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Infinite Ergodic Theory for Heterogeneous Diffusion Processes
Statistical Mechanics · Submitted 8 Aug 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system.
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The abstract, sentence by sentence
We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as ${D(x)}\sim |x-\tilde{x}|^{2-2/\alpha}$ in the vicinity of a point $\tilde{x}$, where $\alpha$ can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables, are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation; It\^o, Stratonovich, or H\"anggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. | contribution 0.93 | 0.90 | 0.38 | asserted 1.00 |
| s1 | We concentrate on a spatially dependent diffusion coefficient that behaves as ${D(x)}\sim |x-\tilde{x}|^{2-2/\alpha}$ in the vicinity of a point $\tilde{x}$, where $\alpha$ can be either positive or negative. | method 0.59 | 0.34 | 0.06 | asserted 0.96 |
| s2 | We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. | result 0.99 | 0.94 | 0.34 | asserted 1.00 |
| s3 | For processes under investigation, the time averages of a wide class of observables, are obtained using an ensemble average with respect to the nonnormalized density. | result 0.97 | 0.89 | 0.36 | asserted 1.00 |
| s4 | A Langevin equation which involves multiplicative noise may take different interpretation; It\^o, Stratonovich, or H\"anggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$. | result 0.68 | 0.45 | 0.13 | hedged possibility 0.72 |
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The paper
Original abstract
We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as ${D(x)}\sim |x-\tilde{x}|^{2-2/\alpha}$ in the vicinity of a point $\tilde{x}$, where $\alpha$ can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables, are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation; It\^o, Stratonovich, or H\"anggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$.
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How the literature uses it
2 arXiv papers cite this in 4 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 1 paper
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This way looks similar to the treatment of a Langevin equation with multiplicative noise $D(x)$ in non-Stratonovich sense (e.g., Itô or Hänggi-Klimontovich sense) in a recent paper [ref] .
Ergodic properties of heterogeneous diffusion processes in a potential well · Summary and conclusions · 0.94
Cites it as background 1 paper
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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
What it cites · 2 sentences without a confident label
Compares with 1 paper
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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] .
From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory · Infinite Ergodic Theory · 1.00
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