535f9e942d55628e00dc39db34bbf51e pmc_10528319.pdf 85663bbff5b0fbc7fd0cbc9aa603a5c01ea471c1 pmc_10528319.pdf 60b25a2d8cf7b8eda9499e96da9389bbd936c258900823cdc5453e41ea9dd151 pmc_10528319.pdf Title: Non-Local Vectorial Internal Variables and Generalized Guyer-Krumhansl Evolution Equations for the Heat Flux Subject: In this paper, we ask ourselves how non-local effects affect the description of thermodynamic systems with internal variables. Usually, one assumes that the internal variables are local, but that their evolution equations are non-local, i.e., for instance, that their evolution equations contain non-local differential terms (gradients, Laplacians) or integral terms with memory kernels. In contrast to this typical situation, which has led to substantial progress in several fields, we ask ourselves whether in some cases it would be convenient to start from non-local internal variables with non-local evolution equations. We examine this point by considering three main lengths: the observation scale R defining the elementary volumes used in the description of the system, the mean free path l of the microscopic elements of the fluid (particles, phonons, photons, and molecules), and the overall characteristic size L of the global system. We illustrate these ideas by considering three-dimensional rigid heat conductors within the regime of phonon hydrodynamics in the presence of thermal vortices. In particular, we obtain a generalization of the Guyer–Krumhansl equation, which may be of interest for heat transport in nanosystems or in systems with small-scale inhomogeneities. Keywords: continuum thermodynamics; heat transport; classical irreversible thermodynamics; internal variables; phonon hydrodynamics Author: Liliana Restuccia and David Jou Creator: LaTeX with hyperref Producer: pdfTeX-1.40.21 CreationDate: Thu Aug 24 14:28:56 2023 CEST ModDate: Thu Aug 24 14:31:31 2023 CEST Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 21 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 431897 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- PNBMPR+URWPalladioL-Bold Type 1 Custom yes yes no 10 0 HTVOVV+URWPalladioL-Roma Type 1 Custom yes yes no 15 0 VRERLM+URWPalladioL-Ital Type 1 Custom yes yes no 19 0 WFMTKF+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 38 0 CDWAAS+CMSY10 Type 1 Builtin yes yes no 60 0 SOSTRQ+CMR10 Type 1 Builtin yes yes no 64 0 HZZRQG+PazoMath-Italic Type 1 Builtin yes yes no 71 0 YLPUFZ+CMEX10 Type 1 Builtin yes yes no 78 0 HVSBKL+PazoMath Type 1 Builtin yes yes no 82 0 XJNKAP+PazoMath-BoldItalic Type 1 Builtin yes yes no 86 0 YVTZRI+MSAM10 Type 1 Builtin yes yes no 90 0 SYFPBV+CMMI10 Type 1 Builtin yes yes no 100 0 FDDSXG+URWPalladioL-BoldItal Type 1 Custom yes yes no 110 0 PDXFNG+PazoMath-Bold Type 1 Builtin yes yes no 123 0 PWTQTX+CMBX10 Type 1 Builtin yes yes no 127 0 OMKIDU+CMBSY10 Type 1 Builtin yes yes no 135 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2025-03-12 02:11:17 CET RepresentationInformation: pmc_10528319.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2025-03-12 02:04:11 CET Size: 431897 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 544 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Destination: section.1 Item: Title: Some Remarks about the Internal Variables and Some Versions of Non-Equilibrium Thermodynamics Destination: section.2 Item: Title: Gyarmati Wave Approach: Fundamentals Destination: section.3 Item: Title: A Model Describing Special Thermal Motions of Phonons Destination: section.4 Item: Title: Phenomenological Equations Destination: section.5 Item: Title: Isotropic Case Destination: section.6 Item: Title: A Generalized Guyer-Krumhansl Heat Transfer Equation in Rigid Media Destination: section.7 Item: Title: Proposal I for the Internal Variable: = (J(q) + R2J(q)) Destination: section.8 Item: Title: Proposal II for the Internal Variable: = J(q) Destination: section.9 Item: Title: The Role of the Entropy Flux in Non-Local Formulations Destination: section.10 Item: Title: Conclusions Destination: section.11 Item: Title: Appendix A Destination: section.A. Item: Title: Appendix B Destination: section.B. Item: Title: References Destination: section.C. Info: Title: Non-Local Vectorial Internal Variables and Generalized Guyer-Krumhansl Evolution Equations for the Heat Flux Author: Liliana Restuccia and David Jou Subject: In this paper, we ask ourselves how non-local effects affect the description of thermodynamic systems with internal variables. Usually, one assumes that the internal variables are local, but that their evolution equations are non-local, i.e., for instance, that their evolution equations contain non-local differential terms (gradients, Laplacians) or integral terms with memory kernels. In contrast to this typical situation, which has led to substantial progress in several fields, we ask ourselves whether in some cases it would be convenient to start from non-local internal variables with non-local evolution equations. We examine this point by considering three main lengths: the observation scale R defining the elementary volumes used in the description of the system, the mean free path l of the microscopic elements of the fluid (particles, phonons, photons, and molecules), and the overall characteristic size L of the global system. We illustrate these ideas by considering three-dimensional rigid heat conductors within the regime of phonon hydrodynamics in the presence of thermal vortices. In particular, we obtain a generalization of the Guyer Krumhansl equation, which may be of interest for heat transport in nanosystems or in systems with small-scale inhomogeneities. Keywords: continuum thermodynamics; heat transport; classical irreversible thermodynamics; internal variables; phonon hydrodynamics Creator: LaTeX with hyperref Producer: pdfTeX-1.40.21 CreationDate: Thu Aug 24 14:28:56 CEST 2023 ModDate: Thu Aug 24 14:31:31 CEST 2023 ID: 0x10b5eecbfad954fb241ab16b35210be1, 0x10b5eecbfad954fb241ab16b35210be1 Filters: FilterPipeline: FlateDecode Fonts: Type1: Font: BaseFont: SOSTRQ+CMR10 FontSubset: true FirstChar: 40 LastChar: 93 FontDescriptor: FontName: SOSTRQ+CMR10 Flags: Symbolic FontBBox: -40, -250, 1009, 750 FontFile: true Font: BaseFont: SYFPBV+CMMI10 FontSubset: true FirstChar: 60 LastChar: 62 FontDescriptor: FontName: SYFPBV+CMMI10 Flags: Symbolic FontBBox: -32, -250, 1048, 750 FontFile: true Font: BaseFont: HZZRQG+PazoMath-Italic FontSubset: true FirstChar: 35 LastChar: 182 FontDescriptor: FontName: HZZRQG+PazoMath-Italic Flags: Symbolic FontBBox: -70, -277, 902, 733 FontFile: true Font: BaseFont: OMKIDU+CMBSY10 FontSubset: true FirstChar: 1 LastChar: 114 FontDescriptor: FontName: OMKIDU+CMBSY10 Flags: Symbolic FontBBox: -26, -940, 1332, 825 FontFile: true Font: BaseFont: PNBMPR+URWPalladioL-Bold FontSubset: true FirstChar: 3 LastChar: 122 FontDescriptor: FontName: PNBMPR+URWPalladioL-Bold Flags: Symbolic FontBBox: -152, -301, 1000, 935 FontFile: true EncodingDictionary: Differences: true Font: BaseFont: YLPUFZ+CMEX10 FontSubset: true FirstChar: 16 LastChar: 105 FontDescriptor: FontName: YLPUFZ+CMEX10 Flags: Symbolic FontBBox: -24, -2960, 1454, 772 FontFile: true Font: BaseFont: FDDSXG+URWPalladioL-BoldItal FontSubset: true FirstChar: 74 LastChar: 113 FontDescriptor: FontName: FDDSXG+URWPalladioL-BoldItal Flags: Symbolic FontBBox: -170, -300, 1073, 935 FontFile: true EncodingDictionary: Differences: true Font: BaseFont: HTVOVV+URWPalladioL-Roma FontSubset: true FirstChar: 1 LastChar: 252 FontDescriptor: FontName: HTVOVV+URWPalladioL-Roma Flags: Symbolic FontBBox: -166, -283, 1021, 943 FontFile: true EncodingDictionary: Differences: true Font: BaseFont: HVSBKL+PazoMath FontSubset: true FirstChar: 68 LastChar: 229 FontDescriptor: FontName: HVSBKL+PazoMath Flags: Symbolic FontBBox: -40, -283, 878, 946 FontFile: true Font: BaseFont: VRERLM+URWPalladioL-Ital FontSubset: true FirstChar: 39 LastChar: 150 FontDescriptor: FontName: VRERLM+URWPalladioL-Ital Flags: Symbolic FontBBox: -170, -305, 1010, 941 FontFile: true EncodingDictionary: Differences: true Font: BaseFont: XJNKAP+PazoMath-BoldItalic FontSubset: true FirstChar: 97 LastChar: 122 FontDescriptor: FontName: XJNKAP+PazoMath-BoldItalic Flags: Symbolic FontBBox: -64, -272, 868, 740 FontFile: true Font: BaseFont: YVTZRI+MSAM10 FontSubset: true FirstChar: 61 LastChar: 62 FontDescriptor: FontName: YVTZRI+MSAM10 Flags: Symbolic FontBBox: 8, -463, 1331, 1003 FontFile: true Font: BaseFont: PDXFNG+PazoMath-Bold FontSubset: true FirstChar: 68 LastChar: 68 FontDescriptor: FontName: PDXFNG+PazoMath-Bold Flags: Symbolic FontBBox: -10, -266, 829, 695 FontFile: true Font: BaseFont: CDWAAS+CMSY10 FontSubset: true FirstChar: 0 LastChar: 114 FontDescriptor: FontName: CDWAAS+CMSY10 Flags: Symbolic FontBBox: -29, -960, 1116, 775 FontFile: true Font: BaseFont: PWTQTX+CMBX10 FontSubset: true FirstChar: 40 LastChar: 61 FontDescriptor: FontName: PWTQTX+CMBX10 Flags: Symbolic FontBBox: -56, -250, 1164, 750 FontFile: true Pages: Page: Sequence: 1 Annotations: Annotation: Subtype: Link Rect: 82, 293, 152, 302 Annotation: Subtype: Link Rect: 34, 282, 94, 290 Annotation: Subtype: Link Rect: 34, 164, 96, 186 Annotation: Subtype: Link Rect: 126, 90, 154, 99 Annotation: Subtype: Link Rect: 34, 78, 147, 87 Annotation: Subtype: Link Rect: 34, 756, 190, 792 Annotation: Subtype: Link Rect: 515, 756, 560, 792 Annotation: Subtype: Link Rect: 125, 661, 136, 674 Annotation: Subtype: Link Rect: 115, 17, 243, 28 Annotation: Subtype: Link Rect: 409, 17, 560, 28 Annotation: Subtype: Link Rect: 34, 366, 87, 385 Page: Sequence: 2 Annotations: Annotation: Subtype: Link Rect: 541, 597, 548, 606 ActionDest: 154 Annotation: Subtype: Link Rect: 548, 597, 555, 606 ActionDest: 154 Annotation: Subtype: Link Rect: 538, 585, 545, 594 ActionDest: 154 Annotation: Subtype: Link Rect: 548, 585, 555, 594 ActionDest: 154 Annotation: Subtype: Link Rect: 481, 535, 488, 544 ActionDest: 154 Annotation: Subtype: Link Rect: 491, 535, 503, 544 ActionDest: 157 Annotation: Subtype: Link Rect: 420, 522, 432, 531 ActionDest: 157 Annotation: Subtype: Link Rect: 435, 522, 447, 531 ActionDest: 157 Annotation: Subtype: Link Rect: 241, 497, 253, 506 ActionDest: 157 Annotation: Subtype: Link Rect: 256, 497, 268, 506 ActionDest: 160 Annotation: Subtype: Link Rect: 422, 469, 429, 481 ActionDest: 68 Annotation: Subtype: Link Rect: 248, 459, 260, 468 ActionDest: 157 Annotation: Subtype: Link Rect: 250, 384, 262, 393 ActionDest: 157 Annotation: Subtype: Link Rect: 543, 321, 555, 330 ActionDest: 157 Annotation: Subtype: Link Rect: 248, 283, 255, 293 ActionDest: 154 Annotation: Subtype: Link Rect: 447, 208, 454, 217 ActionDest: 154 Annotation: Subtype: Link Rect: 512, 183, 524, 192 ActionDest: 157 Annotation: Subtype: Link Rect: 246, 170, 258, 180 ActionDest: 157 Annotation: Subtype: Link Rect: 210, 70, 222, 79 ActionDest: 160 Annotation: Subtype: Link Rect: 222, 70, 234, 79 ActionDest: 160 Page: Sequence: 3 Annotations: Annotation: Subtype: Link Rect: 447, 736, 459, 744 ActionDest: 160 Annotation: Subtype: Link Rect: 459, 736, 471, 744 ActionDest: 160 Annotation: Subtype: Link Rect: 383, 670, 390, 682 ActionDest: 68 Annotation: Subtype: Link Rect: 321, 648, 333, 657 ActionDest: 157 Annotation: Subtype: Link Rect: 552, 645, 559, 657 ActionDest: 94 Annotation: Subtype: Link Rect: 522, 635, 534, 644 ActionDest: 157 Annotation: Subtype: Link Rect: 203, 620, 210, 632 ActionDest: 104 Annotation: Subtype: Link Rect: 213, 620, 220, 632 ActionDest: 114 Annotation: Subtype: Link Rect: 320, 582, 328, 594 ActionDest: 117 Annotation: Subtype: Link Rect: 411, 570, 418, 582 ActionDest: 120 Annotation: Subtype: Link Rect: 441, 570, 448, 582 ActionDest: 132 Annotation: Subtype: Link Rect: 351, 544, 363, 556 ActionDest: 139 Annotation: Subtype: Link Rect: 219, 348, 231, 357 ActionDest: 157 Annotation: Subtype: Link Rect: 231, 348, 243, 357 ActionDest: 157 Annotation: Subtype: Link Rect: 244, 348, 256, 357 ActionDest: 157 Annotation: Subtype: Link Rect: 256, 348, 268, 357 ActionDest: 160 Annotation: Subtype: Link Rect: 269, 348, 280, 357 ActionDest: 160 Annotation: Subtype: Link Rect: 281, 348, 293, 357 ActionDest: 160 Annotation: Subtype: Link Rect: 296, 348, 308, 357 ActionDest: 160 Annotation: Subtype: Link Rect: 308, 348, 320, 357 ActionDest: 160 Annotation: Subtype: Link Rect: 531, 323, 543, 332 ActionDest: 160 Annotation: Subtype: Link Rect: 544, 323, 556, 332 ActionDest: 160 Annotation: Subtype: Link Rect: 311, 298, 323, 307 ActionDest: 157 Annotation: Subtype: Link Rect: 457, 110, 469, 119 ActionDest: 160 Annotation: Subtype: Link Rect: 470, 110, 482, 119 ActionDest: 160 Annotation: Subtype: Link Rect: 528, 110, 540, 119 ActionDest: 160 Annotation: Subtype: Link Rect: 540, 110, 552, 119 ActionDest: 160 Annotation: Subtype: Link Rect: 512, 97, 524, 106 ActionDest: 157 Page: Sequence: 4 Annotations: Annotation: Subtype: Link Rect: 209, 736, 221, 744 ActionDest: 160 Annotation: Subtype: Link Rect: 224, 735, 236, 744 ActionDest: 160 Annotation: Subtype: Link Rect: 229, 723, 240, 732 ActionDest: 160 Annotation: Subtype: Link Rect: 468, 723, 480, 732 ActionDest: 160 Annotation: Subtype: Link Rect: 528, 723, 540, 732 ActionDest: 160 Annotation: Subtype: Link Rect: 540, 723, 552, 732 ActionDest: 160 Annotation: Subtype: Link Rect: 213, 698, 225, 707 ActionDest: 160 Annotation: Subtype: Link Rect: 379, 698, 391, 707 ActionDest: 160 Annotation: Subtype: Link Rect: 437, 698, 449, 707 ActionDest: 160 Annotation: Subtype: Link Rect: 452, 698, 464, 707 ActionDest: 160 Annotation: Subtype: Link Rect: 523, 685, 535, 694 ActionDest: 160 Annotation: Subtype: Link Rect: 188, 673, 200, 682 ActionDest: 160 Annotation: Subtype: Link Rect: 249, 597, 260, 606 ActionDest: 160 Annotation: Subtype: Link Rect: 483, 497, 495, 506 ActionDest: 157 Annotation: Subtype: Link Rect: 295, 459, 307, 468 ActionDest: 157 Annotation: Subtype: Link Rect: 245, 434, 257, 443 ActionDest: 157 Annotation: Subtype: Link Rect: 258, 434, 270, 443 ActionDest: 157 Annotation: Subtype: Link Rect: 273, 434, 285, 443 ActionDest: 157 Annotation: Subtype: Link Rect: 285, 434, 297, 443 ActionDest: 157 Annotation: Subtype: Link Rect: 298, 434, 310, 443 ActionDest: 160 Annotation: Subtype: Link Rect: 310, 434, 322, 443 ActionDest: 160 Annotation: Subtype: Link Rect: 263, 381, 275, 390 ActionDest: 157 Annotation: Subtype: Link Rect: 275, 380, 287, 390 ActionDest: 157 Annotation: Subtype: Link Rect: 288, 380, 300, 390 ActionDest: 163 Annotation: Subtype: Link Rect: 285, 177, 297, 186 ActionDest: 157 Annotation: Subtype: Link Rect: 297, 177, 309, 186 ActionDest: 157 Annotation: Subtype: Link Rect: 309, 177, 321, 186 ActionDest: 157 Page: Sequence: 5 Annotations: Annotation: Subtype: Link Rect: 262, 695, 269, 707 ActionDest: 75 Annotation: Subtype: Link Rect: 182, 657, 189, 669 ActionDest: 75 Annotation: Subtype: Link Rect: 199, 660, 211, 669 ActionDest: 157 Annotation: Subtype: Link Rect: 211, 660, 223, 669 ActionDest: 157 Annotation: Subtype: Link Rect: 226, 660, 238, 669 ActionDest: 157 Annotation: Subtype: Link Rect: 239, 660, 251, 669 ActionDest: 157 Annotation: Subtype: Link Rect: 251, 660, 263, 669 ActionDest: 163 Annotation: Subtype: Link Rect: 266, 660, 278, 669 ActionDest: 163 Annotation: Subtype: Link Rect: 354, 657, 361, 669 ActionDest: 75 Annotation: Subtype: Link Rect: 503, 660, 515, 669 ActionDest: 163 Annotation: Subtype: Link Rect: 515, 660, 527, 669 ActionDest: 163 Annotation: Subtype: Link Rect: 215, 647, 227, 657 ActionDest: 163 Annotation: Subtype: Link Rect: 224, 610, 236, 619 ActionDest: 157 Annotation: Subtype: Link Rect: 236, 610, 248, 619 ActionDest: 160 Annotation: Subtype: Link Rect: 249, 610, 266, 619 ActionDest: 163 Annotation: Subtype: Link Rect: 347, 597, 359, 606 ActionDest: 157 Annotation: Subtype: Link Rect: 359, 597, 371, 606 ActionDest: 157 Annotation: Subtype: Link Rect: 372, 597, 384, 606 ActionDest: 157 Annotation: Subtype: Link Rect: 384, 597, 396, 606 ActionDest: 160 Annotation: Subtype: Link Rect: 327, 469, 334, 481 ActionDest: 94 Annotation: Subtype: Link Rect: 506, 459, 518, 468 ActionDest: 160 Annotation: Subtype: Link Rect: 518, 459, 535, 468 ActionDest: 163 Annotation: Subtype: Link Rect: 538, 459, 555, 468 ActionDest: 163 Annotation: Subtype: Link Rect: 249, 371, 261, 380 ActionDest: 157 Annotation: Subtype: Link Rect: 262, 371, 274, 380 ActionDest: 157 Annotation: Subtype: Link Rect: 274, 371, 286, 380 ActionDest: 160 Annotation: Subtype: Link Rect: 308, 321, 320, 330 ActionDest: 157 Annotation: Subtype: Link Rect: 385, 296, 397, 305 ActionDest: 157 Annotation: Subtype: Link Rect: 321, 70, 328, 82 ActionDest: 75 Page: Sequence: 6 Annotations: Annotation: Subtype: Link Rect: 381, 685, 393, 694 ActionDest: 157 Annotation: Subtype: Link Rect: 486, 685, 498, 694 ActionDest: 157 Annotation: Subtype: Link Rect: 292, 530, 299, 542 ActionDest: 94 Annotation: Subtype: Link Rect: 466, 530, 474, 542 ActionDest: 75 Annotation: Subtype: Link Rect: 354, 399, 371, 408 ActionDest: 163 Annotation: Subtype: Link Rect: 372, 387, 384, 396 ActionDest: 157 Annotation: Subtype: Link Rect: 203, 271, 210, 286 ActionDest: 75 Annotation: Subtype: Link Rect: 211, 259, 218, 271 ActionDest: 94 Annotation: Subtype: Link Rect: 205, 172, 217, 181 ActionDest: 157 Annotation: Subtype: Link Rect: 234, 69, 241, 84 ActionDest: 97 Page: Sequence: 7 Annotations: Annotation: Subtype: Link Rect: 248, 710, 260, 719 ActionDest: 157 Annotation: Subtype: Link Rect: 543, 660, 555, 669 ActionDest: 157 Annotation: Subtype: Link Rect: 316, 610, 333, 619 ActionDest: 163 Annotation: Subtype: Link Rect: 382, 497, 389, 506 ActionDest: 154 Annotation: Subtype: Link Rect: 390, 497, 397, 506 ActionDest: 154 Annotation: Subtype: Link Rect: 397, 497, 404, 506 ActionDest: 154 Annotation: Subtype: Link Rect: 405, 497, 417, 506 ActionDest: 157 Annotation: Subtype: Link Rect: 497, 484, 509, 493 ActionDest: 160 Annotation: Subtype: Link Rect: 510, 484, 522, 493 ActionDest: 160 Annotation: Subtype: Link Rect: 288, 456, 296, 468 ActionDest: 120 Annotation: Subtype: Link Rect: 318, 456, 326, 468 ActionDest: 132 Annotation: Subtype: Link Rect: 550, 255, 557, 267 ActionDest: 94 Annotation: Subtype: Link Rect: 188, 242, 195, 254 ActionDest: 94 Annotation: Subtype: Link Rect: 253, 174, 265, 187 ActionDest: 104 Annotation: Subtype: Link Rect: 213, 117, 222, 129 ActionDest: 151 Annotation: Subtype: Link Rect: 486, 117, 498, 129 ActionDest: 104 Annotation: Subtype: Link Rect: 526, 117, 538, 129 ActionDest: 104 Page: Sequence: 8 Annotations: Annotation: Subtype: Link Rect: 194, 733, 206, 745 ActionDest: 104 Annotation: Subtype: Link Rect: 241, 678, 248, 690 ActionDest: 94 Annotation: Subtype: Link Rect: 283, 678, 295, 690 ActionDest: 107 Annotation: Subtype: Link Rect: 499, 678, 506, 690 ActionDest: 75 Annotation: Subtype: Link Rect: 193, 612, 205, 624 ActionDest: 107 Annotation: Subtype: Link Rect: 300, 523, 312, 535 ActionDest: 107 Annotation: Subtype: Link Rect: 206, 497, 217, 506 ActionDest: 157 Annotation: Subtype: Link Rect: 218, 497, 230, 506 ActionDest: 157 Annotation: Subtype: Link Rect: 210, 370, 222, 382 ActionDest: 107 Annotation: Subtype: Link Rect: 436, 370, 448, 382 ActionDest: 107 Annotation: Subtype: Link Rect: 354, 339, 366, 358 ActionDest: 107 Annotation: Subtype: Link Rect: 392, 339, 404, 358 ActionDest: 107 Annotation: Subtype: Link Rect: 312, 214, 324, 223 ActionDest: 157 Annotation: Subtype: Link Rect: 325, 214, 337, 223 ActionDest: 157 Annotation: Subtype: Link Rect: 337, 214, 349, 223 ActionDest: 157 Annotation: Subtype: Link Rect: 350, 214, 362, 223 ActionDest: 160 Annotation: Subtype: Link Rect: 213, 127, 225, 139 ActionDest: 107 Annotation: Subtype: Link Rect: 406, 115, 418, 127 ActionDest: 107 Annotation: Subtype: Link Rect: 446, 115, 458, 127 ActionDest: 107 Annotation: Subtype: Link Rect: 486, 115, 498, 127 ActionDest: 107 Annotation: Subtype: Link Rect: 209, 103, 221, 114 ActionDest: 107 Page: Sequence: 9 Annotations: Annotation: Subtype: Link Rect: 196, 733, 208, 745 ActionDest: 107 Annotation: Subtype: Link Rect: 310, 708, 320, 722 ActionDest: 151 Annotation: Subtype: Link Rect: 433, 694, 448, 709 ActionDest: 151 Annotation: Subtype: Link Rect: 458, 694, 473, 709 ActionDest: 151 Annotation: Subtype: Link Rect: 228, 657, 240, 669 ActionDest: 107 Annotation: Subtype: Link Rect: 465, 428, 477, 442 ActionDest: 104 Annotation: Subtype: Link Rect: 192, 314, 204, 326 ActionDest: 114 Annotation: Subtype: Link Rect: 230, 314, 242, 326 ActionDest: 114 Annotation: Subtype: Link Rect: 194, 268, 201, 283 ActionDest: 94 Annotation: Subtype: Link Rect: 228, 268, 240, 283 ActionDest: 114 Annotation: Subtype: Link Rect: 442, 268, 449, 283 ActionDest: 75 Annotation: Subtype: Link Rect: 194, 199, 206, 211 ActionDest: 114 Annotation: Subtype: Link Rect: 438, 186, 450, 198 ActionDest: 107 Annotation: Subtype: Link Rect: 476, 186, 488, 198 ActionDest: 107 Page: Sequence: 10 Annotations: Annotation: Subtype: Link Rect: 205, 733, 216, 745 ActionDest: 107 Annotation: Subtype: Link Rect: 266, 683, 278, 695 ActionDest: 114 Annotation: Subtype: Link Rect: 305, 683, 317, 695 ActionDest: 114 Annotation: Subtype: Link Rect: 388, 683, 400, 695 ActionDest: 114 Annotation: Subtype: Link Rect: 523, 683, 535, 695 ActionDest: 114 Annotation: Subtype: Link Rect: 275, 628, 287, 640 ActionDest: 117 Annotation: Subtype: Link Rect: 211, 603, 221, 615 ActionDest: 151 Annotation: Subtype: Link Rect: 231, 525, 239, 537 ActionDest: 154 Annotation: Subtype: Link Rect: 545, 474, 557, 486 ActionDest: 114 Annotation: Subtype: Link Rect: 297, 391, 309, 406 ActionDest: 114 Annotation: Subtype: Link Rect: 543, 391, 555, 406 ActionDest: 114 Annotation: Subtype: Link Rect: 237, 291, 249, 305 ActionDest: 117 Annotation: Subtype: Link Rect: 415, 291, 427, 305 ActionDest: 117 Page: Sequence: 11 Annotations: Annotation: Subtype: Link Rect: 248, 719, 260, 733 ActionDest: 117 Annotation: Subtype: Link Rect: 285, 719, 297, 733 ActionDest: 117 Annotation: Subtype: Link Rect: 204, 707, 216, 718 ActionDest: 114 Annotation: Subtype: Link Rect: 413, 694, 425, 706 ActionDest: 117 Annotation: Subtype: Link Rect: 333, 645, 345, 658 ActionDest: 117 Annotation: Subtype: Link Rect: 290, 634, 302, 646 ActionDest: 117 Annotation: Subtype: Link Rect: 179, 536, 191, 548 ActionDest: 117 Annotation: Subtype: Link Rect: 424, 427, 436, 439 ActionDest: 117 Annotation: Subtype: Link Rect: 238, 319, 250, 331 ActionDest: 120 Annotation: Subtype: Link Rect: 260, 319, 272, 331 ActionDest: 120 Annotation: Subtype: Link Rect: 423, 319, 435, 331 ActionDest: 120 Annotation: Subtype: Link Rect: 444, 319, 456, 331 ActionDest: 120 Annotation: Subtype: Link Rect: 253, 293, 260, 305 ActionDest: 120 Annotation: Subtype: Link Rect: 263, 293, 275, 305 ActionDest: 139 Annotation: Subtype: Link Rect: 300, 180, 312, 189 ActionDest: 157 Page: Sequence: 12 Annotations: Annotation: Subtype: Link Rect: 258, 683, 270, 695 ActionDest: 117 Annotation: Subtype: Link Rect: 359, 683, 371, 695 ActionDest: 117 Annotation: Subtype: Link Rect: 381, 683, 393, 695 ActionDest: 120 Annotation: Subtype: Link Rect: 245, 635, 257, 647 ActionDest: 132 Annotation: Subtype: Link Rect: 194, 579, 206, 591 ActionDest: 120 Annotation: Subtype: Link Rect: 215, 579, 227, 591 ActionDest: 132 Annotation: Subtype: Link Rect: 381, 579, 393, 591 ActionDest: 132 Annotation: Subtype: Link Rect: 180, 506, 192, 518 ActionDest: 132 Annotation: Subtype: Link Rect: 528, 506, 540, 518 ActionDest: 132 Annotation: Subtype: Link Rect: 199, 421, 211, 430 ActionDest: 160 Annotation: Subtype: Link Rect: 211, 421, 223, 430 ActionDest: 160 Annotation: Subtype: Link Rect: 507, 370, 519, 385 ActionDest: 120 Annotation: Subtype: Link Rect: 267, 359, 279, 371 ActionDest: 132 Annotation: Subtype: Link Rect: 458, 293, 465, 305 ActionDest: 94 Annotation: Subtype: Link Rect: 280, 131, 292, 143 ActionDest: 132 Page: Sequence: 13 Annotations: Annotation: Subtype: Link Rect: 457, 733, 469, 745 ActionDest: 132 Annotation: Subtype: Link Rect: 520, 733, 532, 745 ActionDest: 117 Annotation: Subtype: Link Rect: 225, 720, 237, 732 ActionDest: 117 Annotation: Subtype: Link Rect: 247, 720, 259, 732 ActionDest: 120 Annotation: Subtype: Link Rect: 210, 673, 222, 685 ActionDest: 139 Annotation: Subtype: Link Rect: 487, 660, 499, 672 ActionDest: 139 Annotation: Subtype: Link Rect: 215, 535, 227, 547 ActionDest: 139 Annotation: Subtype: Link Rect: 211, 497, 223, 509 ActionDest: 139 Annotation: Subtype: Link Rect: 241, 448, 252, 463 ActionDest: 139 Annotation: Subtype: Link Rect: 502, 438, 519, 447 ActionDest: 163 Annotation: Subtype: Link Rect: 241, 335, 252, 349 ActionDest: 120 Annotation: Subtype: Link Rect: 543, 335, 555, 349 ActionDest: 139 Annotation: Subtype: Link Rect: 324, 310, 336, 324 ActionDest: 139 Annotation: Subtype: Link Rect: 288, 213, 300, 226 ActionDest: 139 Annotation: Subtype: Link Rect: 465, 127, 477, 136 ActionDest: 154 Annotation: Subtype: Link Rect: 477, 127, 489, 136 ActionDest: 157 Annotation: Subtype: Link Rect: 490, 127, 502, 136 ActionDest: 157 Annotation: Subtype: Link Rect: 502, 127, 514, 136 ActionDest: 157 Annotation: Subtype: Link Rect: 515, 127, 527, 136 ActionDest: 157 Annotation: Subtype: Link Rect: 527, 127, 539, 136 ActionDest: 157 Annotation: Subtype: Link Rect: 540, 127, 552, 136 ActionDest: 160 Page: Sequence: 14 Annotations: Annotation: Subtype: Link Rect: 278, 584, 285, 596 ActionDest: 75 Annotation: Subtype: Link Rect: 295, 584, 302, 596 ActionDest: 94 Annotation: Subtype: Link Rect: 328, 584, 340, 596 ActionDest: 142 Annotation: Subtype: Link Rect: 230, 348, 242, 359 ActionDest: 142 Annotation: Subtype: Link Rect: 380, 277, 392, 291 ActionDest: 142 Annotation: Subtype: Link Rect: 302, 251, 314, 263 ActionDest: 142 Page: Sequence: 15 Annotations: Annotation: Subtype: Link Rect: 194, 713, 206, 725 ActionDest: 142 Annotation: Subtype: Link Rect: 404, 713, 411, 725 ActionDest: 75 Annotation: Subtype: Link Rect: 421, 713, 427, 725 ActionDest: 94 Annotation: Subtype: Link Rect: 454, 713, 466, 725 ActionDest: 142 Annotation: Subtype: Link Rect: 194, 375, 206, 395 ActionDest: 145 Annotation: Subtype: Link Rect: 233, 375, 245, 395 ActionDest: 145 Annotation: Subtype: Link Rect: 502, 319, 514, 336 ActionDest: 145 Annotation: Subtype: Link Rect: 378, 170, 395, 179 ActionDest: 163 Annotation: Subtype: Link Rect: 398, 170, 415, 179 ActionDest: 163 Annotation: Subtype: Link Rect: 420, 130, 432, 142 ActionDest: 145 Annotation: Subtype: Link Rect: 371, 120, 383, 129 ActionDest: 160 Page: Sequence: 16 Annotations: Annotation: Subtype: Link Rect: 410, 484, 422, 493 ActionDest: 160 Annotation: Subtype: Link Rect: 214, 358, 231, 368 ActionDest: 163 Annotation: Subtype: Link Rect: 231, 359, 248, 368 ActionDest: 163 Annotation: Subtype: Link Rect: 287, 346, 304, 355 ActionDest: 163 Annotation: Subtype: Link Rect: 388, 346, 400, 355 ActionDest: 160 Annotation: Subtype: Link Rect: 235, 281, 247, 293 ActionDest: 139 Page: Sequence: 17 Annotations: Annotation: Subtype: Link Rect: 493, 497, 505, 509 ActionDest: 107 Page: Sequence: 18 Annotations: Annotation: Subtype: Link Rect: 383, 733, 395, 745 ActionDest: 117 Annotation: Subtype: Link Rect: 207, 719, 222, 732 ActionDest: 151 Annotation: Subtype: Link Rect: 415, 719, 430, 732 ActionDest: 151 Annotation: Subtype: Link Rect: 456, 719, 471, 732 ActionDest: 151 Annotation: Subtype: Link Rect: 233, 574, 240, 586 ActionDest: 114 Annotation: Subtype: Link Rect: 266, 560, 278, 575 ActionDest: 114 Annotation: Subtype: Link Rect: 481, 447, 519, 458 Annotation: Subtype: Link Rect: 214, 424, 252, 435 Annotation: Subtype: Link Rect: 292, 412, 329, 423 Annotation: Subtype: Link Rect: 398, 377, 436, 388 Annotation: Subtype: Link Rect: 442, 377, 478, 388 Annotation: Subtype: Link Rect: 95, 355, 133, 365 Annotation: Subtype: Link Rect: 250, 331, 287, 342 Annotation: Subtype: Link Rect: 277, 309, 314, 319 Annotation: Subtype: Link Rect: 219, 285, 257, 296 Annotation: Subtype: Link Rect: 234, 262, 272, 273 Annotation: Subtype: Link Rect: 257, 239, 294, 250 Annotation: Subtype: Link Rect: 59, 217, 96, 227 Annotation: Subtype: Link Rect: 95, 194, 133, 204 Annotation: Subtype: Link Rect: 59, 171, 96, 181 Annotation: Subtype: Link Rect: 108, 148, 146, 158 Annotation: Subtype: Link Rect: 171, 125, 208, 135 Annotation: Subtype: Link Rect: 340, 101, 378, 112 Annotation: Subtype: Link Rect: 519, 90, 557, 101 Annotation: Subtype: Link Rect: 59, 68, 96, 78 Page: Sequence: 19 Annotations: Annotation: Subtype: Link Rect: 59, 722, 96, 733 Annotation: Subtype: Link Rect: 231, 699, 269, 710 Annotation: Subtype: Link Rect: 59, 665, 96, 675 Annotation: Subtype: Link Rect: 131, 642, 168, 652 Annotation: Subtype: Link Rect: 127, 619, 164, 629 Annotation: Subtype: Link Rect: 497, 583, 534, 594 Annotation: Subtype: Link Rect: 199, 537, 236, 548 Annotation: Subtype: Link Rect: 233, 503, 271, 514 Annotation: Subtype: Link Rect: 235, 480, 273, 491 Annotation: Subtype: Link Rect: 259, 457, 296, 468 Annotation: Subtype: Link Rect: 405, 445, 442, 456 Annotation: Subtype: Link Rect: 417, 434, 454, 445 Annotation: Subtype: Link Rect: 372, 376, 410, 387 Annotation: Subtype: Link Rect: 126, 354, 164, 364 Annotation: Subtype: Link Rect: 519, 342, 557, 353 Annotation: Subtype: Link Rect: 467, 330, 505, 341 Annotation: Subtype: Link Rect: 257, 273, 295, 284 Annotation: Subtype: Link Rect: 460, 261, 497, 272 Annotation: Subtype: Link Rect: 59, 239, 96, 249 Annotation: Subtype: Link Rect: 122, 135, 160, 146 Annotation: Subtype: Link Rect: 59, 113, 96, 123 Annotation: Subtype: Link Rect: 82, 78, 119, 88 Page: Sequence: 20 Annotations: Annotation: Subtype: Link Rect: 109, 722, 146, 733 Annotation: Subtype: Link Rect: 108, 699, 146, 710 Annotation: Subtype: Link Rect: 95, 676, 133, 687 Annotation: Subtype: Link Rect: 59, 653, 96, 664 Annotation: Subtype: Link Rect: 515, 641, 553, 652 Annotation: Subtype: Link Rect: 86, 619, 124, 629 Annotation: Subtype: Link Rect: 95, 596, 133, 606 Annotation: Subtype: Link Rect: 95, 573, 133, 583 Annotation: Subtype: Link Rect: 104, 538, 141, 548 Annotation: Subtype: Link Rect: 195, 514, 233, 525 Annotation: Subtype: Link Rect: 118, 492, 155, 502 Annotation: Subtype: Link Rect: 161, 492, 197, 502 Annotation: Subtype: Link Rect: 170, 445, 207, 456 Annotation: Subtype: Link Rect: 59, 423, 96, 433 Annotation: Subtype: Link Rect: 104, 389, 141, 399 Annotation: Subtype: Link Rect: 485, 376, 523, 387 Annotation: Subtype: Link Rect: 59, 354, 96, 364 Annotation: Subtype: Link Rect: 95, 331, 133, 341 Annotation: Subtype: Link Rect: 380, 296, 417, 307 Annotation: Subtype: Link Rect: 263, 227, 301, 238 Annotation: Subtype: Link Rect: 307, 227, 343, 238 Annotation: Subtype: Link Rect: 152, 205, 190, 215 Annotation: Subtype: Link Rect: 451, 192, 488, 203 Annotation: Subtype: Link Rect: 192, 170, 230, 180 Annotation: Subtype: Link Rect: 161, 146, 198, 157 Annotation: Subtype: Link Rect: 146, 123, 183, 134 Annotation: Subtype: Link Rect: 388, 112, 426, 123 Annotation: Subtype: Link Rect: 136, 66, 173, 77 Page: Sequence: 21 Annotations: Annotation: Subtype: Link Rect: 453, 733, 490, 744 Annotation: Subtype: Link Rect: 472, 722, 510, 733 Annotation: Subtype: Link Rect: 376, 699, 414, 710 Annotation: Subtype: Link Rect: 90, 676, 128, 687 Annotation: Subtype: Link Rect: 487, 664, 524, 675 Annotation: Subtype: Link Rect: 59, 642, 96, 652 Annotation: Subtype: Link Rect: 179, 618, 216, 629 Annotation: Subtype: Link Rect: 147, 596, 185, 606 Annotation: Subtype: Link Rect: 140, 573, 177, 583 Annotation: Subtype: Link Rect: 59, 538, 96, 548 Annotation: Subtype: Link Rect: 367, 526, 405, 537 Annotation: Subtype: Link Rect: 126, 492, 164, 502 Annotation: Subtype: Link Rect: 348, 480, 386, 491 Annotation: Subtype: Link Rect: 438, 468, 476, 479 Annotation: Subtype: Link Rect: 59, 435, 96, 445 Annotation: Subtype: Link Rect: 459, 422, 496, 433 Checksum: 085740fd Type: CRC32 Checksum: 535f9e942d55628e00dc39db34bbf51e Type: MD5 Checksum: 85663bbff5b0fbc7fd0cbc9aa603a5c01ea471c1 Type: SHA-1 Checksum: 60b25a2d8cf7b8eda9499e96da9389bbd936c258900823cdc5453e41ea9dd151 Type: SHA-256