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Ìåæäó ðåæèìàìè d = 3 è d = 4 îïðåäåëåíàëèíèÿ êðîññîâåðà.Àïðîáàöèÿ ðàáîòû. Ïîëó÷åííûå ðåçóëüòàòû äîêëàäûâàëèñü è îáñóæäàëèñü íà ìåæäóíàðîäíûõ êîíôåðåíöèÿõ è øêîëàõ. Äàëåå ñëåäóåò ñïèñîêîñíîâíûõ äîêëàäîâ:1. Ìåæäóíàðîäíàÿ øêîëà Advanced Methods of Modern Theoretical Physics:Integrable and Stochastic Systems, oral talk Renormalization Groupapproach to turbulence 16 àâãóñòà - 21 àâãóñòà, 2015, Äóáíà, Ðîññèÿ18http://www.dubnaschool.cz/2015/2.
International conference Models in Quantum Field Theory, oral talkAnomalous scaling of passive scalar elds advected by the Navier-Stokesvelocity ensemble 21 ñåíòÿáðÿ - 25 ñåíòÿáðÿ, 2015, Ïåòåðãîô, Ðîññèÿhttp://hep.phys.spbu.ru/conf/mqft2015/common_e.htm3. 50-ÿ ìåæäóíàðîäíàÿ çèìíÿÿ øêîëà Ñàíêò-Ïåòåðáóðãñêîãî Èíñòèòóòàßäåðíîé Ôèçèêè, óñòíûé äîêëàä Anomalous scaling in magnetohydrodynamics 27 ôåâðàëÿ 4 ìàðòà, 2016, Ðîùèíî, Ðîññèÿhttp://hepd.pnpi.spb.ru/WinterSchool/archive/2016/index.shtml4. 19-é ìåæäóíàðîäíûé ñåìèíàð Quarks 2016, óñòíûé äîêëàä Renormalization Group approach to turbulence 29 ìàÿ 4 èþíÿ, 2016, Ïóøêèí, Ðîññèÿhttp://quarks.inr.ac.ru/2016/5.
51-ÿ ìåæäóíàðîäíàÿ çèìíÿÿ øêîëà Ñàíêò-Ïåòåðáóðãñêîãî Èíñòèòóòàßäåðíîé Ôèçèêè, ïîñòåðíûé äîêëàä Statistical restoration of brokensymmetries in fully developed turbulence 27 ôåâðàëÿ 4 ìàðòà, 2017,Ðîùèíî, Ðîññèÿhttp://hepd.pnpi.spb.ru/WinterSchool/archive/2017/index.shtml6. 10th CHAOS 2017 International Conference, óñòíûé äîêëàä Turbulentadvection of passive scalar eld near two dimensions 30 ìàÿ 2 èþíÿ,2017, Áàðñåëîíà, Èñïàíèÿhttp://www.cmsim.org/chaos2017.htmlÏóáëèêàöèè. Ïî òåìå äèññåðòàöèè îïóáëèêîâàíî 6 íàó÷íûõ ðàáîò â19èçäàíèÿõ, ðåêîìåíäîâàííûõ ÂÀÊ ÐÔ è âõîäÿùèõ â áàçû äàííûõ ÐÈÍÖ,Web of Science è Scopus:1. N.
V. Antonov and M. M. Kostenko Anomalous scaling of passive scalarelds advected by the Navier-Stokes velocity ensemble:Eects of strongcompressibility and large-scale anisotropy, Physical Review E 90, 063016(2014)2. N. V. Antonov and M. M. Kostenko Anomalous scaling in magnetohydrodynamic turbulence: Eects of anisotropy and compressibility in the kinematic approximation, Physical Review E 92, 053013 (2015)3. N. V. Antonov, N.M. Gulitskiy, M. M. Kostenko, and T. LucivjanskyRenormalization group analysis of a turbulent compressible uid near d= 4: Crossover between local and non-local scaling regimes, EuropeanPhysical Journal: Web of Conferences 125, 05006 (2016)4. N. V.
Antonov, N. M. Gulitskiy, M. M. Kostenko, and T. LucivjanskyAdvection of a passive scalar eld by turbulent compressible uid: renormalization group analysis near d = 4, Journal: Web of Conferences 137,10003 (2017)5. N. V. Antonov, N.M. Gulitskiy, M. M. Kostenko, and T. LucivjanskyTurbulent compressible uid: Renormalization group analysis, scalingregimes, and anomalous scaling of advected scalar elds, Physical ReviewE 95, 033120 (2017)6.
N. V. Antonov, N.M. Gulitskiy, M. M. Kostenko, and T. LucivjanskyStochastic Navier-Stokes equation and advection of a tracer eld: One-20loop renormalization near d = 4 EPJ Web of Conferences 164, 07044(2017)Ëè÷íûé âêëàä àâòîðà. Âñå îñíîâíûå ðåçóëüòàòû, èçëîæåííûå âäèññåðòàöèè, ïîëó÷åíû ñîèñêàòåëåì ëè÷íî ëèáî ïðè åå ïðÿìîì íåîòäåëèìîì ó÷àñòèè â ñîàâòîðñòâå.Ñòðóêòóðà è îáúåì ðàáîòû.Äèññåðòàöèÿ ñîñòîèò èç ââåäåíèÿ, ÷åòûð¼õ ãëàâ, çàêëþ÷åíèÿ è ñïèñêàëèòåðàòóðû èç 102 íàèìåíîâàíèé. Ðàáîòà èçëîæåíà íà 159 ñòðàíèöàõ èñîäåðæèò 20 ðèñóíêîâ è 2 òàáëèöû.Ïåðâàÿ ãëàâà ïîñâÿùåíà èññëåäîâàíèþ ñòîõàñòè÷åñêîé çàäà÷è ïåðåíîñà âÿçêîé ñæèìàåìîé æèäêîñòè óðàâíåíèåì Íàâüå-Ñòîêñà.
 íåé ïîäðîáíî ðàññêàçûâàåòñÿ î ïîñòðîåíèè ôóíêöèîíàëüíîé òåîðèè èç íà÷àëüíîéñòîõàñòè÷åñêîé, èññëåäóåòñÿ åå ðåíîðìèðóåìîñòü, ïîêàçàíà ïîñëåäîâàòåëüíîñòü ïîèñêà êðèòè÷åñêèõ ðàçìåðíîñòåé ïîëåé è ïàðàìåòðîâ.Âòîðàÿ ãëàâà ïîñâÿùåíà ïåðåíîñó ïàññèâíîãî ñêàëÿðíîãî ïîëÿ àíñàìáëåì ñêîðîñòè Íàâüå-Ñòîêñà äëÿ âÿçêîé ñæèìàåìîé æèäêîñòè. Ê ìîäåëèïðèìåíÿþòñÿ ìåòîäû ðåíîðìãðóïïû è îïåðàòîðíîãî ðàçëîæåíèÿ.Òðåòüÿ ãëàâà ïîñâÿùåíà ðåíîðìãðóïïîâîìó àíàëèçó ìîäåëè ïåðåíîñà ïàññèâíîãî ìàãíèòíîãî ïîëÿ, ïîèñêó íåïîäâèæíûõ òî÷åê, àíîìàëüíûõðàçìåðíîñòåé.×åòâåðòàÿ ãëàâà ïîñâÿùåíà èññëåäîâàíèþ äèíàìèêè æèäêîñòè äëÿîñîáåííîãî ñëó÷àÿ d = 4. Ãëàâà ðàçäåëåíà íà íåñêîëüêî ÷àñòåé: â ïåðâîéïðîâîäèòñÿ èññëåäîâàíèå äëÿ ñàìîãî ïîëÿ ñêîðîñòè, â îñòàëüíûõ äëÿïåðåíîñà ïàññèâíûõ ïîëåé (ïëîòíîñòè, òðåéñåðà è ìàãíèòíîãî ïîëÿ). çàêëþ÷åíèè ïåðå÷èñëÿþòñÿ îñíîâíûå ðåçóëüòàòû.211.
Ðà àíàëèç ñòîõàñòè÷åñêîãî óðàâíåíèÿ ÍÑ ñ ñèëüíîéñæèìàåìîñòüþ1.1.Îïèñàíèå ìîäåëèÓðàâíåíèå Íàâüå-Ñòîêñà (äàëåå ÍÑ) äëÿ âÿçêîé ñæèìàåìîé æèäêî-ñòè èìååò âèä [58]ρ∇t vi = ν0 [δik ∂ 2 − ∂i ∂k ]vk + µ0 ∂i ∂k vk − ∂i p + ηi ,(1.1)ãäå∇t = ∂t + vk ∂k(1.2)ýòî ëàãðàíæåâà (ãàëèëååâî-êîâàðèàíòíàÿ) ïðîèçâîäíàÿ, ∂t = ∂/∂t, ∂i =∂/∂xi , è ∂ 2 = ∂i ∂i îïåðàòîð Ëàïëàñà.Óðàâíåíèå (1.1) ïîëó÷åíî èç óðàâíåíèÿ:∂t (ρvi ) + ∂k Πik = ηi ,(1.3)Πik = ρvi vk + δik p + {âÿçêèå ÷ëåíû}(1.4)ãäåòåíçîð íàïðÿæåíèé, è óðàâíåíèÿ íåðàçðûâíîñòè∂t ρ + ∂i (ρvi ) = 0.(1.5) ýòèõ óðàâíåíèÿõ vi ñêîðîñòü, ρ ìàññîâàÿ ïëîòíîñòü, p äàâëåíèå, è ηi ïëîòíîñòü âíåøíåé ñèëû (íà åäèíèöó îáú¼ìà). Âñå ýòè âåëè÷èíû22çàâèñÿò îò x = {t, x}, ïðè÷¼ì x = {xi }, i = 1 .
. . d, ãäå d ïðîèçâîëüíàÿ(äëÿ îáùíîñòè) ðàçìåðíîñòü ïðîñòðàíñòâà. Ïîñòîÿííûå ν0 è µ0 ÿâëÿþòñÿäâóìÿ íåçàâèñèìûìè ìîëåêóëÿðíûìè êîýôôèöèåíòàìè âÿçêîñòè; â âÿçêèõ÷ëåíàõ â (1.1) ìû ÿâíî ðàçäåëèëè ïîïåðå÷íóþ è ïðîäîëüíóþ ÷àñòè. Ñóììèðîâàíèå ïî ïîâòîðÿþùèìñÿ çíà÷êàì ïîäðàçóìåâàåòñÿ ñåé÷àñ è áóäåò ïîäðàçóìåâàòüñÿ â äàëüíåéøåì.Ê óðàâíåíèÿì (1.1), (1.5) íåîáõîäèìî äîáàâèòü óðàâíåíèå ñîñòîÿíèÿ,p = p(ρ).
 ñàìîé ïðîñòîé ôîðìå, â ëèíåéíîì ïðèáëèæåíèè îíî âûãëÿäèòêàê ñîîòíîøåíèå(p − p̄) = c20 (ρ − ρ̄)(1.6)ìåæäó ðàçíîñòÿìè äàâëåíèÿ è ïëîòíîñòè ñ èõ ñðåäíèìè çíà÷åíèÿìè. Ïîñòîÿííàÿ âåëè÷èíà c0 èìååò çíà÷åíèå (àäèàáàòè÷åñêîé) ñêîðîñòè çâóêà.Ðóêîâîäñòâóÿñü ñòàòü¼é [64], ìû ïîäåëèì (1.1) íà ρ, à â âÿçêèõ ÷ëåíàõ çàìåíèì ρ å¼ ñðåäíèì çíà÷åíèåì. Ýòî ïðèáëèæåíèå (êîòîðîå è íóæíîäëÿ ïîëó÷åíèÿ ëîêàëüíîé ïîëåâîé ìîäåëè) íåÿâíî îïðàâäàíî â ðàáîòå [61];òàêæå çàìåòèì, ÷òî âÿçêîñòü â èíåðöèîííîì èíòåðâàëå ìåíÿåòñÿ íå ñóùåñòâåííî.
Îñòàâèì òå æå îáîçíà÷åíèÿ ν0 è µ0 äëÿ ïîëó÷åííûõ ïîñòîÿííûõêîýôôèöèåíòîâ êèíåìàòè÷åêîé âÿçêîñòè. Òîãäà óðàâíåíèÿ (1.1), (1.5) ìîæíî ïåðåïèñàòü â âèäå∇t vi = ν0 [δik ∂ 2 − ∂i ∂k ]vk + µ0 ∂i ∂k vk − ∂i φ + fi ,(1.7)∇t φ = −c20 ∂i vi ,(1.8)ãäå ââåäåíî íîâîå ñêàëÿðíîå ïîëåφ = c20 ln(ρ/ρ̄)(1.9)23à fi = fi (x) ÿâëÿåòñÿ ïëîòíîñòüþ âíåøíåé ñèëû (ïðèõîäÿùåéñÿ íà åäèíèöóìàññû). ñòîõàñòè÷åñêîé ôîðìóëèðîâêå çàäà÷è âíåøíþþ ñèëó ñëåäóåò ïîíèìàòü êàê ñëó÷àéíîå âíåøíåå ïîëå, ìîäåëèðóþùåå ïîñòóïëåíèå â ñèñòåìóýíåðãèè, ïîëó÷åííîé ïðè ïåðåìåøèâàíèè íà áîëüøèõ ìàñøòàáàõ.
Ïðèíÿòîñ÷èòàòü, ÷òî äåòàëè å¼ ñòàòèñòèêè íå âàæíû, òàê ÷òî ðàñïðåäåëåíèå áóäåìñ÷èòàòü Ãàóññîâûì ñ íóëåâûì ñðåäíèì, íå êîððåëèðîâàííûì ïî âðåìåíè(äëÿ ãàëèëååâîé ñèììåòðèè), è âêëþ÷àþùèì â ñåáÿ íåêîòîðûé òèïè÷íûéÈÊ ìàñøòàá L (èíòåãðàëüíûé ìàñøòàá). Ñ äðóãîé ñòîðîíû, äëÿ èñïîëüçîâàíèÿ ñòàíäàðòíîé òåõíèêè Ðà âàæíî, ÷òîáû å¼ êîððåëÿöèîííàÿ ôóíêöèÿïðè áîëüøèõ çíà÷åíèÿõ àðãóìåíòà óáûâàëà ñòåïåííûì îáðàçîì. Áîëåå äåòàëüíûå ðàññóæäåíèÿ ìîæíî íàéòè â [42, 43, 79].  íàñòîÿùåé ðàáîòå êîððåëÿöèîííàÿ ôóíêöèÿ âûáðàíà ñëåäóþùèì îáðàçîì [64]00hfi (x)fj (x )i = δ(t − t )dkDijf (k) exp{ikx},dk>m (2π)Z(1.10)ãäåDijf (k)=g0 ν03 k 4−d−yhPij⊥ (k)+ikαPij (k).(1.11)kÇäåñü Pij⊥ (k) = δij − ki kj /k 2 è Pij (k) = ki kj /k 2 ïîïåðå÷íûé è ïðîäîëüíûéïðîåêòîðû, k = |k| âîëíîâîå ÷èñëî, g0 è α ïîëîæèòåëüíûå ïàðàìåòðû;ìíîæèòåëü ν03 âûäåëåí äëÿ óäîáñòâà.
Ïàðàìåòð m = L−1 îáåñïå÷èâàåò ÈÊðåãóëÿðèçàöèþ; å¼ òî÷íàÿ ôîðìà íåñóùåñòâåííà è äëÿ ïðîñòîòû âû÷èñëåíèé áóäåì èñïîëüçîâàòü ðåçêóþ "îòñå÷êó". Âåëè÷èíà 0 < y 6 4 èãðàåòðîëü, ïîäîáíóþ ε = 4 − d â Ðà òåîðèè êðèòè÷åñêîãî ïîâåäåíèÿ [41, 80]:ýòî îáåñïå÷èâàåò ÓÔ ðåãóëÿðèçàöèþ (òàê ÷òî ÓÔ ðàñõîäèìîñòè ÿâëÿþòñÿïîëþñàìè ïî y ), è êîîðäèíàòû óñòîé÷èâûõ òî÷åê, è ðàçëè÷íûå ïàðàìåòðû24âû÷èñëÿþòñÿ êàê ðÿä ïî y . Íàèáîëåå ðåàëüíîå (ôèçè÷åñêîå) çíà÷åíèå äîñòèãàåòñÿ â ïðåäåëå y → 4, ãäå ôóíêöèè â (1.11) ìîãóò áûòü ðàññìîòðåíû (ñíàäëåæàùèì âûáîðîì àìïëèòóäû) êàê ñòåïåííîå ïðåäñòàâëåíèå ôóíêöèèδ(k): ýòî ñîîòâåòñòâóåò èäåàëüíîé êàðòèíå: ýíåðãèÿ ââîäèòñÿ â ñèñòåìó íàáåñêîíå÷íî áîëüøèõ ìàñøòàáàõ.1.2.Òåîðåòèêî-ïîëåâàÿ ôîðìóëèðîâêà è ïðàâèëà ÔåéíìàíàÑîãëàñíî îáùåé òåîðåìå [41, 80], ñòîõàñòè÷åñêàÿ çàäà÷à (1.7), (1.8),(1.10), (1.11), ýêâèâàëåíòíà òåîðåòèêî-ïîëåâîé ìîäåëè ñ óäâîåííûì íàáîðîì ïîëåé Φ = {vi0 , φ0 , vi , φ} è ôóíêöèîíàëîì äåéñòâèÿ1 0 f 0vi Dik vk + vi0 −∇t vi + ν0 [δik ∂ 2 − ∂i ∂k ]vk + u0 ν0 ∂i ∂k vk − ∂i φ +2+ φ0 −∇t φ + v0 ν0 ∂ 2 φ − c20 (∂i vi ) ,(1.12)S(Φ) =ãäå Df êîððåëÿöèîííàÿ ôóíêöèÿ (1.10), (1.11).
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