[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-179852-105":59,"doc-detail-179852-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","viscoelastic-shear-modulus-and-scaling-variables-for-phase-transitions","Viscoelastic Shear Modulus and Scaling Variables for Phase Transitions","","This document presents a detailed analysis of viscoelastic shear modulus and scaling variables, crucial for understanding phase transitions in various dimensions. It introduces a comprehensive set of variables, including Greek letters like μ (viscoelastic shear modulus) and δp (deviation from continuous rigidity transition pc), along with their descriptions and associated locations within equations. The document further defines several scaling variables, such as M, F, F2, and Fd, which are formed by combinations of frequency (w or f) and the deviation from continuous rigidity transition (δp). These variables are instrumental in characterizing the behavior of the system near critical points. Additionally, the document elucidates universal scaling functions for μ, the density of states (D), and the Green's function (G) across different spatial dimensions (d > 2, d = 2, and d \u003C 2). These functions provide insights into the universality of critical phenomena. A comparative table is provided, outlining the critical frequency (fCPA), critical exponent (ν), dynamical exponent (z), frequency scaling (θ ≡ ωτ), and critical exponent (γ) for different dimensions, highlighting how these parameters vary with dimensionality. The document appears to be a section from a research paper or a technical report focusing on theoretical physics, specifically statistical mechanics or condensed matter physics, concerning phase transitions and critical phenomena.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/viscoelastic-shear-modulus-and-scaling-variables-for-phase-transitions/179852/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/viscoelastic-shear-modulus-and-scaling-variables-for-phase-transitions/179852.png","ImageObject",300,407,{"name":92,"@type":93},"Evangeline","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-02",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does the variable μ represent in this document?","Question",{"text":113,"@type":114},"The variable μ represents the viscoelastic shear modulus, a fundamental property describing the material's resistance to shear deformation under time-dependent or dynamic conditions.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is the significance of the scaling variables F, F2, and Fd?",{"text":118,"@type":114},"The scaling variables F, F2, and Fd are designed to collapse experimental or theoretical data for different dimensions onto universal curves, facilitating the study of critical phenomena and the identification of universality classes near phase transitions.",{"name":120,"@type":111,"acceptedAnswer":121},"How do the critical parameters vary with spatial dimension?",{"text":122,"@type":114},"The document provides a table indicating that critical parameters such as the critical frequency (fCPA), critical exponent (ν), dynamical exponent (z), and others change depending on whether the spatial dimension is greater than 2 (d > 2), equal to 2 (d = 2), or less than 2 (d \u003C 2).","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},179852,1788338477,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":46,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":31,"preview_page_count":4},13056703019662,"https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188","| Variable | Description | Location |\n| --- | --- | --- |\n| μ | Viscoelastic shear modulus | – |\n| δp | p − pc: deviation from continuous rigidity transition pc | – |\n| w | ρω2 : (undamped); iγω: (overdamped) | – |\n| f | w/μ: frequency measured relative to shear stiffness | – |\n| M | (μ/μ0 )/|δp|: scaling variable formed between μ and δp | Eq. (26), Eq. (27), Eq. (A9) |\n| F | (f /f0 )/|δp|: scaling variable formed between f and δp in d > 2 | Eq. (26), Eq. (A19) |\n| F2 | Scaling variable formed between f and δp in d = 2 | Eq. (30), Eq. (A29) |\n| Fd | (f /f0d)/|δp|2/d : scaling variable formed between f and δp in d \u003C 2 | Eq. (27), Eq. (A34) |\n| U | Scaling variable for leading irrelevant correction to scaling in d > 2 | Eq. (26), Eq. (A20) |\n| Ud | Scaling variable for leading irrelevant correction to scaling in d \u003C 2 | Eq. (27), Eq. (A35) |\n| M | Universal scaling function for μ in d > 2 | implicitly defined by Eq. (26) |\n| M2 | Universal scaling function for μ in d = 2 | implicitly defined by Eq. (30) |\n| Md | Universal scaling function for μ in d \u003C 2 | implicitly defined by Eq. (27) |\n| D | Universal scaling function for the density of states in d > 2 | Eq. (A47) |\n| D2 | Universal scaling function for the density of states in d = 2 | Eq. (A49) |\n| Dd | Universal scaling function for the density of states in d \u003C 2 | Eq. (A48) |\n| G | Universal scaling function for the Green’s function in d > 2 | Eq. (A54) |\n| Gd | Universal scaling function for the Green’s function in d \u003C 2 | Eq. (A55) |\n\n|  | fCPA | ν | z | θ ≡ ων | γ |\n| --- | --- | --- | --- | --- | --- |\n| d > 2 | 1 | 1/2 | 2 | d/2 − 1 | 2 |\n| d \u003C 2 | 1 | 1/d | 1 + d/2 | 2/d − 1 | 1 + 2/d |","cbCaioWBl5BxTb0C","https://ap.wps.com/l/cbCaioWBl5BxTb0C","pdf",8534659,"English","# Scaling Variables and Functions\n## Universal Scaling Functions\n## Critical Parameters by Dimension","[{\"question\":\"What does the variable μ represent in this document?\",\"answer\":\"The variable μ represents the viscoelastic shear modulus, a fundamental property describing the material's resistance to shear deformation under time-dependent or dynamic conditions.\"},{\"question\":\"What is the significance of the scaling variables F, F2, and Fd?\",\"answer\":\"The scaling variables F, F2, and Fd are designed to collapse experimental or theoretical data for different dimensions onto universal curves, facilitating the study of critical phenomena and the identification of universality classes near phase transitions.\"},{\"question\":\"How do the critical parameters vary with spatial dimension?\",\"answer\":\"The document provides a table indicating that critical parameters such as the critical frequency (fCPA), critical exponent (ν), dynamical exponent (z), and others change depending on whether the spatial dimension is greater than 2 (d \\u003e 2), equal to 2 (d = 2), or less than 2 (d \\u003c 2).\"}]","Viscoelastic Shear Modulus and Scaling Variables for Phase Transitions | PDF"]