Strong-coupling calculation of fluctuation pressure of a membrane between walls

Transcription

1 11 October 1999 Physics Letters A wwwelseviernlrlocaterphysleta Strong-coupling calculation of fluctuation pressure of a ebrane between walls Michael Bachann ), agen Kleinert 1, Axel Pelster Institut fur Theoretische Physik, Freie UniÕersitat Berlin, Arniallee 14, Berlin, Gerany Received June 1999; received in revised for 6 June 1999; accepted 6 June 1999 Counicated by PR olland Abstract We calculate analytically the proportionality constant in the pressure law of a ebrane between parallel walls fro the strong-coupling liit of variational perturbation theory up to third order Extrapolating the zeroth to third approxiations to infinity yields the pressure constant as This result lies well within the error bounds of the ost accurate available Monte Carlo result a MC s00798"00003 q 1999 Elsevier Science BV All rights reserved 1 Mebrane between walls The violent theral out-of-plane fluctuations of a ebrane between parallel walls generate a pressure p following the law k B T psa, Ž 11 3 k Ž dr wx whose for was first derived by elfrich 1 ere, k denotes the elasticity constant of the ebrane, and d the distance between the walls The exact value of the prefactor a is unknown, but estiates have been derived fro crude theoretical approxia- ) Supported by the Studienstiftung des deutschen Volkes, Eail: bach@physikfu-berlinde 1 E-ail: kleinert@physikfu-berlinde E-ail: pelster@physikfu-berlinde tions by elfrich wx 1 and by Janke and Kleinert wx which yielded a th f004, a th f0065 Ž 1 JK More precise values were found fro Monte Carlo siulations by Janke and Kleinert wx and by Gopwx 3 which per and Kroll gave a MC f0079"000, JK agk MC f00798"00003 Ž 13 In previous work wx 4, a systeatic ethod was developed for calculating a with any desired high accuracy Basis for this ethod is the strong-coupling version of variational perturbation theory wx 5 In that theory, the free energy of the ebrane is expanded into a su of connected loop diagras, which is eventually taken to infinite coupling strength to account for the hard walls As a first approxiation, an infinite set of diagras was calculated, others were estiated by invoking a atheatical analogy with a r99r$ - see front atter q 1999 Elsevier Science BV All rights reserved PII: S

2 18 M Bachann et alrphysics Letters A siilar one-diensional syste of a quantu echanical particle between walls The result of this procedure was a pressure constant p th a K s s , very close to Ž 13 It is the purpose of this Letter to go beyond this estiate by calculating all diagras up to four loops exactly In this way, we iprove the analytic approxiation Ž 14 and obtain a value a th f , Ž 15 which is in excellent agreeent with the precise MC MC value a in Eq Ž 13 GK The sooth potential odel of the ebrane between walls To set up the theory, we let the ebrane lie in the x-plane and fluctuate in the z-direction with vertical displaceents wž x The walls at z s"dr restrict the displaceents to the interval w g Ž ydr,dr Near zero teperature, the theral fluctuations are sall, wž x f 0 The curvature energy E of the ebrane has the haronic approxiwx ation 1 1 Es k dx E w x 1 The therodynaic partition function Z of the ebrane is given by the su over all Boltzann factors of field configurations wž x Zs Ł x qdr ( dw Ž x p k Trk ydr B k ½ B 5 = exp y d x E w Ž x k T This siple haronic functional integral poses the proble of dealing with a finite range of fluctuations This proble is solved by the strong-coupling theory of Ref wx 4 as follows If the area of the ebrane is denoted by A, the partition function 6 deterines the free energy per area as 1 fsy ln Z 3 A By differentiating f with respect to the distance d of the walls, we obtain the pressure psye fre d 1 Sooth potential adapting walls We introduce soe sooth potential restricting the fluctuations wž x to the interval Ž ydr,dr, for instance d p 4 V Ž w Ž x s tan w Ž x p d p 4 ' w x q VintŽ w x, 4 d where we have split the potential into haronic and interacting part p V Ž w Ž x s wž x q w Ž x d int 4 6ž / 8ž / p 4 8 q w Ž x q, 5 d with 4sr3, 6s17r45, 8s6r315, Thus we are left with the functional integral 1 ZsEDw Ž x exp y d x E w Ž x ž ½ p 4 q w x q VintŽ w x, 6 d 5/ where we have set kskbts1 After truncating the Taylor expansion around the origin, the periodicity of the trigonoetric function is lost and the integrals over wž x in can be taken fro y` to q` The interacting part is treated perturbatively Then, the haronic part of VŽwŽ x)) leads to an exactly

3 M Bachann et alrphysics Letters A integrable partition function Z The ass parae- ter is arbitrary at the oent, but will eventually taken to zero, in which case the potential VŽwŽ x)) describes two hard walls at ws"dr We shall now calculate a perturbation expansion for Z up to four loops This will serve as a basis for the liit 0, which will require the strong-couwx pling theory of Ref 4 Perturbation expansion for free energy The perturbation expansion proceeds fro the haronic part of Eq 0 : E Z s Dw x e ya w w xr se yaf, 7 with 4 ½ 5 A ww xs d x E w Ž x q w Ž x 8 Fro Refs w,4 x, the haronic free energy per unit area f is known as 1 f s 8 9 The haronic correlation functions associated with 7 are ² O w Ž x O w Ž x PPP : s Dw Ž x O w Ž x E 1 1 Z =O w x PPP e ya w w xr, 10 where the functions O ŽwŽ x i j ay be arbitrary polynoials of wž x j The basic haronic correla- tion function G Ž x, x s² w Ž x w Ž x : deterines, by Wick s rule, all correlation functions 10 as sus of products of 11 : ² w Ž x PPP w Ž x : 1 n Ý PŽ1 P Ž PŽ ny1 P Ž n pairs s G Ž x, x PPP G Ž x, x, 1 where the su runs over all pair contractions, and P denotes the associated index perutations The haronic correlation function 11 is in oentu space 1 i 1 1 G Ž k s s y, 4 4 k q k qi k yi 13 thus being proportional to the difference of two Ž ordinary correlation functions p y y1 with an iaginary square ass s"i Fro their known x-space for we have iediately G x, x sg Ž 1 Ž x1yx i s K ' i< x yx < 0 1 4p yk ' yi< x yx <, where K Ž z is a odified Bessel function wx 0 6 At zero distance, the ordinary haronic correlations are logarithically divergent, but the difference is finite yielding G 0 s1r8 We now expand the partition function 6 in Ž powers of gvint f x, where g'p rd, and use 10 to obtain a perturbation series for Z Going over to the cuulants, we find the free energy per unit area g fsf q d x² V Ž w Ž x : int A,c g 1 y d x 1 d x ² V int w Ž x 1! 4A,c = V w Ž x : q, 15 int where the subscript c indicates the cuulants Inserting the expansion 5 and using 10 and 1, the series can be written as ž / ` n g fs a0 q Ý a n, 16 ns1 where the coefficients an are diensionless real nubers, starting with a0s 1r8 The higher expan-

4 130 M Bachann et alrphysics Letters A sion coefficients an are cobinations of integrals over the connected correlation functions: 4 a s d x² w x :, A,c 6 6 a s d x² w Ž x : 6 A,c 10 ² 4 4 : 4 1 1,c y 8 A d x d x w x w x, a s d x² Õ Ž x : 3 8 A,c 1 ² 6 4 : ,c y d x d x w x w x 4 A 16 3 ² q d x d x d x w Ž x 48 A = w 4 Ž x w 4 Ž x : 19 3,c To find the free energy 16 between walls, we ust go to the liit 0 Following w4,5 x, we substitute by the variational paraeter M, which 4 is introduced via the trivial identity '( M ygr Ž 4 with rs M y 4 rg, and expand this in powers of g up to the order g N In the liit 0, this expansion reads 1 r 1 r Ž M sm y gy g 6 M 8 M 1 r 3 3 y g y 0 16 M 10 Inserting this into 16, reexpanding in powers of g, and truncating after the Nth ter, we arrive at the free energy per unit area N n Ž1yn N 0 0 Ý n n ns1 f M,d sm a b q a g M b, with Nyn / 1 k Ž 1yn r bn s Ý Ž y1 ž k ks0 being the binoial expansion of Ž 1 y 1 Ž1ynr truncated after the Ž Nyn th ter wx 4 The optiization of 1 is done as usual wx 5 by deterining the Ž iniu of f M, d N with respect to the variational paraeter M, ie by the condition E fn Ž M,d! s 0, 3 E M whose solution gives the optial value M Ž d N Resubstituting this result into Eq 1 produces the optiized free energy f Ž d sf ŽM Ž d,d N N N, which only depends on the distance as fn d s4anrd Its derivative with respect to d yields the desired pressure law with the Nth order approxiation for the constant a N : / y3 d pnsa Nž 4 We ust now calculate the cuulants occuring in the expansion 16 3 Evaluation of the fluctuation pressure up to four-loop order The correlation functions appearing in are conveniently represented by Feynan graphs Green functions are pictured as solid lines and local interactions as dots, whose coordinates are integrated over: x - x 'G 1 Ž x 1, x, Ž 31 Ø' d x Ž 3 These rules can be taken over to oentu space in the usual way One easily verifies that the integrals over the connected correlation functions in 17 Ž nqvy1 19 have a diension Ar, where V is the nuber of the vertices of the associated Feynan diagras Thus we paraetrize each Feynan diagra by ÕAr Ž nqvy1, with a diensionless nuber Õ, which includes the ultiplicity In Table 1, we have listed the values Õ for all diagras up to four loops No divergences are encountered Exact results are stated as fractional nubers The other nubers are obtained by nuerical integration, which are reliable up to the last written digit The right-hand colun shows nubers ÕK obtained by the earlier approxiation wx 4, where all the Feynan diagras

5 M Bachann et alrphysics Letters A Table 1 Feynan diagras with loops L, ultiplicities s, and their diensionless values Õ The last colun shows the values Õ K s L Õ r4 used in Ref wx 4 PB in M To see how the results evolve fro order to order, we start with the first order 1 p f1ž M,d s a0m qa 1, Ž 33 d with a0s 1r8 and a1s 1r64 ere, an optial value of M does not exist Thus we siply use the perturbative result for s0 which is equal to Ž 33 for Ms0 Differentiating f Ž 0,d 1 with respect to d yields the pressure constant in 4 : 1 p p a1s a1 s f Ž 34 4 d 56 This value is about half as big as the Monte Carlo estiates Ž 13 and agrees with the value found in wx 4 To second order, the reexpansion 1 reads 3 p p 4 1 fž M,d s a0m qa1 qa Ž d d M with a f1088p10 y3 fro Table 1 Miniizing this energy in M yields an optial value ( 8 a p p M Ž d s f01536, Ž 36 3 a d d and 0 ( p 3 f d s ž a1q a0a / 37 d were estiated by an analogy to the the proble of a particle in a box In Ref wx 4, it was shown that the value Õ of a large class of diagras of the ebrane proble can be obtained by siply dividing the value of the corresponding particle-in-a-box-diagra ÕPB by a factor 1r4 L, where L is the nuber of loops in the diagras Inserting the nubers in Table 1 into 17 19, we obtain the coefficients a 1, a, a3 of the free energy per area 1, which is then extreized Inserting a s 1r8 and a, a obtain 0 1 fro Table 1, we a f , Ž 38 thus iproving drastically the first-order estiate Ž 34 This value is by a factor 106 larger than that obtained in the approxiation of Ref wx 4 Continuing this proceeding to third order, we ust iniize 5 p 3 p 4 1 f3ž M,d s a0m qa1 q a 16 d d M 4 p 6 1 qa 3, Ž d M

6 13 M Bachann et alrphysics Letters A with a3 f7631p10 y5 The optial value of M is 3 a 1 5 a0a3 p M3 Ž d s cos arccos 5 a 3 a 3 d 0 p f Ž 310 d Inserted into Ž 39, we find the four-loop approxiation for the proportionality constant a: a f Ž This result is in very good agreeent of the Monte Carlo results in Ž 13 It differs fro the approxiate value of the ethod presented in Ref wx 4 by a factor 1047 An even better result will now be obtained by extrapolating the sequence a 1, a, a3 to infinite order 4 Extrapolation towards the exact constant Variational perturbation theory exhibits typically an exponentially fast convergence This was exactly proven for the anharonic oscillator wx 5 Other systes treated by variational perturbation theory show a siilar behavior wx 7 Assuing that an exponential convergence exists also here, we ay extrapolate the sequence of values a 1, a, a3 calculated above to infinite order It is useful to extend this sequence by one ore value at the lower end, a0 s 0, which follows fro the one-loop energy 9 at s 0 This sequence is now extrapolated towards a hypothetical exact value aex by paraetrizing the ap- proach as a ya sexpž yhyj N e Ž 41 ex N The paraeters h, j,, and the unknown value of aex are deterined fro the four values a 0,,a 3, with the result hs5998, js , es197607, Ž 4 and the extrapolated value for the exact constant: a s Ž 43 ex This is now in perfect agreeent with the Monte Carlo values 13 Fig 1 Difference between the extrapolated pressure constant a ex and the optiized N-th order value an obtained fro variational perturbation theory for the ethod presented in this paper Žsolid line and the first four values of the approxiation schee introduced in Ref wxž 4 dashed line Dots represent the values to order N in these approxiations The approach is graphically shown in Fig 1 where the optiized values a 0,,a3 all lie on a straight line Ž solid line For coparison, we have also extrapolated the first four values a K 0,,aK 3 in the approach of Ref wx 4 yielding a value aex K f , which is 49% saller than Ž 43 5 Suary We have calculated the universal constant a occuring in the pressure law Ž 11 of a ebrane fluctuating between two walls This has been done by replacing the walls by a sooth potential with a paraeter This potential approaches the wall potential in the liit 0 The anharonic part of the sooth potential was treated perturbatively The liit 0 corresponds to a strong-coupling liit of the power series, and was calculated by variational perturbation theory Extrapolating the lowest four approxiations to infinity yields a pressure constant a, which is in very good agreeent with Monte Carlo values References wx 1 W elfrich, Z Naturforsch A 33 Ž ; W elfrich, RM Servuss, Nuovo Ciento D 3 Ž wx W Janke, Kleinert, Phys Lett A 117 Ž Žhttp: rrwwwphysikfu yberlinde r ; kleinert r kleiner_ e133 wx 3 W Janke, Kleinert, M Meinhart, Phys Lett B 17 Ž 1989

7 M Bachann et alrphysics Letters A Ž berlinder ; kleinertr kleiner_re184 ; G Gopper, DM Kroll, Europhys Lett 9 Ž ; F David, J de Phys 51 Ž 1990 C7-115; RR Netz, R Lipowsky, Europhys Lett 9 Ž w4x Kleinert, cond-atr Ž 1998 Ž physikfuyberlinder ; kleinertrkleiner_re77 wx 5 Kleinert, Path Integrals in Quantu Mechanics, Statistics, and Polyer Physics, World Scientific, Singapore 1995, sec- ond edition, Chap 5 Žthird edition: berlinder ; kleinertrkleiner_reb3r3rdedhtl wx 6 IS Gradstein, IM Ryshik, Tables of Series, Products, and Integrals vol, Verlag arry Deutsch, Thun 1981, first edition wx 7 Kleinert, W Kurzinger, A Pelster, J Phys A: Math Gen Ž 31, 8307 Ž 1998 Ž quant-phr Ž physikfuyberlinder ; kleinertrkleiner_re67