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ÈÖÚÒÚÞ ÙÞßÑÖÓ×ÓÔÔÓÐ JiÐ ØÝi´ÒÓØÝÑÓÐÑ−2×−1µ1/26s13ÓÒÌÓ⃗ ÑÖÒÖÒØÙÓÒÒØÖiÓ¬ÒØÙÞ´ÙÞÚصÐÓÓÙÓÒÖØÙºÂÒÓÒÔÓÒÓÒÚÓ¹Óд½½Ñº±ÙÖÙµD ØÝi¸ÒÓØÝÑ2×−1 ÈкÌÖÙØÚÖÙÍÐÝl −6Ñ2×−1º0.3Ñ2Ñ Ò= 20ÑÔÖÞÙA ÖÙÚ×ØÚÓºÃÓÐÙÖÙÔÖÓÙÒÙÞÒD sacharoza5.2·10⃗∇c i =gradc i =(∂⃗J i = −D i⃗ ∇ci∂x , ∂∂y , ∂ ∂z)c i =(∂ci∂x ,∂c i∂y ,∂c i∂z=)(25 ◦) =½½¼ÙÖÙÚÐØÖÙcw = 110Ñ−3 gradcwcw/l¼Ñ−4D = 5.2·10 −6Ñ2×−1 ¼ºÑJ = Dgradcw = 2.56·10 −7Ñ−2×−1m = JAt = 2.56·10 −7Ñ−2×−1==5.2·10−10Ñ2×−1×0.3·10110Ñ−3−4Ñ2 ×24×60 2×=7.4·10 −7

ÌÓÐ ÙÞßÒ×ØÒÓÚ¹ËÑÓÐÙÓÛ×Ó℄ÖÓÚÒ ØÝ i2/26s13ÌÖÑÓÝÒÑÒ×ØÒÖÝÐÓ×ØÑÓÐÙÐ⃗v×ÐÖÒØÑÑÓÔÓØÒ ÐÙ⃗JØiÚÝÓÒ ´ÔÑØÙÖÓÞÐÑÔÓØÒiÐÖÚÖÞÐÒÔÖ= ⃗v i c i+d⃗rµ ¸ØÖÓÙ ×¹ ÔÔÓÝÙÞÑ×Ø⃗rÓÑ×Ø⃗r×ÑÔÓÙúÐÚÞØÔÖÓ)µi⃗F i = −⃗∇(= − kT ⃗∇c i)ºN A c´ÔÐúÒµÑÖÒ ÖÝÐÓ×ØiÐÒÖÓÞØÓ¸µ i¸Ô×ÓÒÒ×ÐÓÔÓÖÙÔÖÓ×Øi = µ ❡i +RT ln(c i/c stÈÓÝÙ¹Ð×ÑÓÐÙÐÖÝÐÓ×Ø⃗viÓ¬ÒØØÒºÇ×ÐÝ×ÓÙÚÖÓÚÒÓÚFÞ¸⃗třeníi = −λ i ⃗vغiλ ÈÓÖÓÚÒ⃗F itření +F∇ciÓ×ØÒÑÒ×ØÒÓÚÙÖÓÚÒJi ⃗i = 0 λ i ⃗v i = λ i = − kT⃗c i c iÒÑ×⃗ Ji = −D i⃗ D i = kTλ i

Ò×ØÒÓÚ¹ËØÓ×ÓÚÖÓÚÒ ÐÒ¹»×ÙÖÓ×℄ËØÓ×ÚÚÞÓÖiÚÔÐÒÓÚ×ÓÞØηÔÐØ·3/26s13Î×ÓÞØ ÔÐÒÈÖÓÚÐÙÐÓÚØÑÓÐÙÐÝÓÔÓÐÓÑÖÙR ⇒Ò×ØÒÓÚ¹ËØÓ×ÓÚÖÓÚÒ × ØÓÙÖÓ×ØÓÙ Ð× ¸ ÙÞ¹ ØÔÐÓ¹ÖÖÒÓÚÚÞØÙ ÚØÖÓ×ظ≈ÐÇÔÒ߬ÒÙÑÝÖÓÝÒÑ´ËØÓ×ÚµÔÓÐÓÑÖ´ÚºÒÔº×ÓÐÚØÒ×ÐÙÔݵ ØÖÖÓÚÒ×ØØÚÒÚÐÓ×ØÑÓÐÙÐÝÈк ÇÒØ ÚÐÓ×Ø ÑÓÐÙÐÝ ×¹¾◦º−3Ñ−1×−1Ô ÖÓÞݺÎ×ÓÞØÚÓÝ0.891·10ւ λ i⃗F i = 6πηR i ⃗v iD i = kT ⇒ D i = kTλ i 6πηR iR i = kT6πηD iR = 0.47ÒÑ

Æ×ØÓÒ ÖÙÚÞ ÖÒÚ´ÓÒÒØÖ×ÑÒ××ѵÖÓÚ¹ ÓÒÒÚÒØÔÐ∂cÔÐÓØ»ÙÖº×℄4/26s13i∂t = D i∆c×ØÓÙÚÓÓÙ´½¼ÑµºÐÓÙÓÙÓÒÒØÖÙÐÒÝÖÓÚÒÍ ÞºÓ¹ÓÐÙÚÚ Ð´Úõ×ÐÓÙÔ½¼ÑµÓÔØÖÒÔÚÖ×ØÚÑi∆ = ⃗∇· ⃗∇ = ∂2∂x 2 + ∂2∂y 2 + ∂2∂z 2ÔÓÐÓÚÒÓÒÒØÖÙÒ∂c∂t = D ∂ 2 ci∂x 2 c(x,0) =c(x,t) = c 02 + 2c 0π− 1 ( ) (3πx3 cos exp − 32 π 2 )l l 2 Dt[( πxcosl){c0 x < l/20 x > l/2exp(− π2l 2Dt )Ñ×+ 1 ( ) (5πx5 cos exp − 52 π 2 ) ]l l 2 Dt ···

ÙÞÖÓÛÒÚÔÓÝ ØÖ»ÖÓÛÒº×℄5/26Ò Å×ØÓc(⃗r,t)õѾºÓÚÙÖÓÚÒÔÖÓÔÖÚÔÓÓÒÓ×ØÒÐÞÒ0ÚÔÓ ØÙºÓ×ØÒÙÙ××ÓÚÓÖÓÞÐÓúÒs13×ظ¹ÐÚt =½0.5 t=1t=2( )¿c(x,t) = (4πDt) −1/2 exp − x20.4t=3t=44Dt0.3t=5( )c(⃗r,t) = (4πDt) −3/2 exp − r20.24Dt0.1×ÔÓ×ÙÒÙÒ Æ ÓÒ ÔÖÓ ÞÓÑÓÐÖÓÛÒÓÚÔÓÝÙÞÒÓØÙ×Ù1/2´×ÔÖÚÔÓÓÒÓ×ؽ»¾µÓxÓÒÓ∆x = +(2D)ÎÓÓÙÔÔÔÐØ ØÓ×ÑÙ××ÓÚÓÖÓÞÐÓúÒ´ÒØÖ 1/2´×ÔÖÚÔÓÓÒÓ×ؽ»¾µºÈÓÑÒÓÖÓÓ×ØÒÙ ÐÒÐÑØÒÚصº∆x = −(2D)〈x 2 〉 = 2Dt ÒÓÐ〈r 2 〉 1/2 = 6Dtº√c(x,t)-3 -2 -1 0 1 2 3

ÖÓÛÒÚÔÓÝÓÒ ÓÒ ÔÖÓ Þ ÔÐÓØ»ÖÒÓÑÛк×℄´ËÑÓÐÙÓÛ׸Ò×ØÒµßÓ∆x×ÔÖÚÔÓÓÒÓ×ؽ»¾ Þ×∆τ×ÔÓ×ÙÒÙÒ ÓÒÇÚÓÞÒ×ÔÓÙúØÑÒØÖ ßÓ−∆x×ÔÖÚÔÓÓÒÓ×ؽ»¾ ÐÒÐÑØÒÚØÝ⇒Ù××ÓÚÓÒÓÖÑ τ/∆τ¸ØÚÒÓÑÖÓÙVarx =ÚnÖÓ´Þ×τ= n∆τµVarxÐÒÖÓÞÐÒ×σ =∆x 2= n∆x 2√n∆x 2 = ∆x1√ e −x2 /2σ 2 = 1√ [∆τ√2πσ 2πτ ∆x exp − −x22τ√]∆τ∆x 2·6/26s13ÓúÔÖÓ2D =∆x 2 /∆τØÓ×ÑÓc(x,τ)

ÎÝÒÞÒ Þ×2∆τ×ÔÓ×ÙÒÙÒ ØÒØÖ ÐÒÐÑØÒÚØÙßÓ2∆x×ÔÖÚÔÓÓÒÓ×ؽ» ÓÒßÓ0×ÔÖÚÔÓÓÒÓ×ؽ»¾ ßÓ−2∆x×ÔÖÚÔÓÓÒÓ×ؽ»Þ×2n∆τ×ÔÓ×ÙÒÙÒ ÓÒÓ2k∆x×ÔÖÚÔÓÓÒÓ×Ø·7/26s13ÎÝÑÞπ(n,0)ºÈÖÓØÓúπ(n,k) = ( 2nn−k)4−nÑúÑÒÔ×ظÞÒ ( 2n ) (2n)! =ÚÐÒÝÖÙÓn+1 (n−1)!(n+1)! = (2n)!n!/n·n!(n+1) = ( 2n) 2µn ×n n+1Ù´∝ 1/nnlnπ(n,1) = lnπ(n,0)+lnn+1= lnπ(n,0)+ln(1− 1 )n+1≈ lnπ(n,0)+ln(1− 1 )n≈ lnπ(n,0)− 1 n

ÒÐÓÝÓ×ØÒÑ ÖÓÛÒÚÔÓÝÓÒ ÓÒ ÔÖÓ ÞÁÁÁÓÒlnπ(n,2) = lnπ(n,1)+ln(1− 3 )n+2·8/26s13≈ lnπ(n,1)− 3 n ≈ lnπ(n,0)− 1 n − 3 nÆÝÒÒÖÑ×ÙÑÙÒØÖ ÐÑlnπ(n,k) ≈ lnπ(n,0)−k∑j=12k −1nÇÓÒÐÞÚÔÓØÔÖÓÚ×ØÔÖÓÞk∑∫ kÔÓÖÒ kºÎÐÑØÚÐknk jevelké(2k −1) ≈ (2k −1)dk = k(k −1) ≈ k 2j=10( )π(n,k) ≈ π(n,0)exp − k2nÇÔØ∆x = (2D∆τ) 1/2¸k = x/∆x = x/(2D∆τ) 1/2¸n = t/(2∆τ)(π(n,k) = c(x,τ) ≈ c(x,0)exp1µÓ×ØÒÑc(x,τ)º)− x24DτÈÓÒÓÖÑÐÞ´ÔÓÑÒ∫ π(x,τ)dx =

Æ Ò×ØÒÓÚÓÓÚÓÞÒ ÓÒ ÔÖÓ ÞÚ½ÔÖÓÑÒÒ·9/26s13φ(δx)Ù×ØÓØÔÖÚÔÓÓÒÓ×ظú ×ØÞδtÓ×ØÙÓδxÎÚÓÙ×ØÓØÝ´ÔÖÚÔÓÓÒÓ×صρ(x,t)∫ +∞×ØÞ×δt−∞φ(δx)dδx = 1, φ(−δx) = φ(+δx)ρ(x,t+δt) ≈ ρ(x,t)+δt ∂ρ∂t = ∫ +∞ÈÓÒØÖ´ÐÚÝÔÒÓÙ¸ÚÝõõÞÒ Ñµ∂x 2−∞ ρ(x+δx,t)φ(δx)dδxρ(x+δx,t) = ρ(x,t)+δx ∂ρ∂x + δx2 ∂ 2 ρ2+···ρ(x,t)+δt ∂ρ∂t = ρ(x,t)+ ∂2 ∫ρ +∞∂x 2 −∞∂ρ∂t = ρD∂2 ∂x 2 , D = 1 δt∫ +∞−∞δx 22 φ(δx)dδx´ÖÓÞÔØÝеδx 22 φ(δx)dδx

ÄÒÚÒÓÚÖÓÚÒ ×ØÚÚ×ÞÒÑÔÖÓ×Ø·Ò ÓÒÒ ÖÞÝXÒ 6πηR´ËØÓ×ÚÚÞÓÖµmẍ = −λẋ+X(t)Ò ×ÓÑxÙÔÖÚÑ ÓÒ ÃÓÙÐλ =×Ð〈X(t)X(t ′)〉 = Aδ(t−t ′ )¸〈X(t)〉 = 00md 22dt 2(x2 )−mẋ 2 = − λ d2dt (x2 )+XxÔÐÙÑÒÓÒÓÙ×ØÒÓÒÓØÙ〈X(t)x〉 =·10/26s13ÌÓÐÞõØmd 22dt 2〈x2 〉−kT = − λ d2dt 〈x2 〉ÌÓÝÞ ÔÖÓXº ÐÒÚ×дÞÒ ÑTµ¸Ú×ÑÙÐÐÔÓØÙÑÚÞØddt 〈x2 〉 = 2kTλ +conste−λt/m t→∞ kT = 2D, D =λ

ÜÔÐØÒõÒÔÖÓÖÝÐÓ×Ø´ÞÓÒÞÖÚØÚÒ×е)〉 = Aδ(t−t ′ )¸〈X(t)〉 = 0mẍ = f −λẋ+X(t)∞ẋ(t) = ẋ(0)e − m λ t + 1 ∫ tm 0 X(t′ )e −λ m (t−t′) dt ′ËØÒÒØ ÒÖÎÐÑØt →ẋ(0) = 1 ∫ ∞m 0 X(t)e− m λ t dt〈 ∫〈mẋ 2 1 ∞〉 = mm 0 X(t)e− m λ t dt 1 ∫ ∞〉m 0 X(t′ )e −λ m t′ dt ′⇒­ÙØÙØÓÒ¹××ÔØÓÒØÓÖÑ= 1 ∫ ∞ ∞mdt′∫ 0 0 dtAδ(t−t′ )e − m λ (t+t′) = 1 ∫ ∞m 0 dtAe−λ m 2t = A 2λÎ×ÑÙÐδ(t)ÒÖÑÑÔÙÐ×Ñ1/hÞ×ÓÚÖÓhº ÎÞÓÖÙÚõÒÝ×ØÙÔÒÚÓÐÒÓ×ØßÒÞÓÚA =Ú2λkTÝÒÓ×Ø= 2(kT)2DX´Ù××ÓÚ× ÄÒÚÒÚØÖÑÓ×ØØ´ÖÓÛÒÓÚ× µÒ ÓÒ ÝÒѵ ×Ð〈X(t)X(t ′·11/26s13

×ÔØÚÒ ÑÐÚÒÑÓÐÓÚØÚÐÚÖÓÞÔÓÙõØиݴÖÓÑÖÓÛÒÓÚ ×ØÓÚ ÝÒѴȵ12/26s13ÑÙ×ÞÓÚ ∞µÝÐÓ×ÔÖ ÚÒÝÖÓÝÒÑÓÚ ÒºÚк×ÅúÑ×ÑÙÐÓÚØÙÐйØÓÑÑÓÐÓÚÓØÖÙÓÞÖÒÒÓÙ´ÓÖ×¹ ÚØÝÒÓ×غ ÔÓÝÙÔÖÓt →×ØÙÔÒÚÓÐÒÓ×Ø ¹ÖÒµÚÖÞÖÝÐõ×ÑÙиØÖ ÖÚÚÙ×ÓÚÒ

ÐÓÖØÑÙ× ×ÔØÚÒ ×ØÓÚ ÝÒѴȵm¨⃗r i = ∑ j

ÓÖÓÚ¹ÈÐÒÓÚÖÓÚÒ ÍÚúÙÑÙ×ØÓØÙÔÖÚÔÓÓÒÓ×Øρ(x,p)¸p ÈÖÚÔÓÓÒÓ×ØÒÐÞÒ=×ØÚÓÐ×ØA =∫ρ(x,p,t)dxdp = 1mẋ´ÔÖÓ½×ص·14/26s13∫AÑÒÚ×ØÓÔÖÚÔÓÓÒÓ×Ø×ØÒÑAδx×δpρ(x,p,t)dxdp ≈ ρ(x,p,t)δxδp∂ρδxδp = −[(ρẋ)(x+δx)−(ρẋ)(x)]δp−[(ρṗ)(p+δp)−(ρṗ)(p)]δx∂tÆÓÞÐÓÖÑ ÐÒÞÖÓÚÒÓÒØÒÙØÝ´ÔÖÚÔÓÓÒÓ×Ø×ÒÞØÖص∂ρ∂t = −∂ẋρ ∂x − ∂ṗρ∂pdρdt = 0

ÓÖÓÚ¹ÈÐÒÓÚÖÓÚÒ ÈÓÚÓÞÒÖÓÞõÑÒN ×ØÚ¿∂ρ∂t = −∂ẋρ ∂x − ∂ṗρ∂p ,ẋ = p m , ṗ = fC +f D +f R·15/26s13∂ρ∂t = LC ρ+L D+R ρL C = − ∑ i⃗p i ∂− ∑m∂⃗r i i

ÈÓÚÓÞÒ·16/26s13L C = − ∑ i⃗p i ∂− ∑m∂⃗r i i

ÃÒØÚÐÒÝÚÝÖ ÒØÅßÖÝÐÓ×ØÖ×ØÙÖÝ×ØÐÙ¸ØÖÓÒÒØÐÞ ÑÒ ×ÐÙÙ ÐÓ×Ø´ÔÓÞØÓÑÙ¸ÚÝÔÒØÓÑÙ¸Ö¸17/26s13ºººµÔÓÐÔÑÒÔÖÚÔÓÓÒÓ×Ø Ð× ÅÙÞÚظкÚÓÚÓ×ظÚ×ÓÞظØÔÐÒ úÒÑØÓÝ ÚÓÚÓ×غººÒÖÓÚÒÓÚ ´ÆÅÆÓÒ¹ÕÙÐÖÙÑÅÓÐÙÐÖÝÒÑ×µ úÒÑØÓÝÃÓ¬ÒØÝÔ×ØÚÙ´ÐÒ ÖÒµÓÞÚÙÒ´ÑÐÓÙµÔÓÖÙÙ⃗J látkyA = −D⃗∇c A⃗J tepla = −λ ⃗ ∇Tη ∂v x∂y = P xy

ÌÓÖÐÒ ÖÒÓÞÚÝ×ØØ ÔÓÖÙÑÑÚÐÒÙBÚÒÓÒÑ×ÓÙÓÖÙ´×ÔÓÖÙÓÙµ ÔÑÔÓÖÙÙ×ÒÖ∆H¸H ′= H+∆H·18/26s13∫ÈкÃÐ×ÖÑÓÒÓ×Ð〈B〉 ′ B(t)exp(−βH ′ )dpdq= ∫exp(−βH ′ = 〈B〉−β(〈∆HB〉−〈∆H〉〈B〉) 〈B〉=0= −β〈∆HB〉x)dpdqØÓÖH= K 2 x2¸ÔÓÖÙ∆H = gx¸ÑÑB =Óú×ÔÖ ÚÒ¸ÔÖÓØÓúÚÐ×ØÒÒÔÓ×ÙÒÙÑÑÒÑÙÑÔÓØÒ∫x〈x〉 = −β〈∆Hx〉 = −β〈gx 2 2 exp(−β K〉 = −βg2 x2 )dx∫exp(−βK2x 2 )dx = − g ÐÙKH ′ = K 2 x2 +gx = K 2(x+ g K) 2+const

ÀÑÐØÓÒÓÚÝÖÓÚÒ ÌÓÖÐÒ ÖÒÓÞÚÝ×ÓÚÞ Ú×Ð ÔÓÖÙ0˙qÈÓÖÙ´ÑÔÙÐ×µÚ×t =A(q,p)º A p = ∂A∂pA q0º= ∂A∂qÔÖÓ×ØA =ÈÐA = F 1 x 1ÐA = F 1¸A q = 0ÔÖÓq x 1A p =ËÓÓÚ ÞÑÒÐÓÚÒÖÓ= ∂H∂p ≡ p m , ṗ = −∂H ∂q ≡ f˙q = p m −A pδ(t), ṗ = f +A q δ(t)x1≠ṗ 1,x = f 1,x +F 1 δ(t)·19/26s13H t>0 −H t0 −H t

ÌÓÖÐÒ ÖÒÓÞÚÝËÝ×ØÑÒÓÒÔÖÓt0º〈B(t)〉 Aδ(t) =∫B(t)exp(−βHt0 +βȦ(0)]dpdq∫exp[−βHt>0 +βȦ(0)]dpdq

ÌÓÖÐÒ ÖÒÓÞÚÝÖÒ¹ÃÙÓ ÈÓÖÙÔ×ÓÐõÓÙA(t) =ÓÒ×ØÒØÔÖÓt > 0ºÄÑØt ÈÓÞÒºÒÖ´ÓÒÓØÀÑÐØÓÒ ÈÐ ÒÙµÒÒÓÒ×ØÒØÒÔÖÓt >ṗ1ÔÓÞÒ ÈÖÓF 1 = E x q〈B〉 A = β∫ ∞0 〈Ȧ(0)B(t)〉dt→0∞·21/26s131,x = f 1,x +F 1〈ẋ 1 〉 Aiµ∫ ∞= F 1 β 〈ẋ 1(0)ẋ 1 (t)〉0ÒÓдÒ×ØÒβD i = v i /F∫ÒÑÆÖÒ×ØÓÚÙÖÓÚÒÔÖÓÐÑØÒÑÓÐ ×ÓÒÒ∞D 1ÓÑÔÔÓØÒÒÒÑÓÐÓ×ع= 〈ẋ 1(0)ẋÖÒÚÓÚÓ×Ø1 (t)〉dt0Λ ∞ 1 = 〈ẋq 1N A 〉= q2 1 D 1E x RT

ÎÐÓØݹÚÐÓØÝÙØÓÓÖÖÐØÓÒÙÒØÓÒ Î×ÙÚÖÝÐÓ×ØÒÙØÓÓÖÐÒÙÒc v (t) = 〈ẋ 1(0)ẋ 1 (t)〉〈ẋ 1 (0)ẋ 1 (0)·22/26s131c v (t)0.500 0.5 1 1.5t/ps

ÃÒØÓ¬ÒØÝÚÅÁÁ Ò×ØÒκ=∫ ∞0 〈Ẋ(0)Ẋ(t)〉dt·23/26s13Þ∫ t0 〈Ẋ(0)Ẋ(t′ )〉dt ′ = [〈Ẋ(0)X(t ′ )〉] t 0ÑÒt → −t´ÆẊ(0) → −Ẋ(0)µÔÓ×ÙÒÓt⇒∞Ô∫ t0 〈Ẋ(0)Ẋ(t′ )〉dt ′ = 1 d2dt 〈[X(t)−X(0)]2 〉ÎÐÑØt ÆÔºÔÖÓÙÞ→2tκ = 〈[X(t)−X(0)] 2 〉D= 1 3∫ ∞0 〈˙⃗r i (t)· ˙⃗r i (0)〉dt2tD = 1 3 〈|⃗r i(t)−⃗r i (0)| 2 〉

ÎÓÚÓ×Ø ÆŸÞÔÒ×кÔÓÐEºÈÖÓÙÓÚ Ù×ØÓØÖÒßÃÙÓ⃗j = κE⃗·24/26s13Ò×ØÒdκ = lim t→∞ dtκ = V kT16kTV∫ ∞0 〈 ⃗j(t)·⃗j(0)〉〈 ⎧ ⎨∑⎩iq i [⃗r i (t)−⃗r i (0)]⎫2〉⎬⎭

ÈÓÙúØÒ×ØÒÓÚÚÞØÙ25/26s13MSD(t)/nm 2ÎÓÚÓ×ØÖÓÞØÚÒÓÆÐÔÓÑÓÅ0.10.080.060.04Cl -MSD(t)/arb.u.321Kohlrausch Na + + Cl -Na + 0 2 4 6 8 10whole box0.0200 2 4 6 8 10t/ps0t/ps

ÆÅÆÓÒ¹ÕÙÐÖÙÑÑÓÐÙÐÖÝÒÑ× ÆÅ26/26Ö ÐÒÜÔÖÑÒØ´ÞÔÒÑÔÓиÖÒØØÔÐÓØݺººµs13ÔÖÓÐÑÐÒ ÔÖÓÐÑÒÙØÒÓÐØ ÖÒÖúÑ´ÜØÖÔÓÐÒÒÙÐÓÚÓÙÔÓÖÙÙµßËÄÇ´Ä×¹ÛÖ×µ Ú×ÓÞØßÔÒÓ×ÝÒÓ×Ø Ó×Òѵ ßÔÖÓÑÒÒ ×дÑÓÙÐÓÚÒ ✲ ✲