### 数学代写|计算复杂度理论代写Computational complexity theory代考| Curie-Weiss Model

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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

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Curie-Weiss Model

An important aspect of the Ising model is given by its dimension $D$. Notably, for $D=1$, the Ising model has no phase transitions at finite temperature. For $D=2$, according to the Onsager’s solution, there is a phase transition (at a finite temperature). Then, in higher dimensions, although a phase transition can be observed, the definition of an analytical solution still constitutes an open problem. In particular, for $D=3$, the problem has been solved only by a numerical approach,

while for $D>3$ a solution is still required. Here, we briefly present a toy model that allows to describe the behavior of ferromagnetic transitions at infinite dimension, i.e., the Curie-Weiss (CW hereinafter) model. Remarkably the latter, despite being a toy model, has been proven to have a great relevance both in statistical mechanics and in information theory. The infinite dimension of the system entails that, in the CW model, each spin is connected with all the others. Accordingly, its Hamiltonian reads
$$H\left(\sigma_{1}, \ldots \sigma_{n}\right)=-\frac{1}{N} \sum_{(i<j)} \sigma_{i} \sigma_{j}-h \sum_{i=1}^{N} \sigma_{i}$$
As result, using the jargon of graph theory, the CW model defines a complete graph of $N$ nodes and $N(N-1) / 2$ links. Then, using the magnetization $m$ before introduced, and without considering the contribution of external fields, the Hamiltonian of this model takes following form
$$H\left(\sigma_{1}, \ldots \sigma_{n}\right)=-\frac{N}{2} m^{2}+O(1)$$
As above reported, for computing the expected values of physical quantities of systems like those we are considering, we need to compute the partition function of the system that in this case is equal to
$$Z=\sum_{\left{\sigma_{i}\right}} e^{-\beta H\left(\sigma_{1}, \ldots, \sigma_{N}\right)}$$
with $H$ defined in Eq. (2.11). Now, further calculations are required for solving the equation, as for instance, for computing the summation over the spin variables appearing in Eq. (2.13). However, without to show the whole mathematical derivation, we only report the final equation of state of the CW model
$$m=\tanh (\beta J m+\beta h)$$

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Landau Theory of Phase Transitions

The approaches here presented for studying the phenomenology of phase transitions by analytical methods allow to compute the partition function of a system. Therefore, at least in principle, a number of quantities can be computed according to Eq. (2.6). For a reason that will be soon explained, it is now worth to introduce an important thermodynamic potential, named “free energy,” that allows to study the state of equilibrium of a system. In particular, the so-called Helmholtz free energy is defined as $F=U-T S$, with $U$ internal energy, $T$ temperature, and $S$ entropy. So, since the second law of thermodynamics states that a system evolves toward

the state that maximizes its entropy, this law can be re-paraphrased stating that the state of equilibrium of a system corresponds to one that minimizes its free energy $F$. Now, we can motivate why we moved from the brief remark on the role of the partition function to the introduction of the free energy. Notably, a very important relation links the two quantities
$$-k_{b} T \ln Z=U-T S$$
As a result, we have $Z=e^{-\beta F}$. Here, the mean-field theory allows to obtain an approximated phase diagram of the system. However, when the latter is close to the critical point (e.g., the critical temperature), its behavior can be analyzed by using the formulation introduced by Landau, named “Landau theory of phase transitions.” The underlying assumption of this theory is that a system close to the critical point has a small order parameter (i.e., $m$ ), which leads to the expression of the free energy as the following summation of power series:
$$F(T ; m)=f(T ; 0)+\frac{1}{2} a(T) m^{2}+\frac{1}{4 !} b(T) m^{4}+\ldots$$
with $a(T)$ and $b(T)$ coefficients that can be computed analytically. For instance, Fig. $2.2$ shows the free energy of the CW model (with $h=0$ ). In particular, for $T>T_{c}$ there is only one minimum of free energy ( $\left.m=0\right)$, corresponding to the state defined “paramagnetic phase.” Instead, for $T<T_{c}$ there are two possible minima of free energy, and the symmetry $m \rightarrow-m$ is spontaneously broken (phenomenon known as “symmetry breaking”).

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Numerical Simulations

After introducing the Ising model, and some methods of approximation for studying special cases (e.g., $D \geq 4$ ), we can move toward a quick presentation of computational methods for performing the related numerical simulations. This part is of particular interest for achieving a preliminary idea on how to implement numerical simulations using networked agents, i.e., agents arranged in graphs, no matter if regular (as in the case of the Ising model) or random (as in the case of Complex Networks below discussed). In general, if we want to investigate the properties of a system using, for example, the formalism of the canonical ensemble, we have to deal with a system described by the macrostate $(N, V, T)$ (i.e., number of particles, volume, and temperature). Given a microstate $\sigma$, a generic observable can be indicated as $O(\sigma)$, and its average value at equilibrium is
$$\langle O(\sigma)\rangle=\frac{1}{Z} \sum_{\sigma} e^{-\beta H(\sigma)}$$
thus, without the knowledge of the partition function $Z$, we cannot compute the average value of our observable $O(\sigma)$. A further example, previously mentioned, is given by the Ising model, where numerical simulations become mandatory for studying its behavior for $D \geq 3$. So, in order to overcome this limit, we can adopt Monte Carlo (MC hereinafter) methods for computing the value of the quantities we are interested in. The underlying idea of MC methods is to generate subset configurations (from the whole phase space), with a weight given by the Boltzmann statistics, that are representative for the entire ensemble. So, generating, for example, $M$ configurations, we can have an estimate of the observable computing its average value, i.e.,
$$\langle O(\sigma)\rangle=\frac{1}{M} \sum_{i=1}^{M} O_{i}(\sigma)$$
Therefore, we are able to compute the average value of a physical quantity avoiding to deal with the partition function $Z$ of the system (as in Eq. (2.17)). Now, before to proceed, it is worth to remind that the analytical solutions of a system (e.g., the Ising model) usually are computed considering the thermodynamic limit (i.e., $N \rightarrow \infty$ ). So, from a computational point of view, the first problem is how to approximate such limit/condition. In the case of the Ising model, a viable solution is given by the implementation of lattices which size is sufficiently big, removing the finite

size effect using a simple trick, i.e., generating lattices with continuous boundary conditions. Actually, under this condition, a bidimensional lattice takes the form of a toroid. In particular, this transformation can be easily performed by connecting the sites at the edges of the lattice. For instance, Fig. $2.3$ shows an example focusing on the site named $x$, that is, connected to the sites $y$ and $z$, increasing its amount of bonds up to four. Going back to the problem of performing a numerical simulation of the Ising model, we can implement different algorithms. Here, we refer to one of the most famous, i.e., the Metropolis algorithm. We remind that our aim is measuring the value of parameters like the magnetization at equilibrium. As we know from theory, at low temperatures we expect a ferromagnetic phase, i.e., a system close to the order, while at high temperatures a paramagnetic phase, i.e., a disordered system. The Metropolis algorithm is very simple, and its steps are:

1. Randomly select a site $i$ and compute the local $\Delta E$ associated to its spin flip
2. IF $(\Delta E \leq 0)$ : accept the flip;
ELSE: accept the flip with probability $e^{\frac{-\Delta E}{k_{b} T}}$
repeated until an equilibrium state is reached. The $\Delta E$ indicates a local difference in energy, i.e., computed considering only the selected site and its nearest neighbors.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Curie-Weiss Model

Ising 模型的一个重要方面是它的维度D. 值得注意的是，对于D=1，Ising 模型在有限温度下没有相变。为了D=2，根据 Onsager 的解，存在相变（在有限温度下）。然后，在更高维度上，虽然可以观察到相变，但解析解的定义仍然是一个悬而未决的问题。特别是，对于D=3, 该问题仅通过数值方法解决,

H(σ1,…σn)=−1ñ∑(一世<j)σ一世σj−H∑一世=1ñσ一世

H(σ1,…σn)=−ñ2米2+○(1)

Z=\sum_{\left{\sigma_{i}\right}} e^{-\beta H\left(\sigma_{1}, \ldots, \sigma_{N}\right)}Z=\sum_{\left{\sigma_{i}\right}} e^{-\beta H\left(\sigma_{1}, \ldots, \sigma_{N}\right)}

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Landau Theory of Phase Transitions

−ķb吨ln⁡从=在−吨小号

F(吨;米)=F(吨;0)+12一个(吨)米2+14!b(吨)米4+…

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Numerical Simulations

⟨○(σ)⟩=1从∑σ和−bH(σ)

⟨○(σ)⟩=1米∑一世=1米○一世(σ)

1. 随机选择一个站点一世并计算本地Δ和与其旋转翻转相关联
2. 如果(Δ和≤0)：接受翻转；
ELSE：接受概率翻转和−Δ和ķb吨
重复直到达到平衡状态。这Δ和表示能量的局部差异，即仅考虑所选站点及其最近的邻居来计算。

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## MATLAB代写

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