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# 物理代写|统计力学代写Statistical Mechanics代考|PH635 Dynamical Entropies

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## 物理代写|统计力学代写Statistical Mechanics代考|Dynamical Entropies

Since the notion of entropy is central in this book, and since Kolmogorov and Sinai have introduced a notion of entropy for dynamical systems, named after them, we shall briefly discuss this notion; for more information about the Kolmogorov-Sinai entropy, see Billingsley [28], Sinai [296], Cornfeld, Fomin and Sinai [87], or Walters [327]

Let $(\Omega, \Sigma, \mu, T)$ be a dynamical system with $\mu(\Omega)=1$, and let $\tilde{\Omega}=\left(\Omega_{1}, \ldots, \Omega_{k}\right)$ be a finite partition of $\Omega$. The entropy of that partition is, by definition,
$$S(\tilde{\Omega}, \mu)=-\sum_{i=1}^{k} \mu\left(\Omega_{i}\right) \ln \mu\left(\Omega_{i}\right)$$

If one interprets entropy as measuring an amount of information (see Chap. 7), then (4.7.1) is the amount of information that one obtains by knowing to which element of the partition $x \in \Omega$ belongs.

Given two partitions $\tilde{\Omega}=\left(\Omega_{1}, \ldots, \Omega_{k}\right)$ and $\tilde{\Omega}^{\prime}=\left(\Omega_{1}^{\prime}, \ldots, \Omega_{l}^{\prime}\right)$, one defines their common refinement $\tilde{\Omega} \vee \tilde{\Omega}^{\prime}=\left(\Omega_{i} \cap \Omega_{j}^{\prime}\right){i=1, \ldots, k, j=1, \ldots, l}$. The entropy of the partition $\tilde{\Omega} \vee T^{-1} \tilde{\Omega} \vee \cdots \vee T^{-n+1} \tilde{\Omega}$ is: \begin{aligned} &S(\tilde{\Omega}, \mu, T, n) \equiv S\left(\tilde{\Omega} \vee T^{-1} \tilde{\Omega} \vee \cdots \vee T^{-n+1} \tilde{\Omega}, \mu\right) \ &=-\sum{i_{0}, \ldots, i_{n-1} \in{1, \ldots, k}} \mu\left(\Omega_{i_{0}} \cap T^{-1} \Omega_{i_{1}} \cap \cdots \cap T^{-n+1} \Omega_{i_{n-1}}\right) \ &\ln \mu\left(\Omega_{i_{0}} \cap T^{-1} \Omega_{i_{1}} \cap \cdots \cap T^{-n+1} \Omega_{i_{n-1}}\right) \end{aligned}
This represents the amount of information obtained by knowing to which element of the partition all the points $x, T x \ldots T^{n-1} x$ belong.

## 物理代写|统计力学代写Statistical Mechanics代考|Determinism and Predictability

We want to finish this chapter by a short discussion of a frequent confusion that occurs in the popular, but also in the scientific literature, between determinism and predictability, confusion which is often caused by a lack of precise definitions. In Sect. $2.1$ we mentioned Laplace’s very clear expression of the idea of universal determinism. We also remarked that Laplace clearly distinguished between what nature does and the knowledge we have of it or between determinism and our ability to predict the future.

However, determinism is often confused with predictability. So, according to that view, a process is deterministic if we, humans, can predict it, or, maybe, if we, humans, will be able to predict it in the future. For example, in an often quoted lecture ${ }^{20}$ to the Royal Society, on the three hundredth anniversary of Newton’s Principia, the distinguished British mathematician Sir James Lighthill gave a perfect example of how to confuse predictability and determinism:
We are all deeply conscious today that the enthusiasm of our forebears for the marvelous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960 , but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about determinism of systems satisfying Newton’s laws of motion that, after 1960, were to be proved incorrect $[\ldots]$.

James Lighthill, [231], (Italics added by J.B.)

## 物理代写|统计力学代写Statistical Mechanics代考|Dynamical Entropies

$$S(\bar{\Omega}, \mu)=-\sum_{i=1}^{k} \mu\left(\Omega_{i}\right) \ln \mu\left(\Omega_{i}\right)$$

$$S(\bar{\Omega}, \mu, T, n) \equiv S\left(\bar{\Omega} \vee T^{-1} \bar{\Omega} \vee \cdots \vee T^{-n+1} \bar{\Omega}, \mu\right) \quad=-\sum i_{0}, \ldots, i_{n-1} \in 1, \ldots, k \mu\left(\Omega_{i_{0}} \cap T^{-1} \Omega_{i_{1}} \cap \cdots \cap T^{-n+1} \Omega_{i_{n-1}}\right) \ln \mu\left(\Omega_{i 0} \cap T^{-1} \Omega_{i_{1}} \cap \cdots 1\right.$$

## 物理代写|统计力学代写Statistical Mechanics代考|Determinism and Predictability

1960 年之前可能通常倾向于相信，但我们现在认识到这是错䢔的。我们共同为通过传摇关于满足牛顿运动定律的系统确定性的思

James Lighthill，[231]，（雓体由 JB 添加)

## MATLAB代写

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