### 物理代写|统计力学代写Statistical mechanics代考|PHYC90010

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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|统计力学代写Statistical mechanics代考|Poincaré’s Recurrence Theorem, or The Eternal Return

In words, Poincaré’s recurrence theorem says that, for any measure preserving transformation, almost every trajectory comes back arbitrarily close to its initial condition and does that infinitely often.
That property is called recurrence.
Since we know that the Hamiltonian flow on a constant energy surface preserves the Liouville measure on that surface, it follows that, for any mechanical system bounded (in phase space) almost all configurations will come back infinitely often to a configuration arbitrarily close to itself, or to any other configuration that it visits (since one could always define that configuration as an initial condition).

Thus, bounded mechanical systems do not necessarily have only periodic trajectories but almost all their trajectories have a property somewhat similar but weaker than periodicity, namely recurrence.

If one replaced the measure preserving transformation by a deterministic transformation on a finite set, then, obviously, all trajectories must eventually be periodic

since some element of the finite set must be visited twice (over an infinite time) and from then on, the trajectory becomes periodic.

The genius of Poincaré was to extend this result to a weaker notion (recurrence) for measure preserving transformations on infinite sets but bounded in the sense that the measure of the space on which the transformation acts is finite:

Theorem $4.2$ (Poincaré’s recurrence theorem [261]) Let $(\Omega, \Sigma, \mu)$ be a measure space with $\mu(\Omega)<\infty$, where $\mu$ is $T$-imvariant for $T: \Omega \rightarrow \Omega$. Then, $\forall A \in \Sigma$, with $\mu(A)>0, \exists B \subset A$ with $\mu(A \backslash B)=0$ and such that $\forall x \in B, \exists$ sequence $\left(n_{i}\right){i=1}^{\infty}$, $n{i} \in \mathbb{N}$, with $n_{1}<n_{2}<n_{3} \ldots$ and $T^{n_{i}} x \in A$.

## 物理代写|统计力学代写Statistical mechanics代考|Proof of Poincaré’s Recurrence Theorem

Let, for $N \geq 0 \quad A_{N}=\bigcup_{n=N} T^{-n} A$. Elements of $A_{N}$ are those that are sent into $A$ by the map $T^{n}$ for some $n \geq N$. Thus, $B=A \cap\left(\bigcap_{N=0}^{\infty} A_{N}\right)$ are the elements of $A$ that come back to $A$ infinitely often, namely that are such that $\forall x \in B$, there exist a

sequence $\left(n_{i}\right)$ as in the theorem: since, for each $x \in B$, there exist arbitrarily large $n^{\prime} s$ with $T^{n} x \in A$, one may construct the sequence inductively by taking $n_{i+1}$ to be the first integer $n$ with $T^{n} x \in A$ strictly larger than $n_{i}$.

Let us show that $\mu(B)=\mu(A)$, which, since $\mu(\Omega)<\infty$, is equivalent to $\mu(A \backslash B)$ $=0\left(\right.$ write $\mu(A)=\mu(B)+\mu(A \backslash B)$ ). One has $T^{-1} A_{N}=A_{N+1}$ and thus $\mu\left(A_{N}\right)=$ $\mu\left(A_{N+1}\right)$, since $\mu$ is $T$-invariant. So, $\mu\left(A_{0}\right)=\mu\left(A_{N}\right), \forall N$. Since $A_{0} \supset A_{1} \supset$ $\ldots \supset A_{N}$, and since $\mu(\Omega)<\infty$, one has $\mu\left(A_{N}\right) \leq \mu\left(A_{0}\right) \leq \mu(\Omega)<\infty$ and thus $\mu\left(A_{0} \backslash A_{N}\right)=\mu\left(A_{0}\right)-\mu\left(A_{N}\right)=0, \forall N$. Since a countable union of sets of measure zero is of measure zero $\mu\left(A_{0} \backslash\left(\bigcap_{N=0}^{\infty} A_{N}\right)\right)=\mu\left(\bigcup_{N=0}^{\infty}\left(A_{0} \backslash A_{N}\right)\right)=0$. Thus, since $\bigcap_{N=0}^{\infty} A_{N} \subset A_{0}, \mu(B)=\mu\left(A \cap\left(\bigcap_{N=0}^{\infty} A_{N}\right)\right)=\mu\left(A \cap A_{0}\right)$ and $\mu\left(A \cap A_{0}\right)=$ $\mu(A)$, since $A \subset A 0$. So, $\mu(B)=\mu(A)$.

## 物理代写|统计力学代写Statistical mechanics代考|Ergodic Theorems

Although we have not yet defined what unpredictable dynamical systems are, we said that, when a dynamical system is unpredictable, one should study its trajectories by statistical methods. A first step in that direction is to study certain time averages of trajectories, for example the average time spent by the trajectory in a set $A \in \Sigma$. We will study a slightly more general object.

Given a measure space $(\Omega, \Sigma, \mu)$, a map $T: \Omega \rightarrow \Omega$ such that $\mu$ is $T$-invariant, and a $\mu$-integrable function $F: \Omega \rightarrow \mathbb{R}$, one may consider the temporal averages:
$$\frac{1}{N} \sum_{n=0}^{N-1} F\left(T^{n} x\right)$$
and ask whether the limits $N \rightarrow \infty$ of those quantities exist.
If one takes $F=\mathbb{1}{A}$, the indicator function of a set $A \in \Sigma$, formula (4.3.1) gives the average time, up to time $N$, spent by the system in $A$, and the limit $N \rightarrow \infty$, if it exists, gives the average time $\tau{A}$ spent in $A$.

The following theorem, that we will not prove because the proof is a bit long and not particularly illuminating (see Walters [327, Sect. 1.6] or Cornfeld, Fomin and Sinai [87, Appendix 3]) asserts that all these limits actually exist ${ }^{4}$ :

## 物理代写|统计力学代写Statistical mechanics代考|Poincaré’s Recurrence Theorem, or The Eternal Return

Poincaré 的天才是将这个结果扩展到一个较弱的概念（递归），用于在无限集上保留测度变换，但在变换作用于的空间的测度是有限的意义上是有界的：

1ñ∑n=0ñ−1F(吨nX)

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## MATLAB代写

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