Fluctuation pressure of membrane between walls

Transcription

1 5 July 999 Ž Physics Letters A Fluctuation pressure of membrane between walls H Kleinert Freie UniÕersitat Berlin, Institut fur Theoretische Physi, Arnimallee4, D-495 Berlin, Germany Received February 999; received in revised form 7 March 999; accepted April 999 Communicated by PR Holland Abstract For a single membrane of stiffness fluctuating between two planar walls of distance d, we calculate analytically the pressure law p B T ps 3 8 Ž dr The prefactor p r8;00775 is in very good agreement with results from Monte Carlo simulations 0079"000 q 999 Elsevier Science BV All rights reserved A stac of n parallel, thermally fluctuating membranes exerts upon the enclosing planar walls a pressure which depends on the stiffness and the temperature T as follows: n B T psa n, Ž 3 nq drž nq where B is Boltzmann s constant and d the distance between the walls Ž see Fig This law, first deduced from dimensional considwx, is of fundamental impor- erations by Helfrich tance in the statistical mechanics of membranes just as the ideal gas law pv s BT in the statistical leinert@physifu-berlinde, URL wwwphysifu-berlinder ; leinert mechanics of point particles We would therefore lie to now the size of the prefactor, the stac constant an as accurately as possible So far, its value was determined only by extensive Monte Carlo simulations as being w,3x a` s00"000 Ž For a single membrane, the following value was found w4,3 x: a s0079"000 Ž 3 So far, there exists no analytic theory to explain these values The purpose of this note is to fill this gap for the constant a, by calculating analytically the pressure of a single membrane between parallel walls The r99r$ - see front matter q 999 Elsevier Science BV All rights reserved Ž PII: S

2 70 ( H KleinertrPhysics Letters A Fig Membrane fluctuating between walls of distance d, exerting a pressure p theoretical tool for this has only recently become available: A strong-coupling theory developed origiwx 5, was extended suc- nally in quantum mechanics cessfully to quantum field theories wx 6, where it has been used to obtain extremely accurate values for the critical exponents of OŽ n -symmetric scalar fields 4 with w -interactions wx 6 Strong-coupling theory gives direct access to the large-g behavior of divergent truncated power series expansions of the type g f Ž g sv a0 q Ý a ž Ž 4 q / V s Ž The g `-limit of f g, to be denoted by f,is obtained by setting V'cg r q and optimizing the function f Ž c sg f Ž c where r q ž / r q yq 0 0 Ý s 'g ca b q ac b, Ž 5 y / l Ž yq r b s Ý Ž y Ž 6 ž l ls0 is the binomial expansion of Ž y Žyqr truncated after the Ž y th term Optimizing means extremizing f Ž c in c or, if an extremum does not exist, extremizing the derivative fx Ž c 3 We apply this theory to a membrane between walls by proceeding as follows The partition function of the membrane is given by the functional integral H ½ H B T 5 Zs D už x exp y d x E už x 'e yafrbt, Ž 7 where už x is a vertical displacement field of the membrane fluctuating between horizontal walls at usydr and dr The quantities A and f are the wall area and the free energy per unit area, respectively Such a restriction of a field is hard to treat analytically We therefore perform a transformation which maps the interval u gydr,dr Ž to an infinite w-axis, d pw p w p 4 w 4 us arctan sw y q q, ž 4 p d 3d 5d / Ž 8 and add to the fluctuation energy E in the exponent of Ž 7 a potential energy which eeps the membrane between ydr and dr Ž Poschl Teller potential : pot pot int 4 E se0 qe s Hd xmf Ž už x ½ 5 4 ` m už x s Hd x u Ž x q Ý p, d s with expansion coefficients,,, : Ž 9,,,,, Ž 0 The potential energy per area is plotted in Fig Its presence destroys the simple scaling properties of the partition function Ž 7, which depends only on the Fig Smooth Potential replacing box walls

3 ( H KleinertrPhysics Letters A dimensionless variable d r T The new partition B function Z associated with the modified energy E q E pot has an additional dependence on the dimensionless variable gsp rm d The original hard-wall system is obtained in the strong-coupling limit g ` In the opposite limit where g goes to zero, the energy EqE pot becomes harmonic, 4 H ½ 5 E0 s d x E už x qm už x, Ž leading to a partition function A Ž 4 4 y Trlog E qm y m 8 Z se sconst=e, Ž 0 where A is the area of the walls For a finite distance d, the interaction energy E int is treated perturbatively order by order in g, expanding the exponential e ye int r B T in a power series, and each power in a sum of all pair contractions These are pictured by loop diagrams whose lines represent the correlation function d ² : iž x yx už x už x s H e 4 4 BT Ž p qm Ž 3 The free energy density fsybta y log Z is obtained from all connected loop diagrams For simplicity, we shall use natural units with rbts The lowest contribution to the free energy density comes from the expectation value of the u 4 -interaction or the loop diagram 3 which is of the order rd : m 4 m 4 4 ² u : s 3² u :, Ž 4 d d the line representing the pair expectation d ² u : sh s Ž Ž p qm 8m Ž Together with the exponent in, we thus obtain first-order free energy density m p f s q Ž md Continuing the perturbation expansion, yields an expansion of the general form ž / p p f sm 8 q qa q 64 md md p ž md / qa, Ž 7 where a,,a are dimensionless numbers By Ž comparison with 4 we identify psqs, Vsm, gsp rd The function f Ž c of Eq Ž 5 describing the limiting large-g behavior is obtained by setting V'cp r d, and reads p c a f Ž c s b0 q 64 q b q d ž 4 c a / q b Ž 8 c y According to the above-described strong-coupling theory, we must optimize the expression f Ž c in parentheses Since the second term does not contain c, we separate this term out, and write c a fž c s 64 qdfž c ' 64 qž b0 q b q 4 c a / q b, Ž 9 c y with only the remainder Df Ž c to be optimized Let Df be ist optimal value If we now only a,we find the approximation D f s( 3ar6 Ignoring Df for a moment, the first term in Ž 9 yields the lowest estimate for the free energy density of the original system p f s, Ž 0 64 d implying a pressure law E f p psy s Ž E d 3 d 3 By comparison with the general pressure law Ž, we identify the prefactor as being p a s = f = Ž 8

4 7 ( H KleinertrPhysics Letters A Without the prefactor factor r, this would agree perfectly with the Monte Carlo value Ž 3 Thus we expect the contribution of D f for ` to be equal or almost equal to r64 The calculation of the higher-order terms a, a 3, is tedious, and will be presented in a separate detailed publication wx 7 In this note we shall circumvent it by exploiting a close relationship of the present problem with a closely analogous exactly solvable one, which may be treated in precisely the same way: The euclidean version of a quantummechanical point particle in a one-dimensional box u gydr,dr Ž 4 The partition function of a particle in a box is H yž r BT HdxŽ E u yafrbt Zs Due 'e Ž 3 The quantum-mechanical ground state energy of this Ž system is exactly nown: BTr pr d, corresponding to a free energy density B T p fs Ž 4 d The path integral Ž 3 may now be treated as before, ie, we transform u to w via Ž 8, and separate the field energy into a Gaussian energy Ž in natural units 4 H ½ 5 E0 s dx E už x qm už x Ž 5 and an interaction energy which loos the same as Ž 9, except that the integration Hd x runs now only over one dimension, Hdx The first-order contribution to the free energy density is now Ž in natural units with r Ts m 4 m 4 4 ² u : s 3² u :, Ž 6 d d with the pair expectation dq ² u : sh s, Ž 7 4 p q qm m leading to a first-order free energy density m p f s q, Ž 8 d and a full perturbation expansion of the form p p 4 fsm q qa q Ž md md ž / B From this we find the function f Ž c defined in Eq Ž 5 governing the strong-coupling limit d 0 by setting Vsm 'cp r d : p f Ž c s f Ž c, Ž 30 d with c a fž c s 4qDfŽ c ' 4qž b0 q b q 4 c a / q b Ž 3 c y Here the first term yields the lowest approximation p f s, Ž 3 4 d which is precisely half the exact result Thus we conclude that the optimal value of the neglected expression Df Ž c must be once more equal to r4 in the limit ` In order to see how this happens, we extend the Bender Wu recursion relation for the perturbation coefficients of the anharmonic oscillator wx 8 It yields for the ground state energy an expansion 3p p 4 q 4y Ž 4 y d 8d p y360 4 q q 6d 6 Ž p 8 4 Ž y y q d 8 q5 q 3780 q Ž Inserting the coefficients Ž 0 we find a,a 4,a 6,: 5 7,y,,y,,y,, Ž whereas the odd coefficients a, a, a, vanish Fig 3 Plots of the functions Df Ž c of Eq Ž 35, all being optimal exactly at cs with Df sr4

5 ( H KleinertrPhysics Letters A Table The functions Df Ž c of Eq Ž 35, and their optimal values Df Ž Df c Df c cy 4 6c 4 3c 3cy cy c 5cy 5c3 cy c 35cy 35c3 7cy cy 05c3 63cy5 45cy7 7cy q q y 3 q y q 4 q y q y 5 q y q y q To account for this fact, we resum the series containing only the even terms c a a4 a D f s q b q b q b, Ž c c c taing the coefficients b of Eq Ž i 6 with the param- eters ps, qs This yields the functions Df Ž c plotted in Fig 3 and listed in Table For all D f Ž c, optimization yields a strong-coupling value Df equal to r4, thus raising the initial value r4 in Ž 3 to the correct final value r 5 To exploit this property of a particle in a box for the system at hand, the membrane between walls, we mae the following crucial observation: The Feynman integrals determining the first two terms in the free energy densities in Eq Ž 6 for a membrane and in Eq Ž 8 for a particle are related to each other by a simple transformation of the integration variables The membrane integrals d m 4 4 H logž qm s, Ž p 4 d s Ž 36 Ž p qm 8m H 4 4 go over into those of the particle in the box dq log q dq qm 4 sm H Ž, H s 4 p p q qm m Ž 37 by the transformation d ` dq q, H 4 H Ž p y` p Thus, if we multiply each loop integral by a factor r4, we find immediately the free energy density f of the membrane in Eq Ž 6 from that of the particle in the box in Eq Ž 8 But the analogy carries further: By differentiation Ž Ž 36 and 37 with respect to m, we see that also all Feynman integrals Hd rž p Ž 4 q m 4 n are re- Ž Ž lated to the Hdqr p q qm 4 n by the same factor r4 This property has the consequence that most of the connected loop diagrams contributing to the perturbation expansion of the free energy density, shown in Fig 4 up to five loops, are related by a factor Ž r4 L, where L is the number of loops In particular, all such diagrams coincide which are usually summed in the Hartree Foc approximation Žchain diagrams, daisy diagrams, etc Only the topological more involved diagrams 3, 4, 4, 4 5, 5, 5 3, 5 5, 5 6, 5 7, 5, 5, 5 5 in Fig 4 do not follow this pattern For a particle in a box, we can easily calculate the associated Feynman integrals in x-space as described in Chapter 3 of Ref wx 5, and find that they contribute less than 5% to the sum of all diagrams at each loop level This implies that the corresponding results for the membrane between walls will differ at most by this relative amount from those for the particle in the box We therefore con- Ž c in Eq clude that since the optimal value of Df Ž 3 doubles the initial value for `, the analogous function for the membrane between walls in Eq Ž 9 will double approximately For the quantitawx 7 A tive deviations see the forthcoming publication precise doubling of the result Ž leads to a very good agreement with the Monte Carlo number Ž Fig 4 Vacuum diagrams up to five loops

6 74 ( H KleinertrPhysics Letters A The alert reader will have noted that the field transformation Ž 8 is rather special We may, for instance, chose any mapping us w 4 4 n r n wq8p w r3d qw4w rd qqž wrd x p sw y w qq OŽ w, Ž 38 d which has a doubled coefficient of w 3 with respect to the expansion Ž 8 As a consequence, the functions f Ž c in Ž 9 and Ž 3 would have a doubled first term Since this would be the correct final value, the remaining functions D f Ž c would have to converge to a vanishing optimal value for ` Žin the particle case exactly, in the membrane case approximately To reach this goal, the coefficients w,w, in Ž can be chosen rather arbitrarily, although there are a few convenient ways for which the speed of convergence is fast A preferred choice is one in which all coefficients a,a 3,a 3, of the perturbation expansion vanishes for a particle in a box This and other possibilities will be studied separately w9,0 x Acnowledgements The author is grateful to Prof A Chervyaov, Prof B Hamprecht, Dr A Pelster, and M Bachmann for numerous useful discussions References wx W Helfrich, Z aturforsch A 33 Ž ; W Helfrich, RM Servuss, uovo Cimento D 3 Ž wx W Jane, H Kleinert, Phys Lett 58 Ž wx 3 W Jane, H Kleinert, M Meinhardt, Phys Lett B 7 Ž The simulations were repeated by G Gompper, DM Kroll, Europhys Lett 9 Ž with the results a f00798,a fž 4r3 ` af006 A further simulation of a` was recently reported by RR etz, R Lipowsi, Europhys Lett 9 Ž , with the result a ` f Ž 3r p r8f09, which is half as big as Helfrich s original estimate 3p r8 in Ref wx Another estimate a f0080 was given by F David, J de Phys 5 Ž 990 ` C7-5 wx 4 W Jane, H Kleinert, Phys Lett A 7 Ž wx 5 H Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific, Singapore, 995, sec ed, pp 850 wx 6 H Kleinert, Phys Rev D 57 Ž Žaps997- jun5_00, Addendum Ž condmatr ; Phys Lett B 434 Ž Ž cond-matr98067 wx 7 M Bachmann, H Kleinert, A Pelster, Strong-Coupling Calculation of Pressure for Fluctuating Membrane between Walls, Berlin preprint Ž in preparation wx 8 CM Bender, TT Wu, Phys Rev 84 Ž 969 3; Phys Rev D 7 Ž wx 9 H Kleinert, A Chervyaov, B Hamprecht, Variational Perturbation Expansion for Particle in a Box, Berlin preprint Žin preparation w0x H Kleinert, A Chervyaov, Inverse Variational Perturbation Theory for Hard-Wall Systems, Berlin preprint Žin preparation