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Statistical Mechanics of Thermal Fluctuations of Nearly Spherical Membranes: the Influence of Bending and Stretching Elasticities

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Abstract

Theoretical studies of nearly spherical vesicles and microemulsion droplets, that present typical examples for thermally-excited systems that are subject to constraints, are reviewed. We consider the shape fluctuations of such systems constrained by fixed area \(A\) and fixed volume \(V\), whose geometry is presented in terms of scalar spherical harmonics. These constraints can be incorporated in the theory in different ways. After an introductory review of the two approaches: with an exactly fixed by delta-function membrane area \(A\) [Seifert, Z. Phys. B, 97, 299 (1995)] or approximatively by means of a Lagrange multiplier \(\sigma \) conjugated to \(A\) [Milner and Safran, Phys. Rev. A, 36, 4371 (1987)], we discuss the determined role of the stretching effects, that has been announced in the framework of a model containing stretching energy term, expressed via the membrane vesicle tension [Bivas an d Tonchev, Phys. Rev. E, 100, 022416 (2019)]. Since the fluctuation spectrum for the used Hamiltonian is not exactly solvable an approximating method based on the Bogoliubov inequalities for the free energy has been developed. The area constraint in the last approach appears as a self-consistent equation for the membrane tension. In the general case this equation is intractable analytically. However, much insight into the physics behind can be obtained either imposing some restrictions on the values of the model parameters, or studying limiting cases, in which the self-consistent equation is solved. Implications for the equivalence of ensembles have been discussed as well.

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ACKNOWLEDGMENTS

This review is based on my lecture at the Conference in memory of Vyatcheslav Borisovich Priezzhev held in the Bogoliubov Laboratory of Theoretical Physics (JINR-Dubna, 10 September 2019). I am grateful to the Organizing Committee and especially to V.P. Spiridonov and A.M. Povolotsky, for the invitation and hospitality.

I am grateful to I. Bivas for numerous stimulating discussions on the physics of vesicles concerning theory and experiment. Many of the ideas presented in this review are based on our previous common works. I would like to thank A.G. Petrov for his useful comments on the manuscript.

Funding

This work is partly supported by the JINR (Dubna)–ISSP-BAS (Bulgaria) collaborative Grant “Investigation of the influence of nanoparticles on the properties of biologically relevant systems” 2019/2021.

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Correspondence to N. S. Tonchev.

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This paper is dedicated to the memory of my friend and colleague Slava Priezzhev, an outstanding scientist and man, whose contribution to statistical mechanics and probability theory is original and remarkable in many ways. His comments on physicsand things of life were always nontrivial and impactful.

Appendices

APPENDIX A

LIMIT CASE ANALYTICAL SOLUTIONS OF EQS. (46) AND (123)

Equations (46), (123) and (154) could be studied analytically by replacing the sum in its r.h.s. with an integral. In order to validate the corresponding approximation we shall use the Euler–McLaurin summation formula

$$\begin{gathered} \sum\limits_{n = 0}^{{{n}_{{{\text{max}}}}} - 1} F\left( {n + \frac{1}{2}} \right) = \int\limits_0^{{{n}_{{{\text{max}}}}}} {F\left( {t + \frac{1}{2}} \right)dt} \\ - \,\,\frac{1}{2}\left[ {F\left( {{{n}_{{{\text{max}}}}} + \frac{1}{2}} \right) - F\left( {\frac{1}{2}} \right)} \right] \\ + \,\,\frac{1}{{12}}\left[ {F{\kern 1pt} '\left( {{{n}_{{{\text{max}}}}} + \frac{1}{2}} \right) - F{\kern 1pt} '\left( {\frac{1}{2}} \right)} \right] + \ldots , \\ \end{gathered} $$
(A.1)

where

$$F(x) = {{F}_{1}}(x) = \frac{{2x}}{{{{x}^{2}} + \Sigma - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}$$
(A.2)

with \(\Sigma = {{\bar {\Sigma }}_{{{\text{MS}}}}}\) in Eq. (46) and \(\Sigma = {{\bar {\Sigma }}_{{{\text{app}}}}}\) in Eq. (123), and

$$F(x) = {{F}_{2}}(x) = \frac{{2x}}{{{{{[{{x}^{2}} + \Sigma - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}]}}^{2}}}}$$
(A.3)

with \(\Sigma = {{\bar {\Sigma }}_{{{\text{app}}}}}\) in Eq. (154).

Let us ignore: (i) the higher order terms in Eq. (A.1), and ii.) approximate \(F(x) \approx F(0) + xF{\kern 1pt} '(0)\) in the interval \([0,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}]\). The approximations made are consistent only for large \(\mathop {\overline \Sigma }\nolimits_{{\text{app}}} \gg 1\) since then the relative change of \(F(x)\) is small when \(n \to n + 1\). With these approximations the Euler–Maclaurin formula Eq. (A.1) reduces to

$$\begin{gathered} \sum\limits_{n = 0}^{{{n}_{{{\text{max}}}}}} F\left( {n + \frac{1}{2}} \right) \approx \int\limits_0^{{{n}_{{{\text{max}}}}} + \tfrac{1}{2}} {F(x)dx} + \frac{1}{{24}}F{\kern 1pt} '(0) \\ + \,\,\frac{1}{2}\left[ {F\left( {{{n}_{{{\text{max}}}}} + \frac{1}{2}} \right) + \frac{1}{6}F{\kern 1pt} '\left( {{{n}_{{{\text{max}}}}} + \frac{1}{2}} \right)} \right] \\ \end{gathered} $$
(A.4)

(c.f. with Eq. (59.10), p. 173 [75]). Using Eq. (A.4) the summation in Eqs. (46) and (123) can be performed easily. The result is

$$\begin{gathered} \sum\limits_{n = 2}^{{{n}_{{{\text{max}}}}}} {{F}_{1}}\left( {n + \frac{1}{2}} \right) \approx \ln\frac{{N + {{N}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} + \Sigma }}{{\Sigma - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}} + \frac{1}{{12}}\frac{1}{{\Sigma - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}} \\ + \,\,\frac{1}{2}\frac{{2{{N}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} + 1}}{{N + {{N}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} + \Sigma }} - \frac{1}{6}\frac{{N + {{N}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} - \Sigma + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}{{{{{({{N}^{2}} + {{N}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} + \Sigma )}}^{2}}}}, \\ \end{gathered} $$
(A.5)

and from where it follows

$$\begin{gathered} \sum\limits_{n = 2}^{{{n}_{{{\text{max}}}}}} {{F}_{1}}\left( {n + \frac{1}{2}} \right) \approx \ln\frac{N}{\Sigma } + \frac{\Sigma }{N} \\ + \,\,O\left( {\frac{1}{{{{N}^{{1/2}}}}}} \right) + O\left( {\frac{1}{\Sigma }} \right) + O\left( {\mathop {\left[ {\frac{\Sigma }{N}} \right]}\nolimits^2 } \right), \\ \end{gathered} $$
(A.6)

in the case \({\text{of}}\) Eqs. (46) and (123), and

$$\sum\limits_{n = 2}^{{{n}_{{{\text{max}}}}}} {{F}_{2}}\left( {n + \frac{1}{2}} \right) \approx \frac{1}{{\mathop {\overline \Sigma }\nolimits_{{\text{app}}} }} - \frac{1}{{\mathop {\overline \Sigma }\nolimits_{{\text{app}}} + N}} + O\left( {\frac{1}{{\mathop {\overline \Sigma }\nolimits_{{\text{app}}}^2 }}} \right),$$
(A.7)

in the case of Eq. (157). In the above expressions it is used that \({{n}_{{{\text{max}}}}} \approx \sqrt N \).

Solution of Eq. (46)

Let us introduce the notation

$${{x}_{0}} = - \frac{{{{{\bar {\Sigma }}}_{{MS}}}}}{N}.$$
(A.8)

With the help of Eq. (A.6) and the definition of \({{\bar {\sigma }}_{0}}\) (see Eq. (71)), Eq. (46) may be presented (up to the used approximations) in the form:

$${{x}_{0}}{{e}^{{{{x}_{0}}}}} = - {{e}^{{ - \tfrac{\Delta }{\gamma }}}}.$$
(A.9)

Eq. (A.9) can be solved in terms of the Lambert function \({\mathbf{W}}(x)\). A review of its mathematical properties and physical applications can be found in [7680] and refs. therein. Recall that by definition

$${\mathbf{W}}(x{{e}^{x}}) = x.$$
(A.10)

The Lambert function can take two possible real values for \( - \tfrac{1}{e} \leqslant x \leqslant 0\). Values satisfying \({\mathbf{W}}(x) \geqslant - 1\) belong to the principal branch denoted as \({{{\mathbf{W}}}_{0}}(x)\), while values satisfying \({\mathbf{W}}(x) \leqslant - 1\) belong to the \({{{\mathbf{W}}}_{1}}(x)\) branch. The two branches meet at the branch point for \(x = - \tfrac{1}{e}\), where \({{{\mathbf{W}}}_{0}}\left( { - \tfrac{1}{e}} \right) = {{{\mathbf{W}}}_{{ - 1}}}\left( { - \tfrac{1}{e}} \right)\). All values of \({\mathbf{W}}\) for \(x \geqslant 0\) belong to the principal branch \({{{\mathbf{W}}}_{0}}(x)\).

The solution of Eq. (A.9) now reads

$${{x}_{0}} = {\mathbf{W}}\left( { - {{e}^{{ - \tfrac{\Delta }{\gamma }}}}} \right),$$
(A.11)

or finally

$$\mathop {\overline \Sigma }\nolimits_{{\text{MS}}} = - N{\mathbf{W}}\left( { - {{e}^{{ - \tfrac{\Delta }{\gamma }}}}} \right).$$
(A.12)

In the interval \( - {{e}^{{ - 1}}} \leqslant - {{e}^{{ - \tfrac{\Delta }{\gamma }}}} < 0\) the equation has two solutions given by \({{{\mathbf{W}}}_{0}}\) and \({{{\mathbf{W}}}_{{ - 1}}}\), respectively.

For large \(x\), the function \({\mathbf{W}}(x)\) is approximated by

$${\mathbf{W}}(x) = \ln x - \ln\ln x + o(1).$$
(A.13)

For small \(x\), the Taylor series around \(x = 0\) is given by

$${\mathbf{W}}(x) = x - {{x}^{2}} + ...$$
(A.14)

The first few terms of the series expansion of \({\mathbf{W}}(x)\) near the branching point are

$${\mathbf{W}}(x) = - 1 + p - \frac{1}{3}{{p}^{2}} + ...,$$
(A.15)

where \(p = \pm \sqrt {2(e.x + 1)} \) for \({\mathbf{W}}{{(x)}_{{0,1}}}.\)

Thus, using Eq. (A.14) for \(x = {{e}^{{ - \tfrac{\Delta }{\gamma }}}} \ll 1\), one gets Eq. (47):

$$\mathop {\overline \Sigma }\nolimits_{{\text{MS}}} = N{{e}^{{ - \tfrac{\Delta }{\gamma }}}}.$$
(A.16)

Using the expansion near the branching point of the Lambert function, i.e. \(x = {{e}^{{ - \tfrac{\Delta }{\gamma }}}} \approx {{e}^{{ - 1}}}\), one obtains:

$$\mathop {\overline \Sigma }\nolimits_{{\text{MS}}} = N\left[ {1 - \sqrt {2(1 - {{e}^{{ - \tfrac{\Delta }{\gamma } + 1}}})} } \right].$$
(A.17)

Solution of Eq. (123)

For \(\mathop {\overline \Sigma }\nolimits_{{\text{app}}} \gg 1\), Eq. (123) can be treated in the same way. Let us introduce the notation

$$x = \left( {\frac{1}{{\overline C }} - \frac{1}{N}} \right)\mathop {\overline \Sigma }\nolimits_{{\text{app}}} {\kern 1pt} .$$
(A.18)

Using Eq. (A.6) the self-consistent Eq. (123) may be presented (up to the used approximations) in the form:

$$x{{e}^{x}} = \left( {\frac{1}{{\overline C }} - \frac{1}{N}} \right)N{{e}^{{{{{\bar {\sigma }}}_{0}}/\overline C }}}.$$
(A.19)

In terms of the Lambert \({\mathbf{W}}(x)\) function the solution reads:

$$\mathop {\overline \Sigma }\nolimits_{{\text{app}}} = {{\left( {\frac{1}{{\overline C }} - \frac{1}{N}} \right)}^{{ - 1}}}{\mathbf{W}}\left[ {\left( {\frac{1}{{\overline C }} - \frac{1}{N}} \right)N\exp\left( {\frac{{{{{\bar {\sigma }}}_{0}}}}{{\overline C }}} \right)} \right].$$
(A.20)

Thus, if \(\tfrac{1}{{\overline C }} - \tfrac{1}{N} < 0\) there will be two solutions or none (or only one solution if the argument of W is exactly \( - \tfrac{1}{e}\)). If \(\tfrac{1}{{\overline C }} - \tfrac{1}{N} > 0\) there will be one solution.

With the help of bout expansions the Lambert \({\mathbf{W}}(x)\) function (A.13) and (A.14) one easily obtains Eqs. (127) and (128).

Not that if

$$ - \frac{\Delta }{\gamma } = \frac{{{{{\bar {\sigma }}}_{0}}}}{{\overline C }},$$
(A.21)

combining Eqs. (A.20) and (A.9), the more general relation takes place

$$\begin{gathered} \mathop {\overline \Sigma }\nolimits_{{\text{app}}} = {{\left( {\frac{1}{{\overline C }} - \frac{1}{N}} \right)}^{{ - 1}}} \\ \times \,\,{\mathbf{W}}\left[ {\left( {\frac{1}{{\overline C }} - \frac{1}{N}} \right)\mathop {\overline \Sigma }\nolimits_{{\text{MS}}} \exp\left( { - \frac{{\mathop {\overline \Sigma }\nolimits_{{\text{MS}}} }}{N}} \right)} \right]. \\ \end{gathered} $$
(A.22)

From the above result, if \({{K}_{s}} \to \infty \), immediately follows Eq. (144) where use has been made of definitions (125) and (A.10).

APPENDIX B

THE GRIFFITS–FISHER LEMMA

There is a mathematical statement known as Griffits–Fisher lemma [81, 82], which asserts that if a sequence of convex function converges pointwise to a limit function, then the sequence of its derivatives converges to the derivative of the limit function at the points of its continuous differentiabilty. More precisely, if all functions \({\text{\{ }}{{f}_{n}}(x){\text{\} }}\) and the limit function \({{f}_{\infty }}(x)\) are differentiable at a point \({{x}_{0}} \in I \subset R\), then

$$\mathop {lim}\limits_{n \to \infty } f_{n}^{'}({{x}_{0}}) = f_{\infty }^{'}({{x}_{0}}).$$
(B.1)

More general result due to Fisher consider the case of non-differentiable functions with left and right derivatives at any point \(x \in I\). The latter is relevant if the systems undergo thermodynamic phase transitions with spontaneously symmetry breaking. These statements are useful in proving the asymptotic closeness of certain average values in the model and approximating system, see e.g. [39]. In our case, we consider both Hamiltonian \(H\), Eq. (110), and \({{H}_{{{\text{app}}}}}\) (Eq. (111)), and introduce the following auxiliary Hamiltonians

$$\mathcal{H}(h) = H + h\sigma ({\text{v}})$$
(B.2)

and

$${{\mathcal{H}}_{{{\text{app}}}}}(h) = {{H}_{{{\text{app}}}}} + h\sigma ({\text{v}}),$$
(B.3)

where \(h\) is an auxiliary real parameter which at the end of the calculations will sent to zero. Further, we obtain that

$${{\left\langle {\sigma ({\text{v}})} \right\rangle }_{H}} = \frac{\partial }{{\partial h}}{{\left. {F[\mathcal{H}(h)]} \right|}_{{h = 0}}}$$
(B.4)

and

$${{\left\langle {\sigma ({\text{v}})} \right\rangle }_{{{{H}_{{{\text{app}}}}}}}} = \frac{\partial }{{\partial h}}F{{\left. {[{{\mathcal{H}}_{{{\text{app}}}}}(h)]} \right|}_{{h = 0}}}.$$
(B.5)

In the limit when the analog of the correlator Eq. (157) in the r.h.s. of the Bogoliubov inequalities with Hamiltonians (B.2) and (B.3) tends to zero as a function of its parameters:

$$F[{{\mathcal{H}}_{{{\text{app}}}}}(h)] \to F[\mathcal{H}(h)].$$
(B.6)

Since \(F[{{\mathcal{H}}_{{{\text{app}}}}}(h)]\) and \(F[H(h)]\) are convex differentiable functions of \(h\) from the lemma follows that,

$${{\left\langle {\sigma ({\text{v}})} \right\rangle }_{{{{H}_{{{\text{app}}}}}}}} \to {{\left\langle {\sigma ({\text{v}})} \right\rangle }_{H}}.$$
(B.7)

For the above proof to be correct the definition of the thermodynamic (or other) limit should be scrutinized.

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Tonchev, N.S. Statistical Mechanics of Thermal Fluctuations of Nearly Spherical Membranes: the Influence of Bending and Stretching Elasticities. Phys. Part. Nuclei 52, 290–314 (2021). https://doi.org/10.1134/S1063779621020064

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