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# 物理代写|统计力学代写STATISTICAL MECHANICS代写|PHYSICS7546 Dynamical Systems

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## 物理代写|统计力学代写STATISTICAL MECHANICS代写|Introduction

Before going into all that, we refer the reader to the definitions of Appendix 2 .A.5 about maps preserving measures and list a few examples of measure spaces $(\Omega, \Sigma, \mu)$ and maps $T: \Omega \rightarrow \Omega$ that preserve the measure $\mu$. Here, $\Sigma$ will always be the Borel sets in $\mathbb{R}^{n}$ or the cylinder sets defined in (2.A.5).

1. Rotations of the circle or of the torus. Let $\Omega=[0,1$ [with addition $\bmod 1$, which can be identified with the unit circle, $d \mu=d x$ the Lebesgue measure, and let $T=T_{\alpha}$, for $\alpha \in[0,1)$ :
$$T_{\alpha} x=x+\alpha \quad \bmod 1$$
If one considers $\Omega=\left[0,1\left[^{n}\right.\right.$, which can be identified with the $n$-dimensional torus, with $d \mu$ the Lebesgue measure, one can generalize the map (4.1.1), for $\boldsymbol{\alpha} \in[0,1)^{n}$ to:
$$T_{\alpha} \mathbf{x}=\mathbf{x}+\boldsymbol{\alpha} \bmod 1$$
for $\mathbf{x} \in\left[0,1\left[{ }^{n}\right.\right.$ and the addition is modulo 1 for each component.
In both cases, the invariance of the Lebesgue measure is obvious since the transformation is just a translation (modulo 1).
2. Let $\Omega=[0,1[, d \mu=d x$ and let (Fig. 4.1)
$$T x=2 x \quad \bmod 1 .$$

## 物理代写|统计力学代写STATISTICAL MECHANICS代写|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.

## 物理代写|统计力学代昃STATISTICAL MECHANICS代写|Introduction

$$T_{\alpha} x=x+\alpha \quad \bmod 1$$

$$T_{\alpha} \mathbf{x}=\mathbf{x}+\boldsymbol{\alpha} \bmod 1$$

$$T x=2 x \quad \bmod 1 .$$

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