
percolation siddhant

saankhyikiya bhautiki aur ganit mein, pratiksharan siddhant ack network key vyavhaar kaa varnan kartaa haye jaba nodes yaa link jode jaate hain. yeh ack jyaamitiya prakaar kaa charan sankraman haye, kyonki jodd key ack mahatvapurna ansh parr chhote, disconnect kiye gaye samoohon kaa network kaafee badey judey hue, tathaakathit faile hue samoohon mein vilay hoe jaataa haye. samagri vigyaan aur kaee anya vishayon mein percolation siddhant key anuprayogon kii charcha yahaan aur lekhon mein network siddhant aur percolation mein kii gaee haye.
parichay.
ack pratinidhi prashn (aur naam kaa srot) iss prakaar haye. maan leejiye kii kuchh chhidrapurna samagri key oopar kuchh taral daala jaataa haye. kyaa taral chhed seey chhed taka apana raasta banaa paayega aur neeche taka pahunch paayega? iss bhautik prashn koo ganitiya roop seey sheershon key tri-ayami network key roop mein model kiya jaataa haye, jise aamtaur parr "site" kahaa jaataa haye, jisamen pratyek dou padosiyon key beach kaa kinara yaa "bandhan" sambhavana key saath khula (taral koo anumati dene vaalaa) hoe sakata haye, yaa sambhavana key saath bund hoe sakata haye, aur unhein swatantra maanaa jaataa haye. isaliye, diye gaye key liye, ack khula maarg (jiska arth haye ack maarg, jiska pratyek link ack "khula" bandhan haye) oopar seey neeche taka maujood honey kii kyaa sambhavana haye? badey key liye vyavhaar praathamik ruchi kaa haye. iss samasya koo, jise abb bandhan pratichhedan kahaa jaataa haye, ganit saahitya mein taba seey ganitvidon aur bhautikavidon dwara gahan adhyayan kiya gayaa haye.
yaadrichhik graph praapt karane key liye thode alag ganitiya model mein, ack site koo sambhavana key saath "kabja" kiya jaataa haye yaa "khaali" (jiss mamle mein isake kinaaron koo hataa diyaa jaataa haye) sambhavana key saath; sambandhit samasya koo site percollation kahaa jaataa haye. sawal vahee haih diye gaye "pee" key liye, oopar aur neeche key beach ack maarg maujood honey kii kyaa sambhavana haye? isi tarah, koee bhee yeh poochh sakata haye kii ack judey hue graph koo dekhate hue kii viphaltaon key kiss ansh parr graph disconnect hoe jaayega (koee badaa ghatak naheen).
kissy bhee jaali ayam key liye samaan prashn poochhe jaa sakate hain. jaisa kii kaafee vishisht haye, vaastav mein kewal badey network kii tulanaa mein anant network kii jaanch karna aasaan haye. iss mamle mein sambandhit prashn haih kyaa ack anant khula samooh maujood haye? yaani, kyaa network key madhyam seey "anant lambai key judey hue binduon kaa koee maarg haye? kolmogorov key shoonya-ack niyam key anusaar, kissy bhee diye gaye niyam key anusaar, ack anant samooh key maujood honey kii sambhavana yaa tou shoonya yaa ack haye. choonki yeh sambhavana (yugman tark key madhyam seey pramaan) kaa ack badhata huaa kaary haye, isaliye ack mahatvapurna (dwara niroopit) honah chaahiye jisake neeche sambhavana hamesha 0 hoti haye aur jisake oopar sambhavana hamesha 1 hoti haye. vyavhaar mein, iss mahatvapurnata kaa nireekshan karna bahut aasaan haye. 100 key roop mein chhote key liye bhee, oopar seey neeche taka ack khule maarg kii sambhavana shoonya key bahut karib seey shoonya key bahut karib seey ack key bahut karib taka teji seey badh jaati haye.
itihas.
flory-stockmayer siddhant percolation prakriyaon kii jaanch karane vaalaa pahala siddhant thaa.
risaav model kaa itihas jaisa kii humm jaanate hain kii isaki jadein koyla udyog mein hain. audyogik kranti key baad seey, oorjaa key iss srot key aarthik mahatva ney isaki sanrachana koo samajhane aur isake upayog koo anukoolit karane key liye kaee vaigyaanik adhyayanon koo badhaava diyaa. 30 'aur 40' key dauran, carbonic rasayan vigyaan dwara gunaatmak vishleshan ney adhik seey adhik matratmak adhyayan key liye adhik seey adhik jagah chhod dee. 
iss sandarbh mein, british koyla upayog anusandhaan sangh (bee. see. yuu. aar.) 1938 mein banaaya gayaa thaa. yeh koyla khadanon key maalikon dwara vitt poshit ack shodh sangh haye. 1942 mein, rosalind franklin, jinhone haal hii mein cambridge vishvavidyalaya seey rasayan vigyaan mein snaatak kiya, bee. kyu. aar. mein shamil hoe gaye. unhone koyale key ghanatva aur chhidrata parr shodh shuru kiya. dwiteeya vishva yuddh key dauran, koyla ack mahatvapurna ranneetik sansaadhan thaa. iska upayog oorjaa key srot key roop mein kiya jaataa thaa, lekin yeh gais mask kaa mukhya ghatak bhee thaa.
koyla ack chhidrapurna madhyam haye. isake 'vaastavik' ghanatva koo maapane key liye, ise ack taral yaa gais mein dubona thaa jiska anu isake sookshm chhidron koo bharane key liye paryaapt chhota haye. kaee gasses (helium, methanol, hexen, benzene) kaa upayog karke koyale key ghanatva koo maapane kii koshish karate hue aur jaba unhone upayog kii gaee gais key aadhar parr alag-alag moolya paae, tou rosalind franklin ney dikhaaya kii koyale key chhidra vibhinn lambai key sookshm sanrachanaon seey baney hotey hain joe gasses mein bhedbhaav karane key liye ack sookshm chhalani key roop mein kaary karate hain. unhone yeh bhee paaya kii inn sanrachanaon kaa aakar koyla utpaadan key dauran karbanikaran key tapman parr nirbhar kartaa haye. inn shodhon key saath, unhone ack phd kii degree praapt kii aur unhone 1946 mein bee. see. yuu. aar. chhod diyaa.
pachas key dashak key madhya mein, saiman broadbent ney bcura mein ack saankhyikeevid key roop mein kaam kiya. anya hiton key alawa, unhone gais mask mein koyale key upayog kaa adhyayan kiya. ack sawal yeh samajhana haye kii kaise ack taral padaarth koyale key chhidron mein fail sakata haye, jise khuli yaa bund surangon kii yaadrichhik bhulbhulaiya key roop mein model kiya gayaa haye. 1954 mein, monte carlo vidhiyon parr ack sangoshthi key dauran, unhone iss model kaa vishleshan karane key liye sankhyaatmak vidhiyon key upayog parr johnn hammersley seey sawal poochhe.
broadbent aur hammersley ney 1957 key apane lekh mein iss ghatana koo model karane key liye ack ganitiya model pesh kiya, joe kii percolation haye.
mahatvapurna mapdand kii ganana.
adhikansh anant jaali graph key liye, thheek seey ganana naheen kii jaa sakati haye, halanki kuchh mamalon mein ack sateek moolya hotaa haye. udaaharan key liyeh
sootra _ 9 
yeh ingit kartaa haye kii ack diye gaye degree vitaran key liye, clustering ack badee pratiksharan seema kii or leey jaati haye, mukhya roop seey kyonki link kii ack nishchit sankhya key liye, clustering sanrachana vaishvik connections koo kamjor karane kii keemat key saath network key mool koo majboot karati haye. uchch clustering vaale network key liye, majboot clustering core-paridhiya sanrachana koo prerit curr sakati haye, jisamen core aur paridhiya vibhinn mahatvapurna binduon parr ris sakata haye, aur uparokt anumaanit upchaar lagu naheen haye.
saarvbhaumiktaa.
saarvbhaumiktaa siddhant bataata haye kii ank kaa sankhyaatmak moolya graph kii sthaaniya sanrachana dwara nirdhaarit kiya jaataa haye, jabaki mahatvapurna seema key paas kaa vyavhaar, saarvbhaumik mahatvapurna pratipaadak dwara visheshta haye. udaaharan key liye, sabhi 2dee jaali key liye ack hii ghatank key saath ack shakti kaanoon key roop mein sankat parr samoohon key aakar kaa vitaran kshay hoe jaataa haye. iss saarvbhaumiktaa kaa arth haye kii ack diye gaye ayam key liye, vibhinn mahatvapurna pratipaadak, samoohon kaa asthi-vibhaajak ayam jaali prakaar aur pratiksharan prakaar (ee.jee., bonned yaa site). haalaanki, haal hii mein ack bhaarit samtaliya yaadrichhik jaali (doubleu. pee. aiss. ell.) parr risaav kiya gayaa haye aur paaya gayaa haye kii halanki doubleu. pee. aiss. ell. kaa ayam usa sthaan key ayam key saath mell khaata haye jahan yeh antarnihit haye, lekin iska saarvbhaumiktaa varg sabhi gyaat samtaliya jaali seey alag haye.
charan.
upa-aalochanatmak aur atee-aalochanatmak.
upa-mahatvapurna charan mein mukhya tathya "ghaatiya kshay" haye. yaani, jaba, ack vishisht bindu (udaaharan key liye, mool) key ack khule samooh (jiska arth haye graph key "khule" kinaaron kaa ack adhikatam judaa huaa sett) mein nihit honey kii sambhavana shoonya mein teji seey kshay hoe jaati haye. yeh teen aur adhik aayamon mein aur swatantra roop seey parcolation key liye saabit huaa thaa. dou aayamon mein, yeh keston key pramaan kaa hissa thaa.
varg jaali kaa dohra graph bhee varg jaali haye. yeh iss prakaar haye kii, dou aayamon mein, atee-mahatvapurna charan ack upa-mahatvapurna pratiksharan prakriya key liye dohri haye. yeh anivarya roop seey atee mahatvapurna model key baare mein poori jaankaari pradaan kartaa haye. teen aur adhik aayamon mein atee mahatvapurna charan key liye mukhya parinam yeh haye kii, paryaapt roop seey badey key liye, dwi-ayami slab mein ack anant khula samooh haye. yeh saabit huaa.
dou aayamon mein, ack advitiya anant bund samooh honey kii sambhavana haye (ack bund samooh graph key "bund" kinaaron kaa ack adhikatam judaa huaa samooh haye). iss prakaar upa-mahatvapurna charan koo ack anant bund mahasagar mein seemit khule dweepon key roop mein varnit kiya jaa sakata haye. jaba isake thheek vipareet hotaa haye, ack anant khule mahasagar mein seemit bund dweepon key saath. jaba seey aur aur key beach anant khule aur bund samoohon kaa saha-astitva haye, taba seey chitra adhik jatil hotaa haye.
aalochanatmak.
nirnaayak bindu parr pratiksharan kii ack vilakshanta hoti haye aur kaee guna sootra _ 10 key saath ack shakti-niyam key roop mein vyavhaar karate hain, sootra _ 2 key paas. scaling siddhant mahatvapurna ghatakon key astitva kii bhavishyavani kartaa haye, joe aayamon kii sankhya "dee" parr nirbhar kartaa haye, joe ekvachanta key varg koo nirdhaarit kartaa haye. jaba inn bhavishyavaniyon koo anuroop kshetra siddhant aur scram-lovener vikas key tarkon dwara samarthit kiya jaataa haye, aur ghatakon key liye anumaanit sankhyaatmak moolya shamil hotey hain. inmein seey adhikansh bhavishyavaniyaan anumanatmak hain sivaay isake kii jaba aayamon kii sankhya yaa tou yaa santusht hoe. inmein shamil hainh
dekhein. 11 yaa adhik aayamon mein, yeye tathya kaafee hudd taka feeta vistaar key roop mein jaani jaane vaali takneek kaa upayog karke saabit hotey hain. aisaa maanaa jaataa haye kii feeta vistaar kaa ack sanskaran 7 yaa adhik aayamon key liye maanya honah chaahiye, shayed 6 aayamon key threshold mamle key liye bhee nihitarth key saath. feeta key vistaar key liye risaav kaa sambandh paaya jaataa haye.
dou aayamon mein, pahala tathya ("mahatvapurna charan mein koee risaav naheen") dwandwa kaa upayog karate hue kaee jaali key liye saabit hotaa haye. dou ayami risaav parr oded scram key anuman key madhyam seey paryaapt pragati kii gaee haye kii ack badey samooh kii scaling seema koo scram-lovener vikas key sandarbh mein varnit kiya jaa sakata haye. yeh anuman trikoniya jaali parr site key risaav key vishesh mamle mein saabit huaa thaa.
anuprayog.
jeev vigyaan, jaiv rasayan aur bhautik vishaanu vigyaan mein.
hepatitis bee virus capsid key vikhandan seema kii bhavishyavani aur prayogaatmak roop seey pataa lagaane key saath javic virus key khol (capsid) key vikhandan kii safaltapurvak bhavishyavani karane key liye percolation siddhant kaa upayog kiya gayaa haye. jaba nanoscopic khol seey ack mahatvapurna sankhya mein upa-ikaaiyon koo yaadrichhik roop seey hataa diyaa jaataa haye, tou isake tukade aur iss vikhandan kaa pataa anya ekal-kann takneekon key beach charge detection maas spectroscopy (see. dee. aim. aiss.) kaa upayog karke lagaaya jaa sakata haye. yeh samanya board game jenga key liye ack aanvik anuroop haye, aur virus prithakkaran key vyaapak adhyayan key liye prasangikta haye. dilchasp baat yeh haye kii adhik sthir viral kann (adhik vikhandan seema key saath jhukaav) prakriti mein adhik prachurta mein paae jaate hain.
paaristhitiki mein.
paryaavaran vikhandan pashu aavason koo kaise prabhaavit kartaa haye aur plaig bacteria "yersinia pastice" kaise failata haye, isake model key adhyayan key liye percolation siddhant koo lagu kiya gayaa haye.

