On the relaxation problem in statistical mechanics
Abstract
We reformulate the relaxation problem in statistical mechanics by making explicit what are the \emph{operational} objects subject to relaxation: the local time statistics of the recorded signal Z(t). These local time statistics are simply the estimated histograms of observations {Z(ti)}i=1M performed at uniformly random times {ti}i=1M by a clockless observer. The subject of prediction is a belief about a future fresh out-of-sample reading of a measurement outcome whose distribution is inferred from the mathematical model believed to be true. For finite bounded systems of N≥ 1 degrees of freedom global irreversible relaxation of predictions can occur but special initial conditions exist. The form of the predictions depends on certain loss functions whose choice is up to the particular observer. Finally, entropy is given a learning interpretation as mutual information between the observer and the unknown past of the system under consideration and, in complete generality, its stationary value depends on the information available.
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BibTeX
@misc{vecchio2026relaxation,
title = {{On the relaxation problem in statistical mechanics}},
author = {Giuseppe Del Vecchio Del Vecchio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30871},
eprint = {2608.30871},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30871},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the relaxation problem in statistical mechanics
AU - Giuseppe Del Vecchio Del Vecchio
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.30871
UR - https://arxiv.org/abs/2608.30871
AB - We reformulate the relaxation problem in statistical mechanics by making explicit what are the \emph{operational} objects subject to relaxation: the local time statistics of the recorded signal Z(t). These local time statistics are simply the estimated histograms of observations {Z(t_i)}_i=1^M performed at uniformly random times {t_i}_i=1^M by a clockless observer. The subject of prediction is a belief about a future fresh out-of-sample reading of a measurement outcome whose distribution is inferred from the mathematical model believed to be true. For finite bounded systems of N≥ 1 degrees of freedom global irreversible relaxation of predictions can occur but special initial conditions exist. The form of the predictions depends on certain loss functions whose choice is up to the particular observer. Finally, entropy is given a learning interpretation as mutual information between the observer and the unknown past of the system under consideration and, in complete generality, its stationary value depends on the information available.
ER -