物理代写|理论力学代写theoretical mechanics代考|PHYC30022

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

物理代写|理论力学代写theoretical mechanics代考|Results of the Measurements

The amplitude characteristics are presented in the Table 2. The analysis of the obtained data shows that obvious filtration properties of first, second and third samples begin after the frequency $0.6 \mathrm{MHz}$. The increase of the distance between the rows, used in the third sample, has no effect on the through-transmitted amplitude in the latter case; however an obvious change of the impulse shape is quite clear, much more notable than for the first two samples. One may conclude that the increase in the distance between the rows complicated the diffraction field inside the sample. The fourth sample begins to demonstrate its filtration properties just at the frequency of $0.4 \mathrm{MHz}$, that is obviously connected with smaller ratio of the US wave length above the size of the obstacle.

The preliminary investigations $[1,2]$ show that after the first filtration strip there is a strip of almost perfect transmission. As can be seen from Fig. 5 and Table 2, for the first three samples such a frequency strip begins from $1.8 \mathrm{MHz}$. This effect is less pronounced for the fourth sample, though the amplitude of the through-transmitted signal is still higher than at the frequency $1.25 \mathrm{MHz}$.

Analyzing the table, one may conclude that the increase of the size of the holes (the fourth sample) results in the worst through-transmission in the meta-material, cutting off more than $90 \%$ of energy, beginning from the frequency $-1 \mathrm{MHz}$. The increase of the distance between the rows along the wave propagation also reduces the carrying capacity for higher frequencies, and the passage to the first filtration band becomes smoother (which is obvious for the frequency equal to $0.6 \mathrm{MHz}$, where the sample 3 demonstrates the best through-transmission). The shift of the rows in the second sample has not so strong effect at low frequencies, and in some cases even improves the through-transmission of the US signal, as can be seen for example, for the frequency $1.25 \mathrm{MHz}$. Nevertheless, for higher frequencies one can see a significant suppression of the transmission, which may be connected with a complex structure of the re-reflections inside the meta-material.

物理代写|理论力学代写theoretical mechanics代考|Simulation Method Using the Wang-Landau Algorithm

Monte-Carlo method use broad class of computational algorithms which are based on random walks. The typical problem in statistical physics that can be solved by these method is calculating mean values of macroscopic variables (energy, order parameter,

etc.) at different temperatures for systems which follows Boltzmann statistics. There are some techniques for Monte-Carlo method: Metropolis [18], Wolff [19], Lee [20], Wang-Landau algorithms [21], parallel tempering [22]. In this section, Metropolis and Wang-Landau algorithms are described and illustrated on the example of twodimensional Ising model.

The Ising model consists of spins which have two possible orientations. Originally developed for simulation of ferromagnetic materials, now, this model has many applications including the simulation of ferroelectrics [23], spin glasses [24], image data processing [25], neuroscience, etc. In 1944 , the two-dimensional Ising model on a square lattice was analytically solved by Onsager [26]. The Hamiltonian of this model is determined by the formula:
$$E=-J \sum_{\langle i, j\rangle} \overrightarrow{S_{i}} \overrightarrow{S_{j}}-\vec{H} \sum_{i} \overrightarrow{S_{i}}$$
where $\overrightarrow{S_{i}}$ is the value of spin located in site $i$, the symbol $\langle i, j\rangle$ denotes the pairs of nearest-neighbor segments, $J$ is a parameter of spin interactions, $\vec{H}$ is the external magnetic field strength.

The Metropolis algorithm generates the sequence of states at a predetermined temperature using the probability distribution for the system. For the Ising model, the Metropolis algorithm should be applied as follows:

1. A random spin is chosen and rotated.
2. The new system configuration is accepted with probability:
$$P=\min \left(-\frac{\Delta E}{k_{\mathbb{B}} T}, 1\right),$$
where $\Delta E$ is energy change due to the spin rotation, $k_{B}$ is the Boltzmann constant, $T$ is the temperature.
3. Steps 1 and 2 are repeated.
The results of simulation for the two-dimensional Ising model with periodic boundary conditions obtained by means of the Metropolis algorithm are presented in Fig. 1. The heat capacity was determined by the formula:
$$C=\frac{\left\langle E^{2}\right\rangle-\langle E\rangle^{2}}{k_{B} T^{2}} .$$

物理代写|理论力学代写theoretical mechanics代考|Investigation of the Influence of Bulk Properties

The surface properties of layers are determined not only by chemical composition of the substance, but also by their physical structure and the orientational order of polymer chains [29]. Intermolecular orientation interactions are much weaker than valence interactions; therefore, the self-organization of the system with the given chemical structure is determined by intermolecular interactions. In this chapter, we consider the equilibrium properties and phase transitions on the surface of ferroelectric polymer system, in which orientational interactions both between the surface molecules and molecules located in the bulk are taken into account.

Model. Usually, polymer chains have predominantly planar orientation relatively to the interphase boundary [30]. Therefore, in this paper, to describe the surface of ferroelectric polymer systems, we use a two-dimensional model, which consist of $M$ freely-jointed chains, each of which is a sequence of $N$ connected rigid segments, located in parallel to the surface (Fig. 4).

The main quantitative characteristic of the polymer chain flexibility is the persistent length $a$, which is related with the energetic constant of intrachain orientation interaction $K_{1}$ by the ratio:
$$K_{1}=\frac{a \cdot k_{B} T}{2}$$

Similar to the persistent length $a$, we introduce the interchain interaction parameter of $b$. The orientation interaction of neighboring polymer chain elements is described by the energy constant $K_{2}$,
$$K_{2}=\frac{b \cdot k_{B} T}{2} .$$
To take into account the interaction of surface molecules with molecules located in the bulk of the film, we use the mean field constant $V$ and the dimensionless mean field parameter $q$ :
$$q=\frac{V}{k_{B} T}$$
The internal energy in the low-temperature approximation can be represented as:
\begin{aligned} H=& \frac{1}{2} K_{1} \sum_{n, m=1}^{N, M}\left(\varphi_{n, m}-\varphi_{n-1, m}\right)^{2}+\frac{1}{2} K_{2} \sum_{n, m=1}^{N, M}\left(\varphi_{n, m}-\varphi_{n, m-1}\right)^{2} \ &-\mu V \sum_{n, m=1}^{N, M} \cos \left(\varphi_{n, m}\right) \end{aligned}
where $\mu$ is the long-range orientation order parameter, which is defined as the average cosine of the angle between the directions of chain rigid element and the director, i.e. $\mu=\left\langle\cos \varphi_{\vec{n}}\right\rangle$.

物理代写|理论力学代写theoretical mechanics代考|Simulation Method Using the Wang-Landau Algorithm

Ising 模型由具有两个可能方向的自旋组成。该模型最初是为模拟铁磁材料而开发的，现在，该模型具有许多应用，包括模拟铁电体 [23]、自旋玻璃 [24]、图像数据处理 [25]、神经科学等。 1944 年，二维伊辛模型Onsager [26] 对正方形晶格进行了解析求解。该模型的哈密顿量由以下公式确定：

Metropolis 算法使用系统的概率分布在预定温度下生成状态序列。对于 Ising 模型，Metropolis 算法应用如下：

1. 选择并旋转随机旋转。
2. 新的系统配置很可能被接受：
磷=分钟(−Δ和ķ乙吨,1),
在哪里Δ和是由于自旋旋转引起的能量变化，ķ乙是玻尔兹曼常数，吨是温度。
3. 重复步骤 1 和 2。
通过 Metropolis 算法获得的具有周期性边界条件的二维 Ising 模型的模拟结果如图 1 所示。热容量由以下公式确定：
C=⟨和2⟩−⟨和⟩2ķ乙吨2.

物理代写|理论力学代写theoretical mechanics代考|Investigation of the Influence of Bulk Properties

ķ1=一个⋅ķ乙吨2

ķ2=b⋅ķ乙吨2.

q=在ķ乙吨

H=12ķ1∑n,米=1ñ,米(披n,米−披n−1,米)2+12ķ2∑n,米=1ñ,米(披n,米−披n,米−1)2 −μ在∑n,米=1ñ,米因⁡(披n,米)

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