# 物理代写|统计物理代写Statistical Physics of Matter代考|KYA322

## 物理代写|统计物理代写Statistical Physics of Matter代考|Helfrich Interaction and Unbinding Transitions

When two membranes are brought to close proximity, less space is allowed for thermal undulations to play in between, resulting in a reduction of entropy. This induces a repulsion called the Helfrich interaction. We use a scaling argument as below to determine the interaction as a function of the inter-membrane distance D. First, noting that the interaction is induced by thermal fluctuation and involves two length scales, $\left\langle h^2\right\rangle^{1 / 2}$ and $D$, we must have
$$U_h \sim k_B T\left[\frac{\left\langle h^2\right\rangle}{D^2}\right]^p$$
which should scale as $\sim L^2$. Using $\left\langle h^2\right\rangle \sim\left(k_B T\right) L^2 / \varkappa$, one finds $p=1$, and
$$U_h \sim k_B T \frac{\left\langle h^2\right\rangle}{D^2} \sim \frac{\left(k_B T\right)^2 L^2}{\chi D^2} .$$
The equation interestingly shows that the repulsion is proportional to the factors $D^2$ and $\left(k_B T\right)^2$. Its exact expression is $U_f=3\left(k_B T\right)^2 L^2 /\left(\pi^2 \nsucc D^2\right)$ (Helfrich 1978). In addition, two membranes experience another fluctuation-induced interaction, that is, the van der Waals attraction, $U_{v d w}=-H /\left(12 \pi D^2\right)$ per unit area (6.33).
If the membranes are not charged, the total free energy change is

$$F=\left{\frac{3\left(k_B T\right)^2}{\pi^2 \chi}-\frac{H}{12 \pi}\right} \frac{L^2}{D^2} .$$
The membranes of the bending rigidity $x$ bind if $F$ is negative, i.e.,
$$T\varkappa_c=\frac{\left(6 k_B T_b\right)^2}{\pi H} \sim 10 k_B T_b$$
where Hamaker constant $H \sim k_B T_b$ is considered. This provides a reason why older blood cells with higher $x$ (less flexibility) tend to aggregate among themselves and adhere to vessel wall more frequently. If $T>T_c$, or $\%<\chi_c$, the membranes unbind. This implies that the thermal fluctuation-induced undulation, although very small in magnitude (12.58), can be an essential feature for cell stability.

## 物理代写|统计物理代写Statistical Physics of Matter代考|Brownian Motion/Diffusion Equation Theory

Understanding that the Brownian motion is an incessant continuation of random jumps, Einstein derived the equation for the probability density $P(\boldsymbol{r}, t)$ of a Brownian particle to be found at a position $r$ and time $t$,
$$\frac{\partial P(\boldsymbol{r}, t)}{\partial t}=D \nabla^2 P(\boldsymbol{r}, t) .$$
Here the $D$ is the diffusivity or the diffusion constant given by
$$D=\frac{\left\langle l^2\right\rangle}{6 \tau} .$$
$\tau$ is the jump time, which is chosen to be macroscopically small but microscopically large enough that the motions after the time are mutually independent. In the time interval $\tau$ the particle is displaced by a distance $l$ that is statistically distributed with the mean-square $\left\langle l^2\right\rangle$. The derivation of the above equations will be given in next chapter within the frame of master equation.

Equation (13.1) written for the number density or concentration of such Brownian particles $n(\boldsymbol{r}, t)=N P(\boldsymbol{r}, t)$,
$$\frac{\partial n(\boldsymbol{r}, t)}{\partial t}=D \nabla^2 n(\boldsymbol{r}, t)$$
is the well-known equation called the diffusion equation with the $D$ same as that in (13.1) if the concentration is low enough to neglect interactions between the Brownian particles.

The diffusion equation is one of the hydrodynamic equations derived from conservation laws and constitutive relations below and in Chap. 19. The total number of the particles $N$ being conserved, the number density, satisfies the continuity equation
$$\frac{\partial n}{\partial t}+\boldsymbol{\nabla} \cdot \boldsymbol{J}_{\boldsymbol{n}}=0,$$
where $\boldsymbol{J}_n$ is the number flux vector: $J_n$ is average number of particles that cross a unit area in the $x y$ plane per unit time. Phenomenologically the flux is given by
$$\boldsymbol{J}_n=-D \boldsymbol{\nabla} n,$$
which is the Fick’s law stating that the particles flow from the region of higher concentration to that of lower concentration. Substituting (13.5) into (13.4) yields the diffusion equation
$$\frac{\partial n}{\partial t}=\boldsymbol{\nabla} \cdot D \boldsymbol{\nabla} n$$

## 物理代写|统计物理代写物质统计物理学代考|Helfrich相互作用和解绑定跃迁

. . .

$$U_h \sim k_B T\left[\frac{\left\langle h^2\right\rangle}{D^2}\right]^p$$
，它应该缩放为$\sim L^2$。使用$\left\langle h^2\right\rangle \sim\left(k_B T\right) L^2 / \varkappa$，可以发现$p=1$和
$$U_h \sim k_B T \frac{\left\langle h^2\right\rangle}{D^2} \sim \frac{\left(k_B T\right)^2 L^2}{\chi D^2} .$$
。有趣的是，这个方程表明斥力与因子$D^2$和$\left(k_B T\right)^2$成正比。它的确切表达是$U_f=3\left(k_B T\right)^2 L^2 /\left(\pi^2 \nsucc D^2\right)$ (Helfrich 1978)。此外，两个膜还经历另一种波动诱导的相互作用，即范德华引力，单位面积$U_{v d w}=-H /\left(12 \pi D^2\right)$(6.33)。如果膜不带电，总自由能变化

$$F=\left{\frac{3\left(k_B T\right)^2}{\pi^2 \chi}-\frac{H}{12 \pi}\right} \frac{L^2}{D^2} .$$

$$T\varkappa_c=\frac{\left(6 k_B T_b\right)^2}{\pi H} \sim 10 k_B T_b$$
，其中考虑Hamaker常数$H \sim k_B T_b$，弯曲刚度的膜$x$绑定。这就解释了为什么$x$较高(灵活性较低)的老龄血细胞往往会彼此聚集，并更频繁地粘附在血管壁上。如果是$T>T_c$或$\%<\chi_c$，则膜解缚。这意味着，热波动引起的波动，尽管幅度非常小(12.58)，但可能是细胞稳定性的一个基本特征

## 物理代写|统计物理代写物质统计物理学代考|布朗运动/扩散方程理论

$$\frac{\partial P(\boldsymbol{r}, t)}{\partial t}=D \nabla^2 P(\boldsymbol{r}, t) .$$

$$D=\frac{\left\langle l^2\right\rangle}{6 \tau} .$$
$\tau$是跃迁时间，它在宏观上很小，但在微观上足够大，使得时间之后的运动是相互独立的。在时间间隔$\tau$中，粒子被一个$l$的距离所取代，该距离在统计上以均方$\left\langle l^2\right\rangle$分布。以上方程的推导将在下一章中在主方程的框架内给出

$$\frac{\partial n(\boldsymbol{r}, t)}{\partial t}=D \nabla^2 n(\boldsymbol{r}, t)$$

$$\frac{\partial n}{\partial t}+\boldsymbol{\nabla} \cdot \boldsymbol{J}_{\boldsymbol{n}}=0,$$
，其中$\boldsymbol{J}_n$是数量通量向量:$J_n$是每单位时间在$x y$平面上穿过单位面积的粒子的平均数量。在现象学上，通量由
$$\boldsymbol{J}_n=-D \boldsymbol{\nabla} n,$$

$$\frac{\partial n}{\partial t}=\boldsymbol{\nabla} \cdot D \boldsymbol{\nabla} n$$

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