{"paper":{"title":"A new discrete calculus of variations and its applications in statistical physics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["physics.chem-ph"],"primary_cat":"physics.gen-ph","authors_text":"Q. H. Liu","submitted_at":"2021-04-21T08:16:48Z","abstract_excerpt":"For a discrete function $f\\left( x\\right) $ on a discrete set, the finite difference can be either forward and backward. However, we observe that if $ f\\left( x\\right) $ is a sum of two functions $f\\left( x\\right) =f_{1}\\left( x\\right) +f_{2}\\left( x\\right) $ defined on the discrete set, the first order difference of $\\Delta f\\left( x\\right) $ is equivocal for we may have $ \\Delta ^{f}f_{1}\\left( x\\right) +\\Delta ^{b}f_{2}\\left( x\\right) $ where $ \\Delta ^{f}$ and $\\Delta ^{b}$ denotes the forward and backward difference respectively. Thus, the first order variation equation for this function "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.11075","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2104.11075/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}