Диссертация (Статистическое описание многочастичных эффектов в классических ион-молекулярных системах), страница 23

Äëÿ ñëó÷àÿ d = 2, êîãäà ïîëèìåðíàÿ öåïü ñèëüíî àäñîðáèðîâàíà íà ïëîñêóþ ïîâåðõíîñòü, ôîðìóëà (5.62) áûëà èñïîëüçîâàíà â ðàáîòàõ[178, 181].Ôóíêöèÿ ðàñïðåäåëåíèÿ ïîòåíöèàëüíîé ýíåðãèèãàóññîâîé ïîëèìåðíîé öåïè âd-ìåðíîéïàðàáîëè÷åñêîé ïîòåíöèàëüíîé ÿìåÏîñòàíîâêà çàäà÷èÐàññìîòðèì ôóíêöèþ ðàñïðåäåëåíèÿ âåðîÿòíîñòè ïîòåíöèàëüíîé ýíåðãèè ãàóññîâîé ïîëèìåðíîé öåïè â d-ìåðíîé ñèììåòðè÷íîé ïîòåíöèàëüíîéÿìå, êîòîðàÿ ìîæåò áûòü çàïèñàíà â ôîðìå ñëåäóþùåãî èíòåãðàëà ïî òðàåêòîðèÿì:ZP () =dDrexp − 2Z02bZN κdτ ṙ2 (τ ) δ  −20ZN0129dτ r2 (τ ) ,(5.63)ãäå κ êîýôôèöèåíò æåñòêîñòè èZDr exp −Z0 =d2b2ZN(5.64)dτ ṙ2 (τ )0 ïîñòîÿííàÿ íîðìèðîâêè ìåðû Âèíåðà.

Ïîëîæèì, ÷òî îäèí èç êîíöîâ ïîëèìåðíîé öåïè çàôèêñèðîâàí â íà÷àëå êîîðäèíàò, ò.å., r(0) = 0. Êðîìåòîãî, ïîëîæèì, ÷òî öåíòð ïàðàáîëè÷åñêîé ÿìû òàêæå íàõîäèòñÿ â íà÷àëåêîîðäèíàò. Èñïîëüçóÿ, êàê è â ïåðâîé ÷àñòè, çàìåíó ïåðåìåííûõ τ = sN ,ãäå 0 ≤ s ≤ 1, ôóíêöèÿ ðàñïðåäåëåíèÿ ïåðåïèñûâàåòñÿ â ñëåäóþùåì âèäåZP () =Drdexp −Z02N b2Z1 dsṙ2 (s) δ  −κN20Z1dsr2 (s) .(5.65)0Äàëåå, ïðèìåíÿÿ ôîðìóëó (5.31) äëÿ äåëüòà-ôóíêöèè â ïîäûíòåãðàëüíîìâûðàæåíèè (5.65), ïîëó÷èì:Z+∞P () =dξK(ξ)e−iξ ,2π(5.66)−∞ãäå ââåäåíà õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿZK(ξ) =Drdexp −Z02N b2Z1dsṙ2 (s) +iξκN20Z1dsr2 (s) .(5.67)0Âû÷èñëåíèå õàðàêòåðèñòè÷åñêîé ôóíêöèèÕàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ (5.67) ìîæåò áûòü ïåðåïèñàíà â ôîðìå:ZK(ξ) =Drdexp −Z02N b2Z1ds ṙ2 (s) − Ω2 r2 (s)  ,(5.68)01/2ãäå Ω = Ω(ξ) =.Çàòåì, âûäåëÿÿ ÿâíîå èíòåãðèðîâàíèå ïî ðàäèóñó-âåêòîðó R íåçàôèêñèðîâàííîãî êîíöåâîãî ìîíîìåðíîãî çâåíà öåïè, ïðèäåì ê âûðàæåíèþiξκN 2 b2dr(1)=RZZK(ξ) =dRr(0)=0130Drexp [−S[r]] ,Z0(5.69)ãäådS[r] =2N b2Z1(5.70)ds ṙ2 (s) − Ω2 r2 (s)0 âñïîìîãàòåëüíûé ôóíêöèîíàë.Êàê è â ïåðâîé ÷àñòè, âû÷èñëÿåì ãàóññîâûé èíòåãðàë ìåòîäîì ñåäëîâîéòî÷êè.

Ïðåäñòàâèì ñëó÷àéíóþ ôóíêöèþ r(s) â âèäå ñóììû:r(s) = vSP (s) + ρ(s).(5.71)Çäåñü ôóíêöèÿ vSP (s) óäîâëåòâîðÿåò óðàâíåíèþ Ýéëåðà-ËàãðàíæàδS[vSP ]d2=v̈(s)+Ωv(s)= 0,SPSPδvSP (s) N b2èëè(5.72)è ñëåäóþùèì ãðàíè÷íûì óñëîâèÿì vSP (0) = 0 è vSP (1) = R. Ñëó÷àéíàÿ âåêòîðíàÿ ôóíêöèÿ ρ(s), îïèñûâàþùàÿ ñëó÷àéíûå ôëóêòóàöèè îêîëî ýêñòðåìàëè vSP (s), óäîâëåòâîðÿåò íóëåâûì ãðàíè÷íûì óñëîâèÿì, ò.å.ρ(0) = ρ(1) = 0.

Ðåøåíèå óðàâíåíèÿ (5.72) äàåòv̈SP (s) + Ω2 vSP (s) = 0vSP (s) =R sin(Ωs).sin Ω(5.73)Äàëåå, âûïîëíÿÿ ïðîñòûå àëãåáðàè÷åñêèå ïðåîáðàçîâàíèÿ, ïîëó÷èìK(ξ) =ãäåd2πN b2d/2ZG(ξ)G(ξ) =èdR exp [−S[vSP ]] ,Ωsin Ωd/2(5.75),dΩR2dvSP (1)v̇SP (1)=.S[vSP ] =2N b22N b2 tan Ωãàóññîâûé èíòåãðàë ïî R, ïîñëå àëãåáðàè÷åñêèõÂû÷èñëÿÿâàíèé, ïîëó÷èìK(ξ) =1cos Ω(ξ)(5.74)d/2(5.76)ïðåîáðàçî(5.77).Ôîðìóëà (5.77) ïîçâîëÿåò âû÷èñëèòü êóìóëÿíòû è ìîìåíòû ðàñïðåäåëåíèÿ. Ñòåïåííîé ðÿäln(cos Ω) =∞X(−1)n+1 B2n 22n−1 (1 − 22n )(2n)!nn=1131Ω2n(5.78)âìåñòå ñ ñîîòíîøåíèåì (5.36) äàåò ñëåäóþùåå âûðàæåíèå äëÿ êóìóëÿíòîâχk =(−1)k (k − 1)!B2k 22k−2 (1 − 22k )(κN 2 b2 )k ,k−1(2k)!d(5.79)êîòîðîå ïîçâîëÿåò ïîëó÷èòü ìîìåíòû 3d + 42κN 2 b2µ1 = χ1 = hi =, µ2 = χ2 + µ21 = 2 =κN 2 b2 , ..448d(5.80)Èíòåðåñíî îòìåòèòü, ÷òî ñðåäíÿÿ ïîòåíöèàëüíàÿ ýíåðãèÿ ãàóññîâîé ïîëèìåðíîé öåïè â d-ìåðíîé ñèììåòðè÷íîé ïîòåíöèàëüíîé ÿìå íå çàâèñèò îòðàçìåðíîñòè ïðîñòðàíñòâà.Îöåíêà ôóíêöèè ðàñïðåäåëåíèÿÒåïåðü ïåðåéäåì ê âû÷èñëåíèþ ôóíêöèè ðàñïðåäåëåíèÿ âåðîÿòíîñòèãàóññîâîé ïîëèìåðíîé öåïè ñ ïîìîùüþ ìåòîäà ïåðåâàëà.

Èñïîëüçóÿ ïîëó÷åííîå âûðàæåíèå (5.77) äëÿ õàðàêòåðèñòè÷åñêîé ôóíêöèè, ïîäñòàâëÿÿ åãîâ (5.66), ïðèäåì ê ñëåäóþùåìó èíòåãðàëóZ∞P (ε) =dξexp [−W (ξ)] ,2π(5.81)−∞ãäåΩ2 (ξ)λ+ ln cos Ω(ξ)2ýíåðãèÿ λ = / hi. ÓðàâíåíèådW (ξ) =2è ââåäåíà áåçðàçìåðíàÿèìååò âèä:(5.82)äëÿ ñåäëîâîé òî÷êè2dΩ(ξ)tanΩ(ξ)00W 0 (ξ0 ) =λ−= 0.4ξ0Ω(ξ0 )(5.83)Òàêèì îáðàçîì, â îêðåñòíîñòè ñåäëîâîé òî÷êèè èìååì ðàçëîæåíèå:1W (ξ) = W (ξ0 ) + W 00 (ξ0 )(ξ − ξ0 )2 + ..,2(5.84)dΩ(ξ0 )Ω(ξ0 )W (ξ0 ) =tan Ω(ξ0 ) −.8ξ02cos2 Ω(ξ0 )(5.85)ãäå âòîðàÿ ïðîèçâîäíàÿ èìååò âèä00Óðàâíåíèå äëÿ ñåäëîâîé òî÷êè ìîæåò áûòü ëåãêî ðåøåíî â äâóõ ïðåäåëüíûõ ñëó÷àÿõ: êîãäà λ 1 è λ 1.

 ñëó÷àå, êîãäà λ 1, èìååìΩ0 = Ω(ξ0 ) ' π/2 − 2/πλ, òàê ÷òîdW (ξ0 ) '2π2λ2− 2 + ln,8πλ132(5.86)èκ2 N 4 b4 2λ.W (ξ0 ) '8d(5.87)Âû÷èñëÿÿ ãàóññîâûé èíòåãðàë, ïðèõîäèì ê àñèìïòîòè÷åñêîìó ñîîòíîøåíèþ√ d/2−1 2 (d−1)/2 dπe dλπ dλP () '.(5.88)exp −d/2−122162κN b00Êàê âèäíî èç óðàâíåíèÿ (5.88), â îáëàñòè áîëüøèõ ýíåðãèé ôóíêöèÿ ðàñïðåäåëåíèÿ íàïîìèíàåò ðàñïðåäåëåíèå Áîëüöìàíà ñ ýôôåêòèâíîé òåìïåðàòóðîé T = (16/dπ2) hi. äðóãîì ïðåäåëüíîì ñëó÷àå λ 1 èìååì Ω0 = Ω(ξ0) ' i/λ, òàê ÷òîdd− ln 24λ 2(5.89)κ2 N 4 b4 λ3W (ξ0 ) '.8d(5.90)W (ξ0 ) 'è00Âû÷èñëåíèå ãàóññîâà èíòåãðàëà äàåò:2d/2+1P () 'κN 2 b2dπλ31/2dexp −.4λ(5.91)Êàê âèäíî èç (5.91), ïðè ìàëûõ ýíåðãèÿõ ôóíêöèÿ ðàñïðåäåëåíèÿ ñòðåìèòñÿ ê íóëþ áûñòðåå ëþáîé ïîëîæèòåëüíîé ñòåïåíè λ.Ñîáèðàÿ âìåñòå ñîîòíîøåíèÿ (5.88) è (5.91), çàïèøåìhi√ π(d−1)/2 ed dλd/2−1 exp − π2 λ , λ 1162d/2−1 κN 2 b2P () 'd/2+11/2d2 2 2 d3exp − 4λ,λ 1.κN bπλ(5.92)Çàêëþ÷èòåëüíûå çàìå÷àíèÿ ýòîì ïðèëîæåíèè ñôîðìóëèðîâàí íîâûé ïîäõîä, îñíîâàííûé íà ôîðìàëèçìå èíòåãðèðîâàíèÿ ïî òðàåêòîðèÿì äëÿ âû÷èñëåíèÿ ôóíêöèé ðàñïðåäåëåíèÿ âåðîÿòíîñòè äâóõ õàðàêòåðèñòèê ãàóññîâîé öåïè êâàäðàòè÷íûõïî êîîðäèíàòàì åå çâåíüåâ â d-ìåðíîì åâêëèäîâîì ïðîñòðàíñòâå: êâàäðàòàðàäèóñà èíåðöèè è ïîòåíöèàëüíîé ýíåðãèè öåïè â ñèììåòðè÷íîé ïàðàáîëè÷åñêîé ïîòåíöèàëüíîé ÿìå.

Äëÿ îáîèõ ñëó÷àåâ ïîëó÷åíû òî÷íûå ñîîòíîøåíèÿ äëÿ õàðàêòåðèñòè÷åñêèõ ôóíêöèé è êóìóëÿíòîâ. Ïîëó÷åíû àñèìïòîòè÷åñêèå ñîîòíîøåíèÿ äëÿ ôóíêöèé ðàñïðåäåëåíèÿ âåðîÿòíîñòè óêàçàííûõ âåëè÷èí â îáëàñòè èõ áîëüøèõ è ìàëûõ çíà÷åíèé. Ïîëó÷åíî îáîáùåíèåêëàññè÷åñêîãî ðåçóëüòàòà Ôèêñìàíà äëÿ ôóíêöèè ðàñïðåäåëåíèÿ âåðîÿòíîñòè ðàäèóñà èíåðöèè ãàóññîâîé öåïè íà ñëó÷àé ïðîñòðàíñòâà ïðîèçâîëüíîé133ðàçìåðíîñòè.

Íà ïðèìåðå âû÷èñëåíèÿ ôóíêöèè ðàñïðåäåëåíèÿ âåðîÿòíîñòèïîòåíöèàëüíîé ýíåðãèè ãàóññîâîé öåïè â ïàðàáîëè÷åñêîé ïîòåíöèàëüíîéÿìå, ïðîäåìîíñòðèðîâàíû ïðåèìóùåñòâà ìåòîäà ïî ñðàâíåíèþ ñ ìåòîäîìÂîíãà-Óëåíáåêà. Îòìåòèì, ÷òî äîïîëíèòåëüíîå ïðåèìóùåñòâî ñôîðìóëèðîâàííîãî ìåòîäà ñîñòîèò â òîì, ÷òî îí ìîæåò áûòü ïðèìåíåí äëÿ âû÷èñëåíèÿ ôóíêöèé ðàñïðåäåëåíèÿ âåðîÿòíîñòè êâàäðàòè÷íûõ õàðàêòåðèñòèêäëÿ áîëåå ñëîæíûõ îáúåêòîâ, òàêèõ êàê äèáëîêñîïîëèìåðû è ïîëóãèáêèåïîëèìåðíûå öåïè, ÷òî ÿâëÿåòñÿ ïðåäìåòîì áóäóùèõ èññëåäîâàíèé.134Ëèòåðàòóðà1. Ben-Yaakov D., Andelman D., Harries D. and Podgornik R. Beyondstandard PoissonBoltzmann theory: ion-specic interactions in aqueoussolutions // J.Phys.: Condens.

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