Information-Theoretic Limits of Time Distinguishability in Diffusion and Anomalous Diffusion
This repository accompanies the paper:
Information-Theoretic Limits of Time Distinguishability in Diffusion and Anomalous Diffusion Alexander Yashin Independent Researcher (Germany)
Motivation
In stochastic systems such as diffusion, time is typically treated as an external parameter rather than an observable quantity.
This work addresses a precise operational question:
What is the minimal time difference that can be distinguished from data generated by stochastic dynamics?
The answer is constrained not by metaphysical assumptions or new physics, but by information-theoretic limits inherent in statistical inference.
Core Idea
Time is treated as an inferred parameter of probability distributions, not as a primitive variable.
Temporal resolution is fundamentally limited by:
statistical sample size
measurement noise
the underlying dynamical law governing state evolution
This leads to sharp, estimator-independent bounds on temporal distinguishability.
Main Results
Normal diffusion
For Brownian motion in d dimensions, Fisher information yields a universal bound of the form:
Delta t_min(t) ~ t * sqrt(2 / (d * N))
where N is the effective number of independent samples.
Hypothesis testing equivalence
The same bound follows from a Kullback-Leibler divergence analysis, establishing local equivalence between:
estimation-theoretic limits (Cramer-Rao bound)
hypothesis-testing distinguishability
Photon-limited regime
In the experimentally relevant photon-limited setting, a self-consistent analysis predicts a nontrivial scaling:
Delta t_min proportional to Phi^(-1/3)
where Phi is the photon flux.
This scaling is analytically derived and numerically verified.
Anomalous diffusion
Subdiffusive and superdiffusive dynamics exhibit qualitatively distinct temporal resolution regimes governed by the anomalous exponent.
All results are:
falsifiable
estimator-independent
require no modification of known physical laws
Repository Structure
. |-- paper/ LaTeX source of the paper | |-- main.tex | |-- preamble.tex | |-- metadata.tex | |-- sections/ | |-- src/ Reproducible simulations | |-- simulate_crlb.py | |-- simulate_photon_limit.py | |-- utils.py | |-- reproducibility/ Notes on numerical verification |-- LICENSE |-- CITATION.cff |-- .zenodo.json
Reproducibility
All numerical claims are independently verified by Monte Carlo simulations.
To reproduce the key results:
cd src python simulate_crlb.py python simulate_photon_limit.py
Expected outcomes:
Empirical variances saturate the Cramer-Rao lower bound
Photon-limited scaling exponent is close to -1/3
Detailed acceptance criteria and reproducibility notes are provided in the reproducibility directory.
Status
Analytical derivations: complete Numerical verification: complete Experimental falsifiability: explicit Reproducibility: full
This repository is suitable for independent verification, citation, and extension.
License
Released under the MIT License.
Citation
If you use this work, please cite the accompanying paper. A machine-readable citation is provided in CITATION.cff.
Contact
Alexander Yashin Leipzig / Halle, Germany Email: alexander.yashin@yahoo.de