# 物理代写|统计物理代写Statistical Physics of Matter代考|PHYSICS7546

## 物理代写|统计物理代写Statistical Physics of Matter代考|Effects of Thermal Undulations

It has been well known to physiologists for more than hundred years that red blood cells flicker, indicative of shape fluctuations or undulations. The origin is the thermal collective motion of lipid molecules. To quantify the fluctuations and their correlations, we consider a planar membrane, where a position of the membrane is specified by $\boldsymbol{r}=(\boldsymbol{x}, h(\boldsymbol{x}))$ where $\boldsymbol{x}=(x, y)$ is a two dimensional position on the reference flat surface and $h(\boldsymbol{x})$ is the height of undulation (Fig. 12.8). In this section we evaluate $\left\langle h^2\right\rangle,\langle h(\boldsymbol{x}) \cdot h(0)\rangle,\left\langle(h(\boldsymbol{x})-h(0))^2\right\rangle$, and $\langle\boldsymbol{n}(\boldsymbol{x}) \cdot \boldsymbol{n}(0)\rangle$, where $\boldsymbol{n}(\boldsymbol{x})$ is a normal vector outward from the surface at $\boldsymbol{x}$. To this end, we begin with constructing the effective Hamiltonian in terms of the height undulation field $h(\boldsymbol{x})$.
Consider a stress-free, square membrane, which projects on the area $A_0=L^2$ (Fig. 12.8). The element of the undulating surface at $\boldsymbol{r}$ is constructed by a cross product of the two surface tangent vectors along $x$ and $y$ axis, $\boldsymbol{u}_x=\partial \boldsymbol{r} / \partial x=$ $(1,0, \partial h / \partial x)$ and $\boldsymbol{u}_y-\partial \boldsymbol{r} / \partial y-(0,1, \partial h / \partial y)$ :
$$d \mathcal{A}=\left|\boldsymbol{u}_x \times \boldsymbol{u}_y\right| d x d y=\left[1+\left(\frac{\partial h}{\partial x}\right)^2+\left(\frac{\partial h}{\partial y}\right)^2\right] d x d y$$
Due to the thermal fluctuations, the area increases to
\begin{aligned} \mathcal{A} &=\int_0^L d x \int_0^L d y\left[1+\left(\frac{\partial h}{\partial x}\right)^2+\left(\frac{\partial h}{\partial y}\right)^2\right]^{1 / 2} \ &=A_0+\Delta \mathcal{A} \end{aligned}
where
$$\Delta \mathcal{A} \approx \frac{1}{2} \int_0^L d x \int_0^L d y\left[\left(\frac{\partial h}{\partial x}\right)^2+\left(\frac{\partial h}{\partial y}\right)^2\right]=\frac{1}{2} \int d^2 x\left(\nabla_x h(\boldsymbol{x})\right)^2$$
is the area increase evaluated to the harmonic order in $h$, where $\nabla_x$ is the two dimensional gradient. For an elastic surface with a uniform surface tension $\gamma$ and no bending rigidity, the surface free energy (12.10) of deformation is
$$\mathcal{F}_S=\gamma \Delta \mathcal{A}=\frac{\gamma}{2} \int d^2 \boldsymbol{x}\left(\boldsymbol{\nabla}_x h(\boldsymbol{x})\right)^2$$

## 物理代写|统计物理代写Statistical Physics of Matter代考|Surface Undulation Fluctuation and Correlation

To facilitate calculating the averages of the quantities associated with undulation, we deal with the Fourier transform for $h$ :
$$h(q)=\int d^2 x e^{-i q \cdot x} h(\boldsymbol{x})$$
and

$$h(\boldsymbol{x})=\frac{1}{L^2} \sum_{\boldsymbol{q}} e^{i q \cdot x} h(\boldsymbol{q}) .$$
We use the periodic BC, so that $q=\left(q_x, q_y\right)$ with each component respectively taking $N$ values, $q_n=2 \pi n / L, n=\pm 1, \pm 2 \ldots, \pm N / 2$, where $N=L / a, a$ is a microscopic length which is in the order of the diameter of the lipid molecule.
The mean square of the undulation amplitude is defined as
\begin{aligned} \left\langle h^2\right\rangle & \equiv \frac{1}{L^2} \int d^2 \boldsymbol{x}\left\langle h^2(\boldsymbol{x})\right\rangle \ &=\frac{1}{L^6} \int d^2 \boldsymbol{x} \sum_q \sum_{q^{\prime}}\left\langle h(\boldsymbol{q}) h^\left(\boldsymbol{q}^{\prime}\right)\right\rangle e^{i\left(\boldsymbol{q}-q^{\prime}\right) \cdot \boldsymbol{x}} \ &=\frac{1}{L^4} \sum_{\boldsymbol{q}}\left\langle|h(\boldsymbol{q})|^2\right\rangle \end{aligned} where we used $$\frac{1}{L^2} \int e^{i\left(q-q^{\prime}\right) \cdot x} d^2 x=\delta_{q q^{\prime} \cdot}$$ The undulation correlation function is now expressed as $$C_h\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right)=\left\langle h(\boldsymbol{x}) h\left(\boldsymbol{x}^{\prime}\right)\right\rangle=\frac{1}{L^4} \sum_{\boldsymbol{q}} \sum_{\boldsymbol{q}^{\prime}} e^{i q \cdot x-i \boldsymbol{q}^{\prime} \cdot x^{\prime}}\left\langle h(\boldsymbol{q}) h^\left(\boldsymbol{q}^{\prime}\right)\right\rangle .$$

## 物理代写|统计物理代写物质统计物理学代考|热波动的影响

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