Geometric approach to Hamiltonian dynamics and statistical mechanics

Abstract

This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of freedom of interest for statistical mechanics. The first part of the paper concerns the applications of methods used in classical differential geometry to study the chaotic dynamics of Hamiltonian systems. Starting from the identity between the trajectories of a dynamical system and the geodesics in its configuration space, a geometric theory of chaotic dynamics can be developed, which sheds new light on the origin of chaos in Hamiltonian systems. In fact, it appears that chaos can be induced not only by negative curvatures, as was originally surmised, but also by positive curvatures, provided the curvatures are fluctuating along the geodesics. In the case of a system with a large number of degrees of freedom it is possible to give an analytical estimate of the largest Lyapunov exponent by means of a geometric model independent of the dynamics. In the second part of the paper the phenomenon of phase transitions is addressed and it is here that topology comes into play. In fact, when a system undergoes a phase transition, the fluctuations of the configuration-space curvature exhibit a singular behavior at the phase transition point, which can be qualitatively reproduced using geometric models. In these models the origin of the singular behavior of the curvature fluctuations appears to be caused by a topological transition in configuration space. This leads us to put forward a Topological Hypothesis (TH). The content of the TH is that phase transitions would be related at a deeper level to a change in the topology of the configuration space of the system.

REVTeX, 81 pages, 36 ps/eps figures (some low-quality figures to save space); review article submitted to Physics Reports

References in corpus (26)

• Dynamical ensembles in stationary states
• Lyapunov instability and finite size effects in a system with long-range forces
• Riemannian theory of Hamiltonian chaos and Lyapunov exponents
• Geometry of dynamics, Lyapunov exponents and phase transitions
• A Geometric, Dynamical Approach to Thermodynamics
• Analytic estimation of Lyapunov exponent in a mean-field model undergoing a phase transition
• Chaos and Statistical Mechanics in the Hamiltonian Mean Field Model
• Hamiltonian reformulation and pairing of Lyapunov exponents for Nose-Hoover dynamics
• Geometry of dynamics and phase transitions in classical lattice phi^4 theories
• Topology and phase transitions: a paradigmatic evidence
• Note on Phase Space Contraction and Entropy Production in Thermostatted Hamiltonian Systems
• Lyapunov spectral analysis of a nonequilibrium Ising-like transition
• Universal Behavior of Lyapunov Exponents in Unstable Systems
• Topological aspects of geometrical signatures of phase transitions
• Hamiltonian dynamics of the two-dimensional lattice phi^4 model
• Topological origin of the phase transition in a mean-field model
• Curvature fluctuations and Lyapunov exponent at Melting
• Hamiltonian dynamics and geometry of phase transitions in classical XY models
• Slow Relaxation at Critical Point of Second Order Phase Transition in a Highly Chaotic Hamiltonian System
• Lyapunov exponents from geodesic spread in configuration space
• Geometric Interpretation of Chaos in Two-Dimensional Hamiltonian Systems
• Geometric dynamical observables in rare gas crystals
• Dynamics and statistics of simple models with infinite-range attractive interaction
• Ergodicity properties of energy conserving single spin flip dynamics in the XY model
• Analytic Lyapunov exponents in a classical nonlinear field equation
• Geometric approach to chaos in the classical dynamics of abelian lattice gauge theory

Cited by in corpus (128)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Phase transitions and configuration space topology
• A Theorem on the origin of Phase Transitions
• Collective modes, chaotic behavior and self-trapping in the dynamics of three coupled Bose-Einstein condensates
• Persistent Homology analysis of Phase Transitions
• Topological Signature of First Order Phase Transitions
• Free motion around black holes with discs or rings: between integrability and chaos - II
• Phase transitions and topology changes in configuration space
• Free motion around black holes with discs or rings: between integrability and chaos - I
• Phase transitions detached from stationary points of the energy landscape
• Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates
• Wavepacket dynamics in energy space of a chaotic trimeric Bose-Hubbard system
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Topology and Phase Transitions II. Theorem on a necessary relation
• Topology and Phase Transitions I. Preliminary Results
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• Weak and strong chaos in FPU models and beyond
• Topological Origin of the Phase Transition in a Model of DNA Denaturation
• Does the brain behave like a (complex) network? I. Dynamics
• Dynamical Glass and Ergodization Times in Classical Josephson Junction Chains
• Chaos in a classical limit of the Sachdev-Ye-Kitaev model
• Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics
• Stepwise structure of Lyapunov spectra for many particle systems by a random matrix dynamics
• Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Phase transitions as topology changes in configuration space: an exact result
• On the inequivalence of statistical ensembles
• Lyapunov exponents as a dynamical indicator of a phase transition
• On the mean-field spherical model
• Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach
• The topological hypothesis on phase transitions: the simplest case
• Chaos suppression in the large size limit for long-range systems
• Dynamical behavior and Jacobi stability analysis of wound strings
• Exploring the energy landscape of XY models
• Topological properties of the mean field phi^4 model
• Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators
• Weak Chaos and the "Melting Transition" in a Confined Microplasma System
• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems
• Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications
• Remarks about the thermodynamics of astrophysical systems in mutual interaction and related notions
• Energy landscape and phase transitions in the self-gravitating ring model
• Curvature of the energy landscape and folding of model proteins
• Topology, phase transitions and the spherical model
• Eisenhart--Duval lift for minisuperspace quantum cosmology
• Theoretical estimates for the largest Lyapunov exponent of many-particle systems
• Universal semiclassical equations based on the quantum metric
• On the relationship between phase transitions and topological changes in one dimensional models
• Topological approach to phase transitions and inequivalence of statistical ensembles
• Chaotic properties of spin lattices near second-order phase transitions
• Topological Theory of Phase Transitions
• Complexity of Pure and Mixed Qubit Geodesic Paths on Curved Manifolds
• Stationary point approach to the phase transition of the classical XY chain with power-law interactions
• On a microcanonical relation between continuous and discrete spin models
• Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background
• Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions
• Regular and chaotic motion in softened gravitational systems
• On the clustering phase transition in self-gravitating N-body systems
• Largest Lyapunov exponent of long-range XY systems
• Theoretical investigations of an information geometric approach to complexity
• $N$-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited
• Topological conditions for discrete symmetry breaking and phase transitions
• Finslerian geometrization of quantum mechanics in the hydrodynamical representation
• Collective relaxation of stellar systems revisited
• Geometrical aspects in the analysis of microcanonical phase-transitions
• Ricci curvature, isoperimetry and a non-additive entropy
• Hopping dynamics for localized Lyapunov vectors in many-hard-disk systems
• Phase transitions in simplified models with long-range interactions
• Dynamics and geometric properties of the k-Trigonometric model
• A geometric interpretation of integrable motions
• Third quantization for scalar and spinor wave functions of the Universe in an extended minisuperspace
• The Geometric Theory of Phase Transitions
• Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem
• Geometry and stability of dynamical systems
• Geometry of the energy landscape and folding transition in a simple model of a protein
• Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems
• Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric
• Mode-locking transitions in nano-structured weakly disordered lasers
• Generalized Lyapunov exponents of the random harmonic oscillator: cumulant expansion approach
• On Integrability of the Geodesic Deviation Equation
• Geometric Approach to Lyapunov Analysis in Hamiltonian Dynamics
• Spinorial Wheeler-DeWitt wave functions inside black hole horizons
• Hamiltonian dynamics of homopolymer chain models
• Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model
• Phase transitions and topology in 2+k XY mean-field models
• What Can Algebraic Topology and Differential Geometry Teach Us About Intrinsic Dynamics and Global Behavior of Robots?
• Coarsening of two dimensional XY model with Hamiltonian dynamics: Logarithmically divergent vortex mobility
• Topology of configuration space of the mean-field phi^4 model by Morse theory
• Information Geometric Complexity of Entropic Motion on Curved Statistical Manifolds under Different Metrizations of Probability Spaces
• Lyapunov exponent of many-particle systems: testing the stochastic approach
• Density of states of continuous and discrete spin models: a case study
• Geometry of the energy landscape of the self-gravitating ring
• Chaotic Dynamics of One-Dimensional Systems with Periodic Boundary Conditions
• Necessary and sufficient conditions for ${\mathbb{Z}}_2$-symmetry-breaking phase transitions
• A simple topological model with continuous phase transition
• Different faces of chaos in FRW models with scalar fields -- geometrical point of view
• Macroscopic detection of the strong stochasticity threshold in Fermi-Pasta-Ulam chains of oscillators
• Thermodynamic properties of the trigonometric Rosen-Morse potential and applications to a quantum gas of mesons
• Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces
• Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems
• Schrodinger dynamics as a two-phase conserved flow: an alternative trajectory construction of quantum propagation
• When topology triggers a phase transition
• Jacobi Metric and Morse Theory of Dynamical Systems
• Complexity Characterization in a Probabilistic Approach to Dynamical Systems Through Information Geometry and Inductive Inference
• Moduli of curve families and (quasi-)conformality of power-law entropies
• Density of states of the $XY$ model: an energy landscape approach
• Geometric approach to chaos in the classical dynamics of abelian lattice gauge theory
• Boltzmann-Gibbs thermal equilibrium distribution for classical systems and Newton law: A computational discussion
• How many is different? Answer from ideal Bose gas
• Phase transitions triggered by dumbbell equipotential hypersurfaces
• Condensation phenomena in gravity
• Critical Exponent for the Lyapunov Exponent and Phase Transitions -- The Generalized Hamiltonian Mean-Field Model
• Lyapunov exponent of the random frequency oscillator: cumulant expansion approach
• Constrained dynamics of maximally entangled bipartite system
• Chaos in violent relaxation dynamics. Disentangling micro- and macro-chaos in numerical experiments of dissipationless collapse
• Long range order in gravity
• Magnetic Reversal Time in Open Long Range Systems
• The Topological Non-connectivity Threshold and magnetic phase transitions in classical anisotropic long-range interacting spin system
• Geometric investigation of chaos unfolding in Hamiltonian systems
• The largest Lyapunov exponent as a tool for detecting relative changes in the particle positions
• Fermionic quantum cosmology as a framework for resolving type IV singularities
• Toward a refining of the topological theory of phase transitions
• On the origin of Phase Transitions in the absence of Symmetry-Breaking
• An information geometric perspective on the complexity of macroscopic predictions arising from incomplete information
• Entropy and curvature variations from effective potentials
• Simplified energy landscape of the ${\varphi }^4$ model and the phase transition
• Phase Transitions, Chaos and Joint Action in the Life Space Foam
• Nonlinear Quantum Neuro-Psycho-Dynamics with Topological Phase Transitions
• Geodesic Motion on Closed Spaces: Two Numerical Examples