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Colloid Physics, Proceedings of the Workshop on Colloid Physics, University of Konstanz, Germany, 1995 [Physica A 235, 1 (1997)].
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0003603422
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41
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0030214492
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Such a generalized cylinder has already been introduced in the different context of the interaction between colloidal particles and flexible polymer chains in a dilute polymer solution, see E. Eisenriegler, A. Hanke, and S. Dietrich, Phys. Rev. E 54, 1134 (1996).
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Eisenriegler, E.1
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42
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85037218829
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Any definition (Formula presented) different from that of (Formula presented) as the true correlation length is related to (Formula presented) by the universal ratio (Formula presented) This leads to a corresponding change of the scaling functions (Formula presented) and of quantities obtained from them so that experimentally observable quantities remain unchanged (for more details see Appendix B and Ref. 14
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Any definition (Formula presented) different from that of (Formula presented) as the true correlation length is related to (Formula presented) by the universal ratio (Formula presented) This leads to a corresponding change of the scaling functions (Formula presented) and of quantities obtained from them so that experimentally observable quantities remain unchanged (for more details see Appendix B and Ref. 14).
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43
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85037226021
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The bars on (Formula presented) serve to distinguish these coefficients from the corresponding ones (Formula presented) (Formula presented) introduced in Ref. 12 which are defined differently from Eq. (2.4)
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The bars on (Formula presented) serve to distinguish these coefficients from the corresponding ones (Formula presented) (Formula presented) introduced in Ref. 12 which are defined differently from Eq. (2.4).
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45
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85037196871
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The first singular contribution proportional to (Formula presented) in Eq. (2.4) can be attributed 10 11 to the leading nontrivial “surface operator” (Formula presented) with the stress tensor operator (Formula presented) whose scaling dimension is equal to the spatial dimension D
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The first singular contribution proportional to (Formula presented) in Eq. (2.4) can be attributed 1011 to the leading nontrivial “surface operator” (Formula presented) with the stress tensor operator (Formula presented) whose scaling dimension is equal to the spatial dimension D.
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47
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85037209508
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The curves for (Formula presented) for (Formula presented) shown in Fig. 2 of Ref. 13 and in Fig. 1(b) of Ref. 14, which do not comply with this statement, are not correct
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The curves for (Formula presented) for (Formula presented) shown in Fig. 2 of Ref. 13 and in Fig. 1(b) of Ref. 14, which do not comply with this statement, are not correct.
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48
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85037189614
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See, e.g., the article of F. David in Ref. 27
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See, e.g., the article of F. David in Ref. 27.
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49
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85037241056
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We briefly outline the argument: Since the curvature contributions must depend on how the surface is embedded in the space (Formula presented) they must be derivable from the local extrinsic curvature tensor (Formula presented) where (Formula presented) is the local unit vector normal to the surface (we only consider orientable surfaces). The principal local radii of curvature (Formula presented) are the inverse of the (Formula presented) eigenvalues of the matrix (Formula presented). Therefore (Formula presented) (Formula presented), and (Formula presented) as defined in Eq. (2.17) are the only independent scalar quantities to first and second order in (Formula presented) which can be deduced from (Formula presented) and which are invariant under permutations of the indices (Formula presented)
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We briefly outline the argument: Since the curvature contributions must depend on how the surface is embedded in the space (Formula presented) they must be derivable from the local extrinsic curvature tensor (Formula presented) where (Formula presented) is the local unit vector normal to the surface (we only consider orientable surfaces). The principal local radii of curvature (Formula presented) are the inverse of the (Formula presented) eigenvalues of the matrix (Formula presented). Therefore (Formula presented) (Formula presented), and (Formula presented) as defined in Eq. (2.17) are the only independent scalar quantities to first and second order in (Formula presented) which can be deduced from (Formula presented) and which are invariant under permutations of the indices (Formula presented).
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50
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85037192728
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The leading regular term captured by the ellipses in Eq. (2.23) which is analytic in (Formula presented) is proportional to (Formula presented). Since all terms in curly brackets depend only on the dimensionless variables (Formula presented) and (Formula presented) this regular term must scale with (Formula presented) where the exponent is larger than two if (Formula presented)
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The leading regular term captured by the ellipses in Eq. (2.23) which is analytic in (Formula presented) is proportional to (Formula presented). Since all terms in curly brackets depend only on the dimensionless variables (Formula presented) and (Formula presented) this regular term must scale with (Formula presented) where the exponent is larger than two if (Formula presented).
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51
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85037216128
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The dashed line in Fig. 33 is known rather accurately by means of the (Formula presented) expansions of (Formula presented) and (Formula presented) in conjunction with the corresponding exact values in (Formula presented) 33
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The dashed line in Fig. 33 is known rather accurately by means of the (Formula presented) expansions of (Formula presented) and (Formula presented) in conjunction with the corresponding exact values in (Formula presented) 33.
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52
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85037220996
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a lattice model such a “cylinder” with microscopically small radius R can be realized by a single line of lattice sites where the orientation of the spins is fixed
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In a lattice model such a “cylinder” with microscopically small radius R can be realized by a single line of lattice sites where the orientation of the spins is fixed.
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53
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85037240137
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It is instructive to interpret Eq. (4.3) as an equation of motion of a particle at “position” m and “time” s subjected to the “potential energy” (Formula presented) 7 8. In general the differential equation cannot be solved analytically due to the time-dependent friction term. One seeks solutions (Formula presented) which come at rest on the top (Formula presented) of the potential hill at time (Formula presented). This problem can be solved numerically, e.g., by means of a “shooting method” as a trial and error approach
-
It is instructive to interpret Eq. (4.3) as an equation of motion of a particle at “position” m and “time” s subjected to the “potential energy” (Formula presented) 78. In general the differential equation cannot be solved analytically due to the time-dependent friction term. One seeks solutions (Formula presented) which come at rest on the top (Formula presented) of the potential hill at time (Formula presented). This problem can be solved numerically, e.g., by means of a “shooting method” as a trial and error approach.
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55
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5844239497
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Ref. 7
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Compare, e.g., P. J. Upton, J. O. Indekeu, and J. M. Yeomans, Phys. Rev. B 40, 666 (1989), and Ref. 7.
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(1989)
Phys. Rev. B
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, pp. 666
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Upton, P.J.1
Indekeu, J.O.2
Yeomans, J.M.3
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56
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85037246935
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(a) M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972); (b) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1965)
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(a) M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972);(b) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1965).
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57
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85037223084
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The curvature parameters for a convex surface as considered here should, of course, be the same as for a concave surface. In the present mean-field case this does indeed hold. This can be confirmed by using the methods outlined in Appendix C
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The curvature parameters for a convex surface as considered here should, of course, be the same as for a concave surface. In the present mean-field case this does indeed hold. This can be confirmed by using the methods outlined in Appendix C.
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59
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0030771336
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T. Gil, M. C. Sabra, J. H. Ipsen, and O. G. Mouritsen, Biophys. J. 73, 1728 (1997).
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Biophys. J.
, vol.73
, pp. 1728
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Gil, T.1
Sabra, M.C.2
Ipsen, J.H.3
Mouritsen, O.G.4
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60
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0000937660
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H.-G. Döbereiner, E. Evans, M. Kraus, U. Seifert, and M. Wortis, Phys. Rev. E 55, 4458 (1997).
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, pp. 4458
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Döbereiner, H.-G.1
Evans, E.2
Kraus, M.3
Seifert, U.4
Wortis, M.5
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