Lack-of-fit reduction in the path-integral formalism

TL;DR

This work reformulates lack-of-fit reduction in non-equilibrium thermodynamics through the Onsager-Machlup variational principle, using a MaxEnt-based generalized Kullback–Leibler discrepancy to quantify the distance between detailed and reduced dynamics. The reduced evolution naturally obtains a GENERIC form consisting of a reversible Hamiltonian part and a dissipative gradient flow, with the dissipation encoded by an extremal action that vanishes at equilibrium. A path-integral perspective introduces stochastic fluctuations in the reduced dynamics, yielding Langevin-type equations consistent with large-deviation principles. The methodology is demonstrated on the Kac–Zwanzig model, where dissipation emerges from purely Hamiltonian dynamics, and on diffusion arising from the Vlasov equation with and without interactions, highlighting the role of the reduced state variables in achieving accurate reductions. The framework provides a systematic, parameter-free coarse-graining route for non-equilibrium systems with potential extensions to boundary problems and continuum thermodynamics.

Abstract

We present a new formulation of the lack-of-fit reduction in non-equilibrium thermodynamics using the path-integral formalism. The formulation is based on the Onsager-Machlup variational principle, and it allows us to find reduced dynamical equations by minimizing information discrepancy with respect to the detailed evolution. The reduced evolution consists of a Hamiltonian vector field and a gradient flow. The reduction method is illustrated on the Kac-Zwanzig model, where we show how irreversibility emerges from purely Hamiltonian evolution by ignoring some degrees of freedom. We also show how to generalize the Fisher information matrix and Kullback-Leibler divergence between two probability distributions to the case when the two distributions are related by the principle of maximum entropy, even in the case when the entropy is not of Boltzmann-Gibbs type (for instance Tsallis-Havrda-Charvat entropy).

Figures (4)

• Figure 1: The lack-of-fit discrepancy is measured as the difference between the detailed vector field $\dot{\mathbf{x}}$ in the vicinity of the MaxEnt submanifold and the MaxEnt image of the reduced vector field $\dot{\mathbf{y}}^*$.
• Figure 2: Comparison of the deterministic (dotted) and stochastic (full) results for two choices of the reduced state variables ( $\mathbf{Q},\mathbf{P}, s$ (red) \ref{['eqn:y_evolutionKZ']} and $\mathbf{Q},\mathbf{P}, s,\Psi_{mi}$ (yellow) \ref{['eq.extra']}) and two different initial positions of the big particle (left: $Q_0=100$, right: $Q_0 = 1000$). The reduced dynamics corresponds to the non-stationary solution of the Hamilton-Jacobi (or Riccati) equation, where the action $\Sigma_e(t,\mathbf{y}^*)$ depends explicitly on time. Parameters of the detailed simulation (blue) were $N=10000$, $\gamma=1.0$, $\alpha = 1.0$, $\omega_i$ sampled from uniformly from the interval $(\sqrt{1.0E-05}, \sqrt{1.0E-01})$, and $M$ was set equal to the total mass of the small particles. In pink is the solution given by the projection operator method \ref{['eq.Q.Markov']} that serves as a benchmark for the detailed simulation.
• Figure 3: Comparison of the deterministic and stochastic results for the initial condition $Q_0 = 1000$ and $\omega_i$ sampled from the uniform distribution $\omega_i^2$ between $10^{-5}$ and $10^{-1}$. The projection operator method gives zero dissipation \ref{['eq.gamma1']}. The only difference between this simulation and that in Figure \ref{['fig:StochasticKleeman']} is the distribution of $\omega_i$.
• Figure 4: Comparison of the reversible evolution generated by the solution of the Hamilton-Jacobi (Riccati) $\Sigma(y,t)$ equation for different choices of the reduced state variables – for two state variables $\mathbf{Q}$, $\mathbf{P}$ (green), three with additional $s$ (red) and four variables with the extra variables $\Psi_{mi}$ (yellow) and $\Psi_\Sigma$ (purple) \ref{['eq.extra']}. Stationary solutions are dashed, non-stationary are dotted lines. The initial positions of the big particle is $Q_0=1000$ and all other reduced variables initiate at $0$. Parameters of the simulation were $N=10000$, $\gamma=1.0$, $\alpha = 1.0$, $\omega_i$ sampled from uniformly from the interval $(\sqrt{1.0E-05}, \sqrt{1.0E-01})$, and $M$ was set equal to the total mass of the small particles.