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This is obviously more than the \ number of lower-mass-level particles contained in the particle swarm with the \ same average momentum composed by lower-mass-level particles.", FontFamily->"Times New Roman", FontWeight->"Regular"] }], "Text", CellChangeTimes->{{3.7560729545468388`*^9, 3.756073100638503*^9}, { 3.75607364213144*^9, 3.756073642706232*^9}, {3.764927060275012*^9, 3.764927070331305*^9}, {3.766298909798955*^9, 3.766298934069401*^9}, { 3.76678849446102*^9, 3.7667885609866743`*^9}, {3.7667885920287113`*^9, 3.766788625434799*^9}, 3.766788656730773*^9, {3.770186420477174*^9, 3.770186426690283*^9}, {3.770186717275547*^9, 3.770186737085937*^9}, { 3.77018677035857*^9, 3.7701868769223433`*^9}, {3.770186909441044*^9, 3.770186934172264*^9}, {3.770443177460298*^9, 3.770443179576709*^9}, { 3.77085909519851*^9, 3.770859121773682*^9}, {3.770859272385634*^9, 3.770859466240917*^9}, {3.77085952419211*^9, 3.770859620289151*^9}, { 3.770859736789328*^9, 3.7708597638254337`*^9}, {3.770859852069563*^9, 3.770859852509177*^9}, {3.7709329684060698`*^9, 3.7709329709427853`*^9}, { 3.794729739479273*^9, 3.794729743902834*^9}, {3.794729784036271*^9, 3.7947297887905207`*^9}, {3.7947351717935333`*^9, 3.794735230420698*^9}, { 3.794786111284349*^9, 3.7947861166459*^9}, {3.794786148358968*^9, 3.794786160635964*^9}, {3.794786239219668*^9, 3.794786240891473*^9}, { 3.7947865991961403`*^9, 3.794786675317934*^9}, {3.794786727578107*^9, 3.79478675723603*^9}, {3.7947868219791737`*^9, 3.794786954939763*^9}, { 3.794787001731368*^9, 3.7947870953835907`*^9}, {3.7947875164358253`*^9, 3.794787521354027*^9}, {3.794787953876657*^9, 3.794787955050716*^9}, 3.7947880499749537`*^9, {3.794788090698449*^9, 3.794788091590272*^9}, { 3.7947881586460037`*^9, 3.794788176801855*^9}, {3.794788212652172*^9, 3.794788222402103*^9}, {3.7947883609223413`*^9, 3.79478852668544*^9}, { 3.794788557156478*^9, 3.794788590422022*^9}, {3.794788638761198*^9, 3.794788647400923*^9}, {3.7947887997991657`*^9, 3.794788926981968*^9}, { 3.794788966891*^9, 3.794789011629299*^9}, {3.794789134906061*^9, 3.794789135848316*^9}, {3.7947891782875137`*^9, 3.794789205298427*^9}, { 3.794789272378778*^9, 3.794789304597063*^9}, {3.794791564175867*^9, 3.794791573151154*^9}, 3.794898423167468*^9, {3.794901499946456*^9, 3.794901518381403*^9}, {3.794902653480158*^9, 3.794902654256024*^9}, { 3.7952172620705214`*^9, 3.7952172855001183`*^9}, {3.795247855318746*^9, 3.795247868146678*^9}, {3.7959429369950657`*^9, 3.795942937573454*^9}, { 3.7959429773247643`*^9, 3.795942989545602*^9}, {3.796511124822124*^9, 3.796511141537737*^9}, {3.810008643993403*^9, 3.810008658838655*^9}, { 3.8100094883132477`*^9, 3.810009616994178*^9}, 3.810009686424934*^9, 3.810009775296175*^9, {3.810009893729196*^9, 3.81000991738619*^9}, { 3.810009951463298*^9, 3.810009975137624*^9}, {3.8100100731586647`*^9, 3.810010085514513*^9}, {3.81001296591975*^9, 3.8100129689419527`*^9}, { 3.810013365988785*^9, 3.810013428891493*^9}, {3.810013535278439*^9, 3.81001361350867*^9}, {3.8100136471177683`*^9, 3.810013648597478*^9}, { 3.810013783236052*^9, 3.810013792358069*^9}, 3.81001382972097*^9, { 3.8100139030139713`*^9, 3.810013904869441*^9}, {3.810014240852715*^9, 3.8100143259277487`*^9}, {3.81001439138632*^9, 3.8100144691706533`*^9}, 3.810014564166662*^9, 3.8101703126059103`*^9, {3.8101704693441343`*^9, 3.810170481525976*^9}, {3.810170518178506*^9, 3.8101705666444483`*^9}, 3.810170621439431*^9, {3.8101706605735083`*^9, 3.810170741431294*^9}, { 3.810170827674882*^9, 3.810170889492941*^9}, {3.8102641764179068`*^9, 3.810264214985662*^9}},ExpressionUUID->"d16d5944-32a0-459d-bf17-\ a364a8ee4cf2"], Cell[TextData[{ StyleBox["Each mass level particle can be seen as being formed by particles \ of lower mass level. Regardless of how much mass aggregation or velocity \ direction aggregation the particles exhibit, it can be regarded as a slight \ one with a lower mass level. This is carried out step by step. Finally, the \ minimal deviation of the aggregation behavior of the position or velocity \ direction for infinitesimal particles can be achieved. The abovementioned \ results show that when the slightest aggregation behavior occurs, the \ difficulty of the two aggregation behaviors is equivalent. Therefore, the two \ aggregation behaviors can be replaced with each other for the statistical \ influence of the diffusion behavior on the infinitesimal particles. When the \ particles of higher mass level are investigated, their dynamic behaviors are \ affected by the dynamic behaviors of the lower-mass-level particles forming \ them (this is not contradictory to the viewpoint that each mass level \ particle can be treated equally, that is, when the behavior of particles of \ higher quality level is investigated, these particles can be regarded as free \ particles with a statistical effect, while the lower-mass-level particles \ forming them can be completely ignored. See Section ", FontFamily->"Times New Roman", FontWeight->"Regular"], StyleBox["3.3.5.4", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" of the main text for more details. When the number of particles \ of higher mass level is small and does not meet the statistical conditions, \ their dynamic behavior should be investigated according to the dynamic \ behaviors of the particles of lower mass level. When the number of particles \ of higher mass level is large enough, their dynamic behaviors can be \ investigated separately, that is, they are not affected by the dynamic \ behaviors of lower-mass-level particles, and they show the minimal deviation \ of the aggregation behavior. However, when there are artificial regulations, \ even if the number of particles of higher mass level is large enough, they \ will still be affected by the dynamic behavior of particles of lower mass \ level), thus following the dynamic behaviors of the lower-mass-level \ particles. Therefore, the two aggregation behaviors are equivalent to that of \ the higher-mass-level particles. In summary, the two aggregation effects are \ interchangeable for any mass level particles and thus affect the diffusion \ behavior as a single statistical effect of the aggregation behavior.", FontFamily->"Times New Roman", FontWeight->"Regular"] }], "Text", CellChangeTimes->{{3.756071957801807*^9, 3.7560719610569363`*^9}, { 3.756072162812707*^9, 3.756072255049954*^9}, {3.7560722881432877`*^9, 3.756072460196063*^9}, {3.756072704965282*^9, 3.756072706427143*^9}, { 3.75607423600876*^9, 3.75607425083676*^9}, {3.7560746441659393`*^9, 3.75607466095191*^9}, {3.762405337695375*^9, 3.7624053590902987`*^9}, 3.7624053952706327`*^9, 3.791782935272449*^9, {3.791783123656888*^9, 3.7917831434426126`*^9}, {3.791783200946151*^9, 3.7917832014660997`*^9}, { 3.791783256348329*^9, 3.791783415576569*^9}, {3.79412999132515*^9, 3.79413001325869*^9}, {3.794869377074155*^9, 3.794869388274761*^9}, { 3.80601606075791*^9, 3.806016099560939*^9}, {3.8060161554873533`*^9, 3.806016187070943*^9}, 3.8060162583083143`*^9, 3.806016347894856*^9, { 3.806017320042679*^9, 3.806017329760049*^9}, {3.806017374465602*^9, 3.806017431338913*^9}, {3.806017561909966*^9, 3.806017601254341*^9}, { 3.806017665478314*^9, 3.806017675533852*^9}, {3.8060177537572527`*^9, 3.8060177777994556`*^9}, {3.806017957999001*^9, 3.806017975726028*^9}, { 3.8060180724847717`*^9, 3.806018100269473*^9}, {3.806018197184699*^9, 3.806018210925969*^9}, {3.8060184297018538`*^9, 3.806018559995391*^9}, { 3.806018610054208*^9, 3.8060186115419283`*^9}, {3.806018647278618*^9, 3.8060187009877*^9}, {3.806020738518737*^9, 3.80602074128559*^9}, { 3.806020809999628*^9, 3.806020891143156*^9}, {3.8061070296326313`*^9, 3.806107141186225*^9}, {3.806107188165708*^9, 3.806107202471026*^9}, { 3.807486196082066*^9, 3.8074862017606983`*^9}, {3.80748815434009*^9, 3.807488243067843*^9}, {3.807488328264798*^9, 3.807488359400373*^9}, { 3.8075060068238153`*^9, 3.807506007599895*^9}, {3.8075070834492073`*^9, 3.807507141789667*^9}, {3.807507208428348*^9, 3.8075072590636253`*^9}, { 3.807507302571464*^9, 3.807507350358467*^9}, {3.807507491663368*^9, 3.8075075385871696`*^9}, {3.80750764454713*^9, 3.8075077138219547`*^9}, { 3.807507768395002*^9, 3.807507776192252*^9}, {3.807508034565571*^9, 3.807508055638721*^9}, {3.8075085033143787`*^9, 3.807508525771216*^9}, 3.807508565071125*^9, {3.807508619362371*^9, 3.807508703868938*^9}, { 3.807508763849641*^9, 3.807508764621539*^9}, {3.807508811870532*^9, 3.807508892216145*^9}, {3.807509035069354*^9, 3.807509049858527*^9}, { 3.807509312831649*^9, 3.807509327853038*^9}, {3.807509386473074*^9, 3.807509415518333*^9}, 3.80750961609804*^9, 3.807509677643639*^9, { 3.807510146267227*^9, 3.8075102048866997`*^9}, {3.8075104379357967`*^9, 3.807510438627081*^9}, 3.807510493607407*^9, {3.8075105467476187`*^9, 3.807510563776847*^9}, 3.807594502664948*^9, {3.808607039747506*^9, 3.8086070475447598`*^9}, {3.808607099661969*^9, 3.80860718095531*^9}, { 3.8086072334799147`*^9, 3.808607275957703*^9}, {3.8086073720754957`*^9, 3.808607374307563*^9}, {3.808607411305975*^9, 3.8086074603492317`*^9}, { 3.80860779684984*^9, 3.8086078149466333`*^9}, {3.8086078488471737`*^9, 3.808607963246183*^9}, {3.808608006830275*^9, 3.8086080305171432`*^9}, { 3.8086080616880903`*^9, 3.808608098587665*^9}, {3.8086081478795652`*^9, 3.808608160291877*^9}, {3.8086082019383917`*^9, 3.808608203361676*^9}, { 3.8086083067456627`*^9, 3.8086083439901543`*^9}, 3.8100146902445793`*^9, { 3.810276620185*^9, 3.81027662387359*^9}},ExpressionUUID->"b92f76f2-221d-4861-8660-\ c8ad8cdbacc5"], Cell[BoxData[ FormBox["", TraditionalForm]], "Input", CellChangeTimes->{{3.794797938721333*^9, 3.794797939625449*^9}},ExpressionUUID->"9acef281-267e-4a43-832e-\ b6694cbcbc27"], Cell[TextData[StyleBox["Part 4. The Norm of the Component Vector is \ Proportional to the Number of Vectors Forming It", FontFamily->"Times New Roman", FontSize->18, FontWeight->"Bold"]], "Text", CellChangeTimes->{{3.7624053609701033`*^9, 3.762405366212669*^9}, { 3.770186407886489*^9, 3.770186418029324*^9}, {3.7701866820903387`*^9, 3.7701866828833227`*^9}, {3.7701872534303017`*^9, 3.770187254213917*^9}, 3.770188653216902*^9, {3.7704432574803457`*^9, 3.770443283879126*^9}, 3.770863292030612*^9, 3.781508953293495*^9, 3.794525637146884*^9, { 3.7947036059191*^9, 3.794703623013896*^9}, {3.794794358358652*^9, 3.794794464000637*^9}, {3.795248462420512*^9, 3.795248515356161*^9}, 3.8061879449709682`*^9}, TextAlignment->Center,ExpressionUUID->"c1a4db39-eef8-421c-9011-15883354b9e0"], Cell[TextData[{ StyleBox["When the total vector value of a specified vector swarm is \ determined, the mean norms between different component vectors should be \ proportional to the number forming them.", FontFamily->"Times New Roman", FontWeight->"Regular"], StyleBox[" ", FontFamily->"Songti SC", FontWeight->"Regular"], StyleBox["The following proves this viewpoint in detail.", FontFamily->"Times New Roman", FontWeight->"Regular"] }], "Text", CellChangeTimes->{{3.7560729545468388`*^9, 3.756073100638503*^9}, { 3.75607364213144*^9, 3.756073642706232*^9}, {3.764927060275012*^9, 3.764927070331305*^9}, {3.766298909798955*^9, 3.766298934069401*^9}, { 3.76678849446102*^9, 3.7667885609866743`*^9}, {3.7667885920287113`*^9, 3.766788625434799*^9}, 3.766788656730773*^9, {3.770186420477174*^9, 3.770186426690283*^9}, {3.770186717275547*^9, 3.770186737085937*^9}, { 3.77018677035857*^9, 3.7701868769223433`*^9}, {3.770186909441044*^9, 3.770186934172264*^9}, {3.770443177460298*^9, 3.770443179576709*^9}, { 3.77085909519851*^9, 3.770859121773682*^9}, {3.770859272385634*^9, 3.770859466240917*^9}, {3.77085952419211*^9, 3.770859620289151*^9}, { 3.770859736789328*^9, 3.7708597638254337`*^9}, {3.770859852069563*^9, 3.770859852509177*^9}, {3.7709329684060698`*^9, 3.7709329709427853`*^9}, { 3.794729739479273*^9, 3.794729743902834*^9}, {3.794729784036271*^9, 3.7947297887905207`*^9}, {3.7947351717935333`*^9, 3.794735230420698*^9}, { 3.794786111284349*^9, 3.7947861166459*^9}, {3.794786148358968*^9, 3.794786160635964*^9}, {3.794786239219668*^9, 3.794786240891473*^9}, { 3.7947865991961403`*^9, 3.794786675317934*^9}, {3.794786727578107*^9, 3.79478675723603*^9}, {3.7947868219791737`*^9, 3.794786954939763*^9}, { 3.794787001731368*^9, 3.7947870953835907`*^9}, {3.7947875164358253`*^9, 3.794787521354027*^9}, {3.794787953876657*^9, 3.794787955050716*^9}, 3.7947880499749537`*^9, {3.794788090698449*^9, 3.794788091590272*^9}, { 3.7947881586460037`*^9, 3.794788176801855*^9}, {3.794788212652172*^9, 3.794788222402103*^9}, {3.7947883609223413`*^9, 3.79478852668544*^9}, { 3.794788557156478*^9, 3.794788590422022*^9}, {3.794788638761198*^9, 3.794788647400923*^9}, {3.7947887997991657`*^9, 3.794788926981968*^9}, { 3.794788966891*^9, 3.794789011629299*^9}, {3.794789134906061*^9, 3.794789135848316*^9}, {3.7947891782875137`*^9, 3.794789205298427*^9}, { 3.794789272378778*^9, 3.794789304597063*^9}, {3.794791564175867*^9, 3.794791573151154*^9}, 3.794898423167468*^9, {3.794901499946456*^9, 3.794901518381403*^9}, {3.794902653480158*^9, 3.794902654256024*^9}, { 3.7952172620705214`*^9, 3.7952172855001183`*^9}, {3.795247855318746*^9, 3.795247868146678*^9}, {3.7959429369950657`*^9, 3.795942937573454*^9}, { 3.7959429773247643`*^9, 3.795942989545602*^9}, {3.796511124822124*^9, 3.796511141537737*^9}},ExpressionUUID->"a4d59ddd-098f-4583-834f-\ f13722ed46ec"], Cell[TextData[{ StyleBox["It has been proven that the degree of slowdown on all three axes \ is the same in ", FontFamily->"Times New Roman", FontWeight->"Regular"], StyleBox["Part 2", FontFamily->"Times New Roman"], StyleBox[" of the Supplementary Information. Then, let ", FontFamily->"Times New Roman", FontWeight->"Regular"], Cell[BoxData[ FormBox[ RowBox[{"\[InvisiblePrefixScriptBase]", StyleBox[ RowBox[{"\[ScriptCapitalM]", StyleBox["k", FontFamily->"Times New Roman", FontWeight->"Regular", FontSlant->"Plain"]}]]}], TraditionalForm]],ExpressionUUID-> "d19e6864-d1e5-49b2-bff5-730ef40e93b3"], StyleBox[" being the norm of momentum of ", FontFamily->"Times New Roman", FontWeight->"Regular"], StyleBox["k", FontFamily->"Times New Roman", FontWeight->"Regular", FontSlant->"Italic"], StyleBox[" particles observed from ", FontFamily->"Times New Roman", FontWeight->"Regular"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["R", FontFamily->"Euclid Math One", FontWeight->"Regular", FontSlant->"Plain"], StyleBox["u", FontFamily->"Times New Roman", FontSize->10.649999999999999`, FontWeight->"Regular"]], TraditionalForm]],ExpressionUUID-> "5cb19740-10e9-4342-8614-89b90deeb7c0"], StyleBox[", ", FontFamily->"Times New Roman", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[InvisiblePrefixScriptBase]", TraditionalForm]],ExpressionUUID-> "a370d80d-c4cf-4ad9-89a3-3069dd2b21e5"], StyleBox["it follows Maxwell distribution with scale parameter ", FontFamily->"Times New Roman", FontWeight->"Regular"], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ SqrtBox["k"], SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "-", SuperscriptBox["u", "2"]}]]}], SqrtBox["3"]], FontFamily->"Times New Roman"], TraditionalForm]], CellChangeTimes->{3.8060178194296713`*^9}, FontSize->21, FontWeight->"Regular",ExpressionUUID-> "ee452c16-ece6-4513-a681-be80b5b54cef"], StyleBox[" when observing from ", FontFamily->"Times New Roman", FontWeight->"Regular"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["R", FontFamily->"Euclid Math One", FontWeight->"Regular", FontSlant->"Plain"], StyleBox["u", FontFamily->"Times New Roman", FontSize->10.649999999999999`, FontWeight->"Regular"]], TraditionalForm]],ExpressionUUID-> "88375e9b-295d-4317-8ab2-b6f2d478f09b"], StyleBox[". 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