TEBURIN BILLIARD MAI SIFFAR LEMO INDA ƘWALLO KE ZUWA KO'INA

Masana lissafi suna da abin wasa da suka fi so don nazarin hargitsi (chaos): billiard. Wani digo yana tafiya a layuka madaidaita a cikin wata siffa a rufe kuma yana tunkuɗewa daga gefe “kamar yadda photon ke yi a cikin madubi”, yana barin wurin da kusurwa ɗaya da wadda ya zo da ita. Babu gogayya, babu juyawa, babu ƙarshe. Tambayar ita ce me irin wannan ƙwallo ke yi a dogon lokaci.

Ga wasu teburai, amsar tana da ban mamaki matuƙa. Motsin yana da ergodic: kusan kowace hanya daga ƙarshe tana ziyartar dukkan wuraren da alƙiblun da za su yiwu, tana ɗaukar lokaci a kowane yanki daidai da girmansa. Wentao Fan da Boris Hasselblatt sun tabbatar da wani abu mafi ƙarfi ma ga teburinsu, halin Bernoulli (Bernoulli property): a ƙididdiga, ƙwallon da ke tunkuɗewa yana nuna hali kamar jerin zaɓe bazuwar masu zaman kansu.

Matsalar gefuna masu zagaye

Teburan da aka daɗe ana sanin suna da hargitsi suna da wata matsala. Wasu sharuɗɗa da masanin lissafi Maciej Wojtkowski ya tsara sun bayyana yadda bangwaye masu lanƙwasa da ke tattara haske za su iya haifar da hargitsi — amma suna buƙatar cikakkiyar da’irar da kowane bango baka ɗaya kawai daga cikinta ne ta shiga gaba ɗaya a cikin teburin — abin da ke tilasta gefen teburin ya shiga ciki a wani wuri. Hakan yana hana teburan da ke lanƙwasa waje sosai (strictly convex), wato gefensu yana kumbura zuwa waje a ko’ina, ba tare da wani sashi miƙaƙƙe ko mai shiga ciki ba.

Don haka wata tambaya ta dabi’a ta kasance a buɗe: shin billiard mai lanƙwasa waje sosai zai iya zama ergodic ko kaɗan?

Lemo

A 2013, Jingyu Chen, Luke Mohr, Hong-Kun Zhang da Pengfei Zhang sun gabatar da wani iyalin ’yan takara: billiards na lemo. Ɗauki faifai biyu, ɗaya mai radius r da wani mafi girma sosai mai radius R, an ajiye su ta yadda babban faifai ya ƙunshi tsakiyoyin biyu. Wurin da suka haɗu shi ne teburin — a aikace, ƙaramin faifai da aka aske gefe ɗaya da bakan wani babban da’ira mai sassauci, da ke haɗuwa da sauran a kusurwoyi biyu. Masu bincike huɗun sun yi hasashen cewa irin waɗannan lemo suna da ergodic idan radius biyun sun bambanta ƙwarai.

Teburin billiard na lemo L(r, R, ζ): wurin haɗuwar ƙaramin da’ira mai radius r da babban da’ira mai radius R, da ke haɗuwa a kusurwoyi A da B. — Figure 1, Fan and Hasselblatt (2026), arXiv:2610.28247.

Hujja

Fan, a Hebei Normal University a China, da Hasselblatt, a Tufts University a Amurka, sun tabbatar da hasashen ga lemo da suka karkata sosai. Babban ka’idarsu tana aiki idan kusurwar ζ da ke tsara girman yankan tana tsakanin 0 da arctan(1/3), wato kusan digiri 18, kuma idan babban radius ya wuce r · 30,000 / sin²ζ — a mafi kyau ninki 300,000 na r, ga yankan mafi faɗi da aka yarda da shi, kuma fiye da haka idan yankan ya ƙanƙanta. “Gwargwadon saninmu,” suka rubuta, “wannan shi ne misali na farko na billiard mai lanƙwasa waje sosai wanda aka tabbatar da ergodicity ɗinsa.”

Aikin ya gina ne a kan wata takardar abokiya ta marubutan iri ɗaya, wadda ta nuna cewa dawowar ƙwallon zuwa wani rukunin tunkuɗewa da aka zaɓa da kyau tana miƙe hanyoyin da ke kusa da juna nesa da juna a ma’auni tsayayye. Abin da ya rage shi ne mafi wuyar sashi: sarrafa layukan singularity (singularity curves), hanyoyin da ke fitowa daga ko ke shiga ɗaya daga cikin kusurwoyi biyu. Ba a kan teburin ake zana waɗannan layuka ba, sai a kan wata taswirar tunkuɗewa, inda ake ajiye kowane tunkuɗewa bisa wurinsa a kan gefen teburin da kusurwar tashinsa; a cikin billiards na lemo, wasu daga cikin waɗannan layuka suna tsaye kusan a miƙe a can kuma suna juya alamar gangarensu — wani siffa da “aka daɗe ana ɗauka a matsayin babbar matsala ga dabarun billiard na yau da kullum”. Marubutan sun nuna cewa bayan tunkuɗewa ɗaya kawai, waɗannan layuka suna daidaita da kyau. Wannan ne ainihin dalilin da ya sa suka zaɓi su kalli ƙwallon ba a kowane tunkuɗewa ba, sai a ƙarshen kowane jerin tunkuɗewa a jere a kan ƙaramin da’irar — na kafin na ƙarshe na jerin, ko shi kaɗai idan jerin ɗaya ne kawai.

Sauran shi ne bincike mai kyau na sharuɗɗa huɗu na fasaha da wata ka’idar “ergodicity ta wuri ɗaya” (local ergodicity) ta 2013 ke buƙata, wadda aka daidaita ga lemo. Wannan ya cika kusan duk takardar mai shafuka 58.

Abin da ke gaba

Sakamakon ya shafi lemo masu karkata sosai: da iyakar da ke sama, babban da’irar ya fi ƙaramin girma da ɗaruruwan dubban ninki (aƙalla 300,000). Lemo marasa karkata sosai har yanzu ba a tabbatar da su ba.

Marubutan sun lissafa abubuwan da suke son cimmawa na gaba — yadda motsin ke cakuɗuwa da sauri, da dokokin ƙididdiga da ke biyo baya — kuma sun ƙare da wata tambaya mai buri. Lemonsu har yanzu yana da kusurwoyi biyu. Shin akwai billiard mai lanƙwasa waje sosai kuma mai ergodic wanda gefensa yake santsi kwata-kwata, ba tare da wani kusurwa ko ɗaya ba? Sun nuna wasu tsoffin ’yan takara da aka yi nazari “tun kusan rabin ƙarni da ya wuce”: teburai da aka yi da murabba’i mai tsawo da aka rufe da bakunan da’ira da ke haɗuwa da shi cikin santsi. Wani layi ɗaya a cikin godiyarsu ma ya fito fili: “Ba a yi amfani da AI a kowane fanni na shirya wannan maƙala ba.”