma)},$$

with $q(\Gamma)$ a reference measure.

• Constraints on dynamical observables (e.g., averages of $F_i[\Gamma]$) and normalization are enforced via Lagrange multipliers.
• The variational solution is

$$p(\Gamma) = \frac{q(\Gamma)}{Z} \exp \left[ -\sum_i \lambda_i F_i[\Gamma] \right],$$

with $Z$ the path-partition function.

• Near the equilibrium limit, MaxCal recovers the Green–Kubo relations, Onsager reciprocity, and the minimum entropy production principle.
• Far from equilibrium, MaxCal systematically generates Markov models, effective transport rates, and network inference algorithms by judicious constraint choice.
• The formalism’s inferential power and generality depend critically on proper selection of constraints and reference path measures (Dixit et al., 2017).

MaxCal thus unifies inference, fluctuation, and control principles in dynamical settings, providing systematic links between micro-level trajectories and macro-level observables.

4. Variational Principles in Random and Complex Dynamical Systems

The dynamical variational principle admits several extensions to random, bundle, and group-indexed systems, as highlighted in recent convex-analytic frameworks (Yang et al., 2022, Lian, 2024):

• For a random dynamical system (RDS) associated to a measurable bundle over a group action $(\Omega, \mathcal{F}, \mathbb{P}, G)$, the topological entropy or random pressure function $P:\mathcal{F}\to\mathbb{R}$ is characterized via duality:

$$P(\phi) = \sup_{\mu \in \mathcal{M}_P(\Omega \times X, G)} \{ s(\mu) + \int \phi\,d\mu \},$$

with $s(\mu)$ an entropy-like functional derived as the Legendre–Fenchel transform of $P$, and $\mathcal{M}_P$ the appropriate class of invariant measures.

• This variational structure persists even in non-invertible and non-amenable scenarios, as long as $P$ satisfies convexity and monotonicity axioms.
• The principle applies to maximal pattern entropy, polynomial entropy, mean dimension, and preimage entropies, recovering classical results in limits and allowing extension to zero/infinite-entropy and high-dimensional settings.
• Amenability, subadditivity, and abstract convex representation theory ensure robustness of the principle across random, group, and non-commutative structures.

These extensions demonstrate that dynamical variational principles provide a systematic and flexible organizing tool for entropy, pressure, and complexity invariants even in highly nonclassical dynamical regimes (Yang et al., 2022, Lian, 2024).

5. Bayesian Dynamical Variational Principles and Information Theory

The theory encompasses generalized Bayesian inference in dynamical and thermodynamic settings via IFS-based and transfer operator constructions (Lopes et al., 2022):

• Given an IFS (iterated function system) $\tau: \Theta \times Y \to Y$ and likelihood $\ell(\theta, y)$, the "posterior" is characterized as the solution to a variational problem:

$$\sup_{T~\text{holo.}} \left\{ \int [\log \ell(\theta, y) + \log T_a(\theta) - \log p(y)] dT + H(T) \right\},$$

where $H(T)$ is the relative entropy and $T$ is a holonomic coupling encoding stationarity under $\tau$.

• For dynamically stationary prior/posterior pairs, the solution is determined by the eigenvector (Gibbs measure) of the lifted Ruelle operator, linking Bayesian inference with thermodynamic pressure.
• This unifies classical Bayesian optimization (Zellner’s minimum information loss principle), dynamical Bayesian updating, and thermodynamic-formalism pressure into a single variational engine (Lopes et al., 2022).

Such structures highlight the deep connections between dynamical variational formulations and information-theoretic inference in both static and stochastic dynamical models.

6. Unified Perspective and Outlook

Dynamical variational principles provide a comprehensive blueprint for encoding physical laws, complexity measures, spectral exponents, and inference rules in a wide landscape of dynamical systems. Key common features include:

• Unified treatment of both reversible and irreversible phenomena via action-like or path-entropy functionals and holonomic/nonholonomic constraints (Fukagawa et al., 2011, Dixit et al., 2017).
• Duality relations expressing central invariants (spectral radius, entropy, pressure) as suprema or infima over invariant measures or couplings, with entropy-like functionals emerging as the Legendre duals (Antonevich et al., 2008, Yang et al., 2022, Lopes et al., 2022).
• Broad applicability to higher-order mechanics, stochastic/nonequilibrium processes, group actions, operator algebras, and statistical inference.
• Explicit technical constructs (t-entropy, MaxCal, holonomic couplings, Ruelle operators) that realize these principles in concrete analytical and computational settings.

Ongoing research leverages these structures for robustness diagnostics, multifractal analysis, system identification, and generalization to nonamenable and nonstationary random processes. The dynamical variational principle thus serves as a universal bridge between variational mechanics, thermodynamic formalism, operator theory, and statistical learning (Fukagawa et al., 2011, Antonevich et al., 2008, Dixit et al., 2017, Lopes et al., 2022, Yang et al., 2022, Lian, 2024).