Infinite ergodic theory meets Boltzmann statistics

Abstract

We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.

arXiv comment21 pages (including Refs), 18 figures

References in corpus (30)

• Active Brownian Particles. From Individual to Collective Stochastic Dynamics
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Holographic optical trapping
• Infinite Ergodic Theory for Heterogeneous Diffusion Processes
• Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem
• The Single Big Jump Principle in Physical Modelling
• Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
• Pesin-Type Identity for Weak Chaos
• Bulk-mediated diffusion on a planar surface: full solution
• Infinite densities for Lévy walks
• Solution of the Fokker-Planck equation with a logarithmic potential
• Superdiffusive motion of membrane-targeting C2 domains
• From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory
• Diffusing Up the Hill: Dynamics and Equipartition in Highly Unstable Systems
• First-passage and first-exit times of a Bessel-like stochastic process
• Aging generates regular motions in weakly chaotic systems
• Anomalous diffusion in random dynamical systems
• Transport in disordered systems: the single big jump approach
• Strange kinetics of bulk-mediated diffusion on lipid bilayers
• Ergodic properties of heterogeneous diffusion processes in a potential well
• Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos
• Large-deviations for spatial diffusion of cold atoms
• Multi-species dynamical density functional theory for microswimmers: derivation, orientational ordering, trapping potentials, and shear cells
• Infinite invariant density in a semi-Markov process with continuous state variables
• Distributional behavior of time averages of non-$L^1$ observables in one-dimensional intermittent maps with infinite invariant measures
• Infinite density for cold atoms in shallow optical lattices
• Statistics of occupation time and connection to local properties of non-homogeneous random walks
• Living on the edge of instability
• The probability distribution of Brownian motion in periodic potentials
• Fluctuations of ring polymers

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• Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory
• Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling
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• Short-time expansion of one-dimensional Fokker-Planck equations with heterogeneous diffusion
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• Transition to anomalous dynamics in a simple random map
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• Non-normalizable quasi-equilibrium states under fractional dynamics
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