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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|统计物理代写Statistical Physics of Matter代考|Two-State Model

As a simple example we revisit the two-state model of independent $N$ subunits that was studied earlier in a microcanonical way. The Hamiltonian is derived from (3.6),
$$\mathcal{H}\left{n_{i}\right}=\sum_{i=1}^{N}\left{\left(1-n_{i}\right) \epsilon_{0}+n_{i} \epsilon_{1}\right}$$
where $n_{i}$, the occupation number of the i-th subunit, can be either 0 or 1 . The probability of the microstate, that is, the joint probability that all subunits are in the state $n_{1}, n_{2}, \ldots n_{N}$ simultaneously is given by
$$P\left{n_{i}\right}=\frac{\exp \left[-\beta \mathcal{H}\left{n_{i}\right}\right]}{Z}=Z^{-1} \exp \left(-\beta \sum_{i=1}^{N}\left(1-n_{i}\right) \epsilon_{0}+n_{i} \epsilon_{1}\right),$$
where
\begin{aligned} Z &\left.=\sum_{\left{n_{i}\right}} \exp \left[-\beta H\left{n_{i}\right}\right]=\sum_{n_{i}=0}^{1} \exp \left(-\beta \sum_{i=1}^{N}\left(1-n_{i}\right) \epsilon_{0}+n_{i} \epsilon_{1}\right)\right) \ &=\prod_{i=1}^{N} \sum_{n_{i}=0}^{1} \exp \left{-\beta\left(1-n_{i}\right) \epsilon_{0}+n_{i} \epsilon_{1}\right} \ &=\left(e^{-\beta \epsilon_{0}}+e^{-\beta \epsilon_{1}}\right)^{N} \end{aligned}
is the partition function. In deriving it, the two summations in the second expression above was exchangeable. The binomial expansion of (3.43) expresses the partition function as $Z=\sum_{N_{1}=0}^{\mathrm{N}} \frac{N !}{N_{0} ! N_{1} !} e^{-\beta\left(\epsilon_{1} N_{0}+\epsilon_{1} N_{1}\right)}$

## 物理代写|统计物理代写Statistical Physics of Matter代考|The Gibbs Canonical Ensemble

Now, a system in contact with a thermal bath is subject to a generalized force $f_{i}$, which is kept at constant, so that the system’s Hamiltonian is modified to
$$\mathcal{H}{g}{\mathcal{M}}=\mathcal{H}{\mathcal{M}}-f{i} \mathcal{X}{i}{\mathcal{M}}$$ Here the generalized displacement $\mathcal{X}{i}{\mathcal{M}}$, the conjugate to the force $f_{i}$, is a thermally fluctuating variable. The system is specified by the macroscopic variables $\left(T, f_{i}, N\right)$ and the underlying microstates constitute the so called “Gibbs canonical ensemble”.
The microstate $\mathcal{M}$ occurs with the canonical probability
$$P{\mathcal{M}}=\frac{e^{-\beta \mathcal{H}{g}{\mathcal{M}}}}{Z{g}\left(T, f_{i}, N\right)}=\frac{e^{-\beta \mathcal{H}{\mathcal{M}}+\beta f_{i} \mathcal{X}{i}{\mathcal{M}}}}{Z{g}\left(T, f_{i}, N\right)}$$
where
$$Z_{g}\left(T, f_{i}, N\right)=\sum_{\mathcal{M}} e^{-\beta \mathcal{H}{\mathcal{M}}+\beta f_{i} \mathcal{X}{i}{\mathcal{M}}}$$ is the Gihhs partition function. Fxamples are a magnet suhject to a constant magnetic field, and a polymer chain subject to a constant force which is discussed below. The average displacement in this ensemble is given by \begin{aligned} X{i}=\left\langle\mathcal{X}{i}{\mathcal{M}}\right\rangle &=\frac{\sum{\mathcal{M}} \mathcal{X}{i}{\mathcal{M}} e^{-\beta \mathcal{H}{\mathcal{M}}+\beta f{i} \mathcal{X}{i}{\mathcal{M}}}}{\sum{\mathcal{M}} e^{-\beta \mathcal{H}{\mathcal{M}}+\beta f_{i} \mathcal{X}{i}{\mathcal{M}}}} \ &=\frac{\partial Z{g}}{\beta \partial f_{i}} / Z_{g}=k_{B} T \frac{\partial}{\partial f_{i}} \ln Z_{g}\left(T, f_{i}, N\right) \end{aligned}
In view of the thermodynamic identity, $X_{i}=-\frac{\partial}{\partial f i} G,(2.23)$, the Gihhs free energy is identified as $$G\left(T, f_{i}, N\right)=-k_{B} T \ln Z_{g}\left(T, f_{i}, N\right),$$
from which all the thermodynamic variables are generated as explained in Chap. $2 .$

# 统计物理代考

## 物理代写|统计物理代写Statistical Physics of Matter代考|Two-State Model

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## 物理代写|统计物理代写Statistical Physics of Matter代考|The Gibbs Canonical Ensemble

$$\mathcal{H} g \mathcal{M}=\mathcal{H M}-f_{i} \mathcal{X}{i \mathcal{M}}$$ 这里是广义位移 $\mathcal{X} i \mathcal{M}$, 与力的共轭 $f{i}$ 是一个热波动变量。系统由宏观变量指定 $\left(T, f_{i}, N\right)$ 底层的微观状态构成了所谓的“吉布斯规范系综“。 微观状态 $\mathcal{M}$ 以典型概率发生
$$P \mathcal{M}=\frac{e^{-\beta \mathcal{H} g \mathcal{M}}}{Z g\left(T, f_{i}, N\right)}=\frac{e^{-\beta \mathcal{H M}+\beta f_{i} X_{i} \mathcal{M}}}{Z g\left(T, f_{i}, N\right)}$$

$$Z_{g}\left(T, f_{i}, N\right)=\sum_{\mathcal{M}} e^{-\beta \mathcal{M}+\beta f_{i} \mathcal{X}{i} \mathcal{M}}$$ 是 Gihhs 分区函数。示例是承受恒定磁场的磁体和承受恒定力的聚合物链，这将在下面讨论。该集合中的平均位移由下式给出 $$X i=\langle\mathcal{X} i \mathcal{M}\rangle=\frac{\sum \mathcal{M} \mathcal{X}{i} \mathcal{M} e^{-\beta \mathcal{H}+\beta f i \mathcal{X}{i} \mathcal{M}}}{\sum \mathcal{M} e^{-\beta \mathcal{H}+\beta f{i} \mathcal{X}{i} \mathcal{M}}} \quad=\frac{\partial Z g}{\beta \partial f{i}} / Z_{g}=k_{B} T \frac{\partial}{\partial f_{i}} \ln Z_{g}\left(T, f_{i}, N\right)$$

$$G\left(T, f_{i}, N\right)=-k_{B} T \ln Z_{g}\left(T, f_{i}, N\right),$$

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