A Fractional Entropy in Fractal Phase Space: Properties and Characterization

<table>We compare the complexity <svg height="14.9456pt" id="M374" style="vertical-align:-5.702869pt" version="1.1" viewbox="-0.0498162 -9.24273 24.5677 14.9456" width="24.5677pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M586 170C521 97 455 62 367 62C210 62 116 171 116 335C116 484 213 600 364 600C448 600 550 570 598 453H617V554C558 645 451 676 363 676C162 676 27 529 27 333C27 132 151 -14 364 -14C476 -14 570 34 637 119L586 170Z" id="g202-14"></path><glyph.data ascent="684" descent="-216" horiz-adv-x="671" vert-adv-y="671"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,9.125,3.264)"><path d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 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366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z" id="g50-114"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="499" vert-adv-y="499"></glyph.data></g><g transform="matrix(.0068,0,0,-0.0068,20.838,-.068)"><path d="M323 549L316 585C302 600 218 636 190 624L84 81L142 54L323 549Z" id="g176-31"></path><glyph.data ascent="3439" descent="-2858" horiz-adv-x="381" vert-adv-y="381"></glyph.data></g></svg> of a five level system computed using the Eulidean distance <svg height="12.8147pt" id="M375" style="vertical-align:-3.31359pt" version="1.1" viewbox="-0.0498162 -9.50111 18.3322 12.8147" width="18.3322pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M227 361C221 360 211 359 198 359C143 359 113 391 113 435C113 483 134 520 176 547C219 574 266 587 319 587C350 587 376 585 397 580V578C376 562 358 541 341 516C325 491 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42 237 55C221 68 220 100 231 158L261 316H357C455 316 455 309 455 235H489L529 437H494C468 366 460 360 369 360H269L305 565C315 609 316 611 357 611H450C525 611 535 599 548 578C559 556 561 526 560 482L595 486L615 650Z" id="g50-70"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="639" vert-adv-y="639"></glyph.data></g></svg> given to the left (has a smooth upper bound) and the complexity using the Wootters’ distance <svg height="12.9289pt" id="M376" style="vertical-align:-3.42779pt" version="1.1" viewbox="-0.0498162 -9.50111 21.7591 12.9289" width="21.7591pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M227 361C221 360 211 359 198 359C143 359 113 391 113 435C113 483 134 520 176 547C219 574 266 587 319 587C350 587 376 585 397 580V578C376 562 358 541 341 516C325 491 317 471 317 455C317 418 328 394 343 377C352 366 354 365 365 354L377 343C446 274 400 218 348 218C334 218 320 221 307 228L297 207C318 196 340 191 365 191C406 191 442 205 473 232S520 290 520 325C520 368 491 407 460 430C431 453 401 484 401 512C401 529 413 549 436 572C524 559 602 523 670 463C739 404 773 338 773 265C773 198 753 144 713 102C674 61 622 40 557 40C445 40 376 176 246 176C181 176 120 132 65 43L83 30C113 61 149 76 192 76C268 80 346 -28 466 -24C579 -24 667 7 730 68C788 124 817 195 817 281C817 360 794 431 749 495C704 560 647 609 577 643C507 678 433 695 356 695C270 695 197 674 138 631S50 534 50 469C50 388 107 333 192 333C198 333 210 334 227 337V361Z" id="g200-5"></path><glyph.data ascent="704" descent="-209" horiz-adv-x="868" vert-adv-y="868"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,11.76,3.264)"><path d="M1013 650H775L768 618L793 615C853 608 859 601 837 545C792 433 708 266 639 131H637L537 639H503L251 132H249L190 552C183 602 190 607 235 614L262 618L269 650H25L18 618C92 610 95 608 107 525L184 -9H221L475 494H479L576 -9H613L884 526C928 607 934 609 1005 618L1013 650Z" 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International Journal of Statistical Mechanics

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Figure 5

Figure 5: A Fractional Entropy in Fractal Phase Space: Properties and Characterization 