Statistical Mechanics of Specular Reflections from Fluctuating Membranes and Interfaces

Abstract

We study the density of specular reflection points in the geometrical optics limit when light scatters off fluctuating interfaces and membranes in thermodynamic equilibrium. We focus on the statistical mechanics of both capillary-gravity interfaces (characterized by a surface tension) and fluid membranes (controlled by a bending rigidity) in thermodynamic equilibrium in two dimensions. Building on work by Berry et al. we show that the statistics of specular points is fully characterized by three fundamental length scales, namely, a correlation length $\xi$, a microscopic length scale $\ell$ and the overall size L of the interface or membrane. By combining a scaling analysis with numerical simulations, we confirm the existence of a scaling law for the density of specular reflection points, $n_{spec}$, in two dimensions, given by $n_{spec}\propto \ell ^{-1}$ in the limit of thin fluctuating interfaces with the interfacial thickness $\ell \ll \xi _I$. The density of specular reflections thus diverges for fluctuating interfaces in the limit of vanishing thickness and shows no dependance on the interfacial capillary-gravity correlation length $\xi _I$. Although fluid membranes under tension also exhibit a divergence in $n_{spec}\propto \left( \xi _M\ell \right) ^{-1/2}$, the number of specular reflections in this case can grow by decreasing the membrane correlation length $\xi _{M}$.

Appendices

Appendices

Appendix A: Correlation Function Calculation

Here we provide details of the correlation function calculations reported in section II. A for the fluctuating interfaces or membranes in $1+1$ dimensions. From Eq. (4) we obtain,

$$\begin{aligned} C(y)=\langle f(x_{0})f(x_{0}+y)\rangle =\frac{1}{2\pi }\int ^{+\infty }_{-\infty }\frac{dq e^{iqy}}{a+b q^2+ c q^4} \end{aligned}$$

(A1)

Upon considering the complex q-plane, the denominator defines four poles at:

$$\begin{aligned} q= & {} \pm i \left[ \frac{b\mp \sqrt{\delta }}{2c}\right] ^{1/2}\nonumber \\ \delta= & {} b^{2}-4ac \end{aligned}$$

(A2)

We assume that $\delta >0$. Upon defining an interfacial capillary-gravity correlation length by $\xi _I\equiv \sqrt{b/a}$ and the ultraviolet cutoff induced by the c-term as $\ell \equiv \sqrt{c/b}$, this condition simply means that $\xi _I>2\ell$. Because all poles reside on the imaginary axis, we also have $b>\sqrt{\delta }$. By completing the contour in the upper half plane, which encloses the poles at $q= i \left[ \frac{b\pm \sqrt{\delta }}{2c}\right] ^{1/2}$, we find,

$$\begin{aligned}&C(y)=\frac{1}{2\pi }\int ^{+\infty }_{-\infty }\frac{dq e^{iqy}}{a+b q^2+ c q^4}\nonumber \\&\quad = i \bigg (\frac{e^{iqy}}{2bq+4cq^3}\bigg )_{q=i \left[ \frac{b+\sqrt{\delta }}{2c}\right] ^{1/2}}+ i \bigg (\frac{e^{iqy}}{2bq+4cq^3}\bigg )_{q=i \left[ \frac{b-\sqrt{\delta }}{2c}\right] ^{1/2}}\nonumber \\&\quad =\frac{1}{\sqrt{2 \delta }}\left[ \frac{e^{-\sqrt{(\frac{b-\sqrt{\delta }}{2c})}|y|}}{\sqrt{\frac{b-\sqrt{\delta }}{c}}}-\frac{e^{-\sqrt{(\frac{b+\sqrt{\delta }}{2c})}|y|}}{\sqrt{\frac{b+\sqrt{\delta }}{c}}}\right] \end{aligned}$$

(A3)

In a similar fashion we find that the second derivative of the correlation function is given by,

$$\begin{aligned} C^{(2)}(y)= & {} \frac{-1}{2\pi }\int ^{+\infty }_{-\infty }\frac{q^2 dq e^{iqy}}{a+b q^2+ c q^4}\nonumber \\= & {} \frac{1}{2\sqrt{2 \delta }}\left[ \sqrt{\frac{b-\sqrt{\delta }}{c}}e^{-\sqrt{(\frac{b-\sqrt{\delta }}{2c})}|y|}-\sqrt{\frac{b+\sqrt{\delta }}{c}}e^{-\sqrt{(\frac{b+\sqrt{\delta }}{2c})}|y|}\right] \end{aligned}$$

(A4)

Note that $C^{(2)}(0)$ diverges as $c\rightarrow 0$, reflecting the sensitivity of this quantity to the effective ultraviolet cutoff $\ell =\sqrt{c/b}$ as $\ell \rightarrow 0$.

Appendix B: Functional Integrals for the Probability Density of a Fluctuating Membrane

Here we calculate, via functional integral methods, the normalized probability of a function f(y) to have a specific value and slope at position x given by $f(x)=g, f'(x)=g_1$. The probability density for these quantities with a fluctuating interface described by our simple quadratic free energy functional $F_{I}\left[ f(y)\right]$, [we use the interface free energy Eq. (5) for concreteness, but identical manipulation apply for the membrane free energy Eq. (6)] is given by,

$$\begin{aligned}&p\left[ f(x)=g,f'(x)=g_1;x\right] \nonumber \\&\quad =\frac{\int \mathcal {D}f(y)\delta \left[ f(x)-g\right] \delta \left[ f'(x)-g_1\right] e^{-F_I\left[ f(y)\right] }}{\int \mathcal {D}f(y) e^{-F_I\left[ f(y)\right] }}\nonumber \\&\quad =\frac{\int \mathcal {D}f(y)\delta \left[ f(x)-g\right] \delta \left[ f'(x)-g_1\right] e^{\frac{-1}{2}\int dy\left[ a f(y)^{2}+b\left( \frac{df(y)}{dy}\right) ^{2}\right] }}{\int \mathcal {D}f(y) e^{-F_I\left[ f(y)\right] }}\nonumber \\&\quad =\int ds\int dt e^{-2\pi i sg}e^{-2\pi i t g_1}\langle e^{2\pi i s f(x)+2\pi i t f'(x) }\rangle \nonumber \\&\quad = \int ds\int dt e^{-2\pi i sg}e^{-2\pi i t g_1}e^{-2\pi ^2 \langle f^2(x)\rangle s^2 -2\pi ^2 \langle f'^2(x)\rangle t^2-4\pi ^2 \langle f(x) f'(x)\rangle st }\nonumber \\&\quad =\int ds\int dt e^{-2\pi i sg}e^{-2\pi i t g_1}e^{-2\pi ^2 C(0) s^2+2\pi ^2 C^{(2)}(0) t^2 } \end{aligned}$$

(B1)

where $\langle .\rangle$ represents a thermal average, and $\left\langle f(x)f'(x)\right\rangle =\frac{1}{2}\left\langle \frac{d}{dx}f^{2}(x)\right\rangle =0$ with our periodic boundary conditions. We also have $C(0)=\langle f^{2}(x)\rangle$ and $C^{(2)}(x)=-\left\langle \left( \frac{df}{dx}\right) ^{2}\right\rangle$. Here we used the identity, valid for any Gaussian probability distribution, $\langle e^{h(x)}\rangle =e^{\frac{1}{2}\langle {h^{2}(x)}\rangle }$ and the integral representation of the delta function,

$$\begin{aligned} \delta \left[ f(x)-g\right] =\int ^{\infty }_{-\infty } ds e^{2\pi i \left[ f(x)-g\right] s} . \end{aligned}$$

(B2)

This probability density for a fluctuating membrane is normalized by construction, $\int ^{\infty }_{-\infty } p\left[ g,g_1\right] dg_1,dg_2=1$. By the aid of standard Gaussian integrals we find from Eq. (B1) the probability density, $p\left[ f(x)=g, f'(x)=g_1;x\right]$ is in fact independent of x,

$$\begin{aligned} p\left[ f(x)=g,f'(x)=g_1;x\right] =\frac{1}{2\pi \left[ -C(0)C^{(2)}(0)\right] ^{1/2}}\exp \left[ -\frac{g^{2}}{2C(0)}+\frac{g^{2}_{1}}{2C^{(2)}(0)}\right] .\nonumber \\ \end{aligned}$$

(B3)

Similarly, one can calculate the normalized probability of a function f(y) to have a given slope and curvature described by $f'=g_1, f''=g_2$ at the position x,

$$\begin{aligned} p\left[ f'(x)=g_1,f(x)''=g_2;x\right]= & {} \frac{\int \mathcal {D}f(y)\delta \left[ f'(x)-g_1\right] \delta \left[ f''(x)-g_2\right] e^{-F\left[ f(y)\right] }}{\int \mathcal {D}f(y) e^{-F\left[ f(y)\right] }}\nonumber \\= & {} \int ds\int dt e^{-2\pi i sg_1}e^{-2\pi i t g_2}\langle e^{2\pi i s f'(x)+2\pi i t f''(x) }\rangle \nonumber \\= & {} \int ds\int dt e^{-2\pi i sg_1}e^{-2\pi i t g_2}e^{2\pi ^2 C^{(2)}(0) s^2-2\pi ^2 C^{(4)}(0) t^2 }\nonumber \\= & {} \frac{1}{2\pi \left[ -C^{(2)}(0)C^{(4)}(0)\right] ^{1/2}}\exp \left[ \frac{g_1^{2}}{2C^{(2)}(0)}-\frac{g_2^{2}}{2C^{(4)}(0)}\right] \nonumber \\ \end{aligned}$$

(B4)

where $C^{(4)}(0)=\lim _{y\rightarrow 0}\left\langle f''(x+y)f''(x)\right\rangle =\left\langle \left( \frac{d^2 f}{dx^2}\right) ^2\right\rangle$. In a similar fashion, we can find the probability density of a stochastic function with fixed higher order derivatives, at position x, with $\frac{d^n f(x)}{dx^n}=g_n$.