A mathematical method for irregular hamiltonian systems

Qiuping A. Wang; Wei Li

Home/Archive/arXiv

arXiv · cond-mat/0410219

A mathematical method for irregular hamiltonian systems

Qiuping A. Wang·Wei Li

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present certain mathematical aspects of an information method which was formulated in an attempt to investigate diffusion phenomena. We imagine a regular dynamical hamiltonian systems under the random perturbation of thermal (molecular) noise and chaotic motion. The random effect is taken into account via the uncertainty of irregular dynamic process produced in this way. This uncertainty due to different paths between two phase points is measured by a path information which is maximized in connection with the action defined originally for the unperturbed regular hamiltonian systems. The obtained transition probability depends exponentially on this action. The usefulness of this information method has been demonstrated by the derivation of diffusion laws without the usual assumptions. In this work, some essential mathematical aspects of this irregular dynamics is reviewed. It is emphasized that the classical action principle for single least action path is no more valid and the formalism of classical mechanics for regular hamiltonian systems is no more exact for irregular hamiltonian dynamics. There is violation of the fundamental laws of mechanics by randomly perturbed hamiltonian systems. However, the action principle is always present for the ensemble of paths through the average action. This average action principle leads to a formalism of stochastic mechanics in which, in spite of the violation of fundamental laws, the mathematical form of classical mechanics can be recovered by a consideration of the statistical averaging of the dynamics.

Explore related subjects

cond-mat.stat-mech

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Qiuping A. Wang, Wei Li. 2004-10-11. A mathematical method for irregular hamiltonian systems. https://arxiv.org/abs/cond-mat/0410219

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Universal sampling of spin systems across quenched disorder

Statistical physics extracts macroscopic laws by averaging over the many microscopic degrees of freedom of a system. Disordered systems demand a second and far harder average, one over the quenched randomness itself. The classic analytical routes, the replica and cavity methods, become uncontrolled outside mean-field or tree-like limits, and conventional numerical algorithms like parallel tempering require expensive, independent equilibration for every disorder realization. In this work, we introduce a universal neural variational framework that amortizes inference across the disorder ensemble, eliminating both the need for per-instance Markov chain equilibration and the cost of retraining instance-specific variational ansatzes. Built on an encoder-decoder Transformer architecture, after training once, it produces an explicit approximation to the Boltzmann distribution given previously unseen disorder realizations without further optimization. We validate this framework on 2D Edwards-Anderson models, and apply it to the random-bond Ising model, successfully capturing the Binder cumulant crossings near the Nishimori multicritical point. These results shift the object of variational inference from the single instance to the disorder ensemble, opening a route to frustrated many-body systems where instance-by-instance computation is prohibitive.

cond-mat.stat-mech↗

arXiv2026

Information-Theoretic Characterization of Macroscopic Chaos Emerging from the Chemical Master Equation

Open chemical reaction networks exhibit stochastic concentration dynamics at finite system sizes, whereas their macroscopic limit is governed by deterministic rate equations that can display chaos. In this Letter, we show theoretically that a rate of information loss constructed from two-time mutual information recovers the Kolmogorov-Sinai entropy in the deterministic limit. We verify this result through numerical simulations of a Markov jump process for a three-species system involving seven reactions.

cond-mat.stat-mech↗

arXiv2026

Orientational order on non-orientable domains

We study the statistical properties of passive and active many-body systems with orientational degrees of freedom on non-orientable domains. By rephrasing topological constraints as non-local symmetry relations on an orientable double-cover, we show that non-orientability eliminates global rotational soft modes without acting like an external field. In a passive XY model, this results in topological caging, where orientational fluctuations that exhibit conventional diffusive behavior on a torus saturate on a Klein bottle to a finite value that we compute exactly in the thermodynamic limit. In models of active self-propelled particles with orientational degrees of freedom, topological caging persists despite continuously changing interaction neighborhoods. In an active Ising spin model, non-orientability enforces the coexistence of ordered anti-parallel domains with vanishing global polar order, a state that is absent on orientable domains.

cond-mat.stat-mech↗