62 lines
2.9 KiB
Markdown
62 lines
2.9 KiB
Markdown
# Topic 5: Introducing dynamics: classical and non-commutative
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## Section 1: Dynamics in classical probability
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### Basic definitions
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#### Definition of orbit
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Let $T:\Omega\to\Omega$ be a map (may not be invertible) generating a dynamical system on $\Omega$. Given $\omega\in \Omega$, the (forward) orbit of $\omega$ is the set $\mathscr{O}(\omega)=\{T^n(\omega)\}_{n\in\mathbb{Z}}$.
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The theory of dynamics is the study of properties of orbits.
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#### Definition of measure-preserving map
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Let $P$ be a probability measure on a $\sigma$-algebra $\mathscr{F}$ of subsets of $\Omega$. (that is, $P:\mathscr{F}\to$ anything) A measurable transformation $T:\Omega\to\Omega$ is said to be measure-preserving if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is:
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$$
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\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
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$$
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Example:
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The doubling map $T:\Omega\to\Omega$ is defined as $T(x)=2x\mod 1$, is a Lebesgue measure preserving map on $\Omega=[0,1]$.
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#### Definition of isometry
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The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure preserving map defined on $\mathscr{H}=L^2(\Omega,\mathscr{F},P)$ is isometry of $\mathscr{H}$ if $\langle U\psi,U\phi\rangle=\langle\psi,\phi\rangle$ for all $\psi,\phi\in\mathscr{H}$.
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#### Definition of unitary
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The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure preserving map defined on $\mathscr{H}=L^2(\Omega,\mathscr{F},P)$ is unitary of $\mathscr{H}$ if $U$ is an isometry and $T$ is invertible with measurable inverse.
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## Section 2: Continuous time (classical) dynamical systems
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### Spring-mass system
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The pure state of the system is given by the position and velocity of the mass. $(x,v)$ is a point in $\mathbb{R}^2$. $\mathbb{R}^2$ is the state space of the system. (or phase space)
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The motion of the system in its state space is a closed curve.
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$$
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\Phi_t(x,v)=\left(\cos(\omega t)x-\frac{1}{\omega}\sin(\omega t)v, \cos(\omega t)v-\omega\sin(\omega t)x\right)
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$$
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Such system with closed curve is called **integrable system**. Where the doubling map produces orbits having distinct dynamical properties (**chaotic system**).
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> Note, some section is intentionally ignored here. They are about in the setting of operators on Hilbert spaces, the evolution of (classical, non-dissipative e.g. linear spring-mass system) system, is implemented by a one-parameter group of unitary operators.
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>
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> The detailed construction is omitted here.
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#### Definition of Hermitian operator
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A linear operator $A$ on a Hilbert space $\mathscr{H}$ is said to be Hermitian if $\forall \psi,\phi\in$ **domain of $A$**, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$.
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It is skew-Hermitian if $\langle A\psi,\phi\rangle=-\langle\psi,A\phi\rangle$.
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