RECORD
From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory
Statistical Mechanics · Submitted 16 Apr 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as $1/x$, for example a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are: normalization, finite averaged energy and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as $1/x$, for example a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. | background 0.53 | 0.34 | 0.02 | asserted 0.92 |
| s1 | As time increases, our system approaches a non-normalizable Boltzmann state. | result 1.00 | 0.78 | 0.18 | asserted 1.00 |
| s2 | We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. | contribution 0.57 | 0.93 | 0.08 | asserted 0.99 |
| s3 | We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are: normalization, finite averaged energy and a mean-squared displacement which increases linearly with time. | contribution 0.57 | 0.95 | 0.18 | asserted 1.00 |
| s4 | Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity. | interpretation 0.90 | 0.89 | 0.75 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as $1/x$, for example a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are: normalization, finite averaged energy and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.
Human review
How the literature uses it
4 arXiv papers cite this in 7 sentences
Papers citing it · 3 sentences without a confident label
Builds on it 1 paper
-
It generalizes previous results for one-dimensional Brownian [ref] and run-and-tumble particles [ref] .
Universal Correlations in Local Measurements Directly Probe Effective Diffusivity · Introduction · 0.99
Uses its method or tool 1 paper
-
Derivation of Eq. [ref].—The derivation follows the eigenstate expansion approach devised for a one-dimensional Brownian particle [ref] .
Universal Correlations in Local Measurements Directly Probe Effective Diffusivity · Introduction · 0.95
Compares with it 1 paper
-
For Itô and Hänggi-Klimontovich interpretations the results, Eqs. [ref] [ref] with [ref], are similar to the ones found in diffusion in Logarithmic potential [ref] .
Infinite Ergodic Theory for Heterogeneous Diffusion Processes · Infinite Ergodic Theory · 1.00
Cites it as background 1 paper
-
Non-normalizable steady solutions have been previously discussed in the literature on physical issues, e.g., Lévy walk and laser-cooled atoms [ref] , the weakly-chaotic maps [ref] , nonlinear oscillators [ref] , the thermodynamics [ref] , and multiplicative noise [ref] .
Ergodic properties of heterogeneous diffusion processes in a potential well · Introduction · 0.82
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.