{"doi":"10.1103/physreva.31.1695","title":"Canonical dynamics: Equilibrium phase-space distributions","abstract":"Nos\\'e has modified Newtonian dynamics so as to reproduce both the canonical and the isothermal-isobaric probability densities in the phase space of an N-body system. He did this by scaling time (with s) and distance (with ${V}^{1/D}$ in D dimensions) through Lagrangian equations of motion. The dynamical equations describe the evolution of these two scaling variables and their two conjugate momenta ${p}_{s}$ and ${p}_{v}$. Here we develop a slightly different set of equations, free of time scaling. We find the dynamical steady-state probability density in an extended phase space with variables x, ${p}_{x}$, V, \\ensuremath{\\epsilon}\\ifmmode \\dot{}\\else \\.{}\\fi{}, and \\ensuremath{\\zeta}, where the x are reduced distances and the two variables \\ensuremath{\\epsilon}\\ifmmode \\dot{}\\else \\.{}\\fi{} and \\ensuremath{\\zeta} act as thermodynamic friction coefficients. We find that these friction coefficients have Gaussian distributions. From the distributions the extent of small-system non-Newtonian behavior can be estimated. We illustrate the dynamical equations by considering their application to the simplest possible case, a one-dimensional classical harmonic oscillator.","journal":"Physical Review A","year":1985,"id":9998,"datarank":23.78597585110013,"base_score":10.04524749757396,"endowment":10.04524749757396,"self_citation_contribution":1.5067871246360942,"citation_network_contribution":22.279188726464035,"self_endowment_contribution":1.5067871246360942,"citer_contribution":22.279188726464035,"corpus_percentile":99.1,"corpus_rank":230,"citation_count":23045,"citer_count":198,"citers_with_citation_signal":198,"citers_with_endowment":198,"datacite_reuse_total":0,"is_dataset":false,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"1985-03-01","authors":[{"id":85460,"name":"William G. Hoover","orcid":"0000-0002-1717-052X","position":0,"is_corresponding":true}],"reference_count":11,"raw_metadata":{"citation_network_status":"fetched"},"created_at":"2026-03-01T18:20:47.508186Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}