{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:38:01Z","timestamp":1760402281220,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt\u2019s and Zerm\u00e9lo\u2019s paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt\u2019s) as well as periodic in the time (Zerm\u00e9lo\u2019s, illustrating Poincar\u00e9 recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals\u2019 information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.<\/jats:p>","DOI":"10.3390\/e24010078","type":"journal-article","created":{"date-parts":[[2022,1,3]],"date-time":"2022-01-03T22:51:50Z","timestamp":1641250310000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Nonequilibrium Time Reversibility with Maps and Walks"],"prefix":"10.3390","volume":"24","author":[{"given":"William Graham","family":"Hoover","sequence":"first","affiliation":[{"name":"Ruby Valley Research Institute, 601 Highway Contract 60, Ruby Valley, NV 89833, USA"}]},{"given":"Carol Griswold","family":"Hoover","sequence":"additional","affiliation":[{"name":"Ruby Valley Research Institute, 601 Highway Contract 60, Ruby Valley, NV 89833, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7434-5912","authenticated-orcid":false,"given":"Edward Ronald","family":"Smith","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, Brunel University London, Uxbridge UB8 3PH, UK"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1103\/PhysRevLett.59.10","article-title":"Resolution of Loschmidt\u2019s Paradox: The Origin of Irreversible Behavior in Reversible Atomistic Dynamics","volume":"59","author":"Holian","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1080\/08927028708080932","article-title":"Dissipative Irreversibility from Nos\u00e9\u2019s Reversible Mechanics","volume":"1","author":"Hoover","year":"1987","journal-title":"Mol. Simul."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1063\/1.166318","article-title":"Chaos and Irreversibility in Simple Model Systems","volume":"8","author":"Hoover","year":"1998","journal-title":"Chaos"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1063\/1.166324","article-title":"An Analytical Construction of the SRB Measures for Baker-Type Maps","volume":"8","author":"Tasaki","year":"1998","journal-title":"Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"153","DOI":"10.12921\/cmst.2019.0000045","article-title":"2020 Ian Snook Prize Problem: Three Routes to the Information Dimensions for One-Dimensional Stochastic Random Walks and Their Equivalent Two-Dimensional Baker Maps","volume":"25","author":"Hoover","year":"2019","journal-title":"Comput. Methods Sci. Technol."},{"key":"ref_6","unstructured":"Hoover, W.G., and Hoover, C.G. (2019). Random Walk Equivalence to the Compressible Baker Map and the Kaplan-Yorke Approximation to Its Information Dimension. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"016115","DOI":"10.1103\/PhysRevE.71.016115","article-title":"Irreversibility in a Simple Reversible Model","volume":"71","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hoover, W.G., and Hoover, C.G. (2012). Time Reversibility, Computer Simulation, Algorithms, Chaos, World Scientific. [2nd ed.].","DOI":"10.1142\/8344"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hoover, W.G., and Hoover, C.G. (2018). Microscopic and Macroscopic Simulation Techniques: Kharagpur Lectures, World Scientific.","DOI":"10.1142\/10777"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1063\/1.447334","article-title":"A Unified Formulation of the Constant Temperature Molecular Dynamics Methods","volume":"81","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1080\/00268978400101201","article-title":"A Molecular Dynamics Method for Simulations in the Canonical Ensemble","volume":"52","year":"1984","journal-title":"Mol. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1103\/PhysRevA.31.1695","article-title":"Canonical Dynamics. Equilibrium Phase-Space Distributions","volume":"31","author":"Hoover","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1304","DOI":"10.1515\/zna-1982-1117","article-title":"Information Dimension and the Probabilistic Structure of Chaos","volume":"37a","author":"Farmer","year":"1982","journal-title":"Z. Naturforschung"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1007\/BF01012932","article-title":"Is the Dimension of Chaotic Attractors Invariant Under Coordinate Changes?","volume":"36","author":"Ott","year":"1984","journal-title":"J. Statstical Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"070901","DOI":"10.1063\/5.0019038","article-title":"From Hard Spheres and Cubes to Nonequilibrium Maps with Thirty-some Years of Thermostatted Molecular Dynamics","volume":"153","author":"Hoover","year":"2020","journal-title":"J. Chem. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/1\/78\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:35:53Z","timestamp":1760362553000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/1\/78"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,1]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["e24010078"],"URL":"https:\/\/doi.org\/10.3390\/e24010078","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,1,1]]}}}