Logical arrow of time, 10

Posted on October 3, 2021 by rpflee

Two distinguishable macrostates can both evolve into one indistinguishable macrostate.

— Me@2013-08-11 11:08 AM

Ultimate Freedom, 2

Posted on September 26, 2021 by rpflee

Imagination is not an empirical or superadded power of consciousness, it is the whole of consciousness as it realizes its freedom.

— Jean-Paul Sartre

Your mental world/software world lies your ultimate freedom.

It is where anything not forbidden can happen.

It is where anything logically consistent can happen.

— Me@2016-11-20 09:14:06 AM

— Me@2021-09-26 09:22:54 PM

改變因果鏈起點

Posted on September 23, 2021 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

所以，我們先企圖進攻那些大題目，無論它們長遠是否，對你直接有用；因為，過程之中，自然會引發很多技術細節。而那些技術細節，很多是你直接可用的。

原初辛苦上山的目標是，傳說中，山頂上的寶藏。但是，上到山頂後，才發現最珍貴的寶藏，反而是沿途找到的那些。

（安：我覺得你對我最大的價值，反而是：

因為你鼓勵我，先企圖進攻的那些大題目中，很多也是你自己，曾經進攻過的，所以，你從它們之中篩選出來，提議我去研究的，必定是你覺得對我來說，最有價值的那些。

如果不是那樣的話，我可能要麼就是，開始研究某個大題目時，就覺得徬徨，而無心機繼續；要麼就是，真的花了一年半載，研究某個大題目後，發覺原來事倍功半，或者價值不大。）

那其實正正就是，「教育」的精髓，「學習」的意思。

為何不事事親身去經歷呢？　

因為人生苦短。

透過吸收他人的經歷，可以改變因果鏈的起點。

人在某個意思之下，其實已經有了永生。個別的人會死亡，但人類整體而然，可存在很久。只要不絶種，就是得永生。對於任何物種，都可以這樣說。

但是，只有人類這個物種，有大規模的知識傳授。而大規模的知識傳授，則自然需要大量時間。

幸而，雖然人生苦短，但仍剛剛足夠長，去行一個「學習、發掘和傳授」循環。在進化接力賽中，人類每一代剛好有足夠時間，去完成交收接力棒的工序。在知識上，每一代人也可以，以上一代人的終點，作為自己的起點。

— Me@2021-09-22 10:24:46 PM

1990s, 5.2

Posted on September 19, 2021 by rpflee

— Photomyne colorization

— Me@2021-09-19 06:13:39 PM

Ex 1.21 The dumbbell, 3.3

Posted on September 18, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

c. Make a change of coordinates to a coordinate system with center of mass coordinates $\displaystyle{x_{cm}}$ , $\displaystyle{y_{cm}}$ , angle $\displaystyle{\theta}$ , distance between the particles $\displaystyle{c}$ , and tension force $\displaystyle{F}$ . Write the Lagrangian in these coordinates, and write the Lagrange equations.

~~~

[guess]


(define (KE-particle m v)
  (* 1/2 m (square v)))

(define ((L-free-constrained m0 m1 l) local)
  (let* ((extract (extract-particle 2))
     (p0 (extract local 0))
     (q0 (coordinate p0))
     (qdot0 (velocity p0))
  
     (p1 (extract local 1))
     (q1 (coordinate p1))
     (qdot1 (velocity p1))
  
     (F (ref (coordinate local) 4)))
 
    (- (+ (KE-particle m0 qdot0)
          (KE-particle m1 qdot1))
       (U-constraint q0 q1 F l))))

(define ((extract-particle pieces) local i)
  (let* ((indices (apply up (iota pieces (* i pieces))))
         (extract (lambda (tuple)
                    (vector-map (lambda (i)
                                  (ref tuple i))
                                indices))))
    (up (time local)
        (extract (coordinate local))
        (extract (velocity local)))))

(define (U-constraint q0 q1 F l)
  (* (/ F (* 2 l))
     (- (square (- q1 q0))
        (square l))))

(let ((L (L-free-constrained 'm_0 'm_1 'l))
      (q-rect (up (literal-function 'x_0)
                  (literal-function 'y_0)
                  (literal-function 'x_1)
                  (literal-function 'y_1)
                  (literal-function 'F))))
  (show-expression
   ((compose L (Gamma q-rect)) 't)))

$\displaystyle{ \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) + \frac{F}{2 l} \left( l^2 - y_1^2 + 2 y_0 y_1 - x_1^2 + 2 x_0 x_1 - y_0^2 - x_0^2 \right) }$

$\displaystyle{ = \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) - \frac{F}{2 l} \left[ (y_1 - y_0)^2 + (x_1 - x_0)^2 - l^2 \right] }$


(define ((q->r m0 m1) local)
  (let ((q (coordinate local)))
    (let ((x_cm (ref q 0))
          (y_cm (ref q 1))
          (theta (ref q 2)) 
          (c (ref q 3))
	      (F (ref q 4))
	      (M (+ m0 m1)))
      (let ((x0 (- x_cm (* (/ m1 M) c (cos theta))))
            (y0 (- y_cm (* (/ m1 M) c (sin theta))))
            (x1 (+ x_cm (* (/ m0 M) c (cos theta))))
            (y1 (+ y_cm (* (/ m0 M) c (sin theta)))))
        (up x0 y0 x1 y1 F)))))

(let ((q (up (literal-function 'x_cm)
	         (literal-function 'y_cm)
	         (literal-function 'theta)
	         (literal-function 'c)
	         (literal-function 'F))))
  (show-expression (q 't)))

 
(show-expression
 (up 't
     (up 'x_cm 'y_cm 'theta 'c 'F)
     (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot)))


(show-expression
  ((q->r 'm_0 'm_1) 
     (up 't
         (up 'x_cm 'y_cm 'theta 'c 'F)
         (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))


(let ((q (up (literal-function 'x_cm)
             (literal-function 'y_cm)
             (literal-function 'theta)
             (literal-function 'c)
             (literal-function 'F))))
  (show-expression ((q->r 'm_0 'm_1) ((Gamma q) 't))))


(show-expression
  ((F->C (q->r 'm_0 'm_1)) 
     (up 't
         (up 'x_cm 'y_cm 'theta 'c 'F)
         (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))

(define (L-cm m0 m1 l)
  (compose
   (L-free-constrained m0 m1 l) (F->C (q->r m0 m1))))

(show-expression
 ((L-cm 'm_0 'm_1 'l)
     (up 't
         (up 'x_cm 'y_cm 'theta 'c 'F)
         (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))

$\displaystyle{ \frac{1}{\mu} = \frac{1}{m_0} + \frac{1}{m_1} }$

$\displaystyle{ L_{cm} }$

$\displaystyle{ = \frac{ ( c^2 \dot \theta^2 + \dot c^2 ) l m_0 m_1 + (l m_0^2 + 2 l m_0 m_1 + l m_1^2) (\dot x_{cm}^2 + \dot y_{cm}^2) + F ( l^2 - c^2 )(m_0 + m_1) }{2 l (m_0 + m_1)} }$

$\displaystyle{ = \frac{ ( c^2 \dot \theta^2 + \dot c^2 ) l m_0 m_1 + l (m_0 + m_1)^2 (\dot x_{cm}^2 + \dot y_{cm}^2) + F ( l^2 - c^2 )(m_0 + m_1) }{2 l (m_0 + m_1)} }$

$\displaystyle{ = \frac{1}{2} ( c^2 \dot \theta^2 + \dot c^2 ) \mu + \frac{1}{2} (m_0 + m_1) (\dot x_{cm}^2 + \dot y_{cm}^2) + \frac{1}{2l} F ( l^2 - c^2 ) }$


(show-expression
 (((Lagrange-equations
    (L-cm 'm_0 'm_1 'l))
   (up (literal-function 'x_cm)
       (literal-function 'y_cm)
       (literal-function 'theta)
       (literal-function 'c)
       (literal-function 'F)))
  't))


(show-expression
 (((Lagrange-equations
    (L-cm 'm_0 'm_1 'l))
   (up (literal-function 'x_cm)
       (literal-function 'y_cm)
       (literal-function 'theta)
       (lambda (t) 'l)
       (literal-function 'F)))
  't))

[guess]

— Me@2021-09-17 06:35:51 AM

Meta numbers

Posted on September 13, 2021 by rpflee

Numbers are meta objects.

Infinities are meta numbers.

— Me@2017-02-03 05:20:51 PM

Year 2000 Reloaded

Posted on September 12, 2021 by rpflee

These are what I kept reminding myself during my Age of University.

1. I think, therefore I am

A lot of times, something seems to be a deadend.

However, once I have faith that a solution may be possible and start to think, most of the time, that “deadline” is actually resolvable.

2. Nothing Less

A lot of times, an action seems to be insignificant.

However, actually, every action may have infinite consequences, especially when you cultivate it.

3. The Aladdin Factor

A key to get a meaningful life is to dare to ask other people or yourself for answers, advice, or favours, when necessary.

Don’t be afraid to ask, as long as a request is ethical, reasonable, and polite.

— Me@2011.08.17

— Me@2021-09-11

相聚零刻 2.1

Posted on September 10, 2021 by rpflee

尋覓 2.2.3.6.1

男士 20 歲時，還找不到女朋友，可能是好事，因為，男士其實要到 28 歲時，思想才帶點成熟，通常。

那是某一個哲學家講的。我當年在 25 歲時，就已經聽到這一點。到 28 歲時，真的明白了一些，以前不明白的東西。（當然，任何年齡也可以這樣說，如果一個人不斷進步的話。）

例如，說話時，為什麼會「食螺絲」（舌頭打結）呢？

是因為，企圖在同一刻講，超過一個字。只要稍為放慢說話速度，就可以化解。

又例如，生活中，為什麼會諸事不順呢？

是因為，企圖在同一格時間中，放多於一件事。

然後，那個哲學家又說… … 千萬不要說出去，以免得罪人。

然後，那個哲學家又說，至於女士，則於 18 歲時，就已經成熟；自此以後，就不再成熟了。

聽不聽得出，他在投訴什麼呢？

他的意思是，女士在思想上，於 18 歲時，就已經成熟；自此以後，就不再成熟。（當然，那只是大概而然；每個人也不同。）

女士，就是介乎於，男士與小孩之間的物體。

那個哲學家，覺得那是缺點。但我覺得不一定；那也可以是優點。

例如，善良之女士，可以永久可愛。男士則不可以。男士青春期時，聲線會變。變聲後，不可能再可愛。

女士要可愛。男士要可靠。

無論在外表或性格上，女士的職責是，保持可愛，尋找可靠；男士的職責則是，保持可靠，保護可愛。

— Me@2021-07-26 05:12:55 PM

— Me@2021-09-07 05:15:56 PM

Pier, 2

Posted on September 1, 2021 by rpflee

— Me@2021-09-01 04:41:10 PM

2.10 A spacetime orbifold in two dimensions, 3

Posted on August 31, 2021 by rpflee

A First Course in String Theory

(b) Draw a spacetime diagram, indicate the $\displaystyle{x^+}$ and $\displaystyle{x^-}$ axes, and sketch the family of curves

$\displaystyle{x^+ x^- = a^2}$ ,

where $\displaystyle{a > 0}$ is a real constant that labels the various curves.

…

~~~

— Me@2021-08-31 08:42:40 PM

Visualizing higher dimensions, 2

Posted on August 23, 2021 by rpflee

Geometry is global.

Space is what we can see at once.

Dynamics is local.

Time is what we cannot see at once.

— Me@2017-02-07 10:11:34 PM

If we could see, for example, several minutes at once, that several minutes would become a spatial dimension.

In other words, that dimension is visualized for us.

— Me@2017-02-03 07:31:25 AM

Broken

Posted on August 22, 2021 by rpflee

A: Before meeting you, my life is chaotic.

M: It is because you missed, me. With me, you are completed.

— Me@2021-06-07 06:16:29 PM

— Me@2021-08-22 05:40:53 PM

上山尋寶, 2

Posted on August 16, 2021 by rpflee

重點副作用 6.2 | The non-side-effect-ness of side-effects, 6.2

這段改編自 2010 年 4 月 24 日的對話。

其實，我們可以考慮改變方案。雖然有些課題，可能幾個小時，就可以完成，但是，我們可以每個星期，也講不同的課題。

留意，雖然個別課題的成果，只會花幾小時，但是那是指，事先已有課題的情況下。那些有趣而深刻的課題，簡稱「勁題目」，並不會從天而降。那些勁題目本身，大部分情況下，只會在你開始研究，將會花「幾年加幾年」的苦功知識時，才會引發得到。

所以，我們先企圖進攻那些大題目，無論它們長遠是否，對你直接有用；因為，過程之中，自然會引發很多技術細節。而那些技術細節，很多是你直接可用。

例如，剛才因為跟你研究，大學力學課本《Structure and Interpretation of Classical Mechanics》（SICM），而提及到一個程式語言 Scheme programming language。而又因為研究該程式語言，而提及了一個，特別的文字編輯程式 Notepad++。雖然，大學力學和程式語言 Scheme 本身，對你而言，只是工餘興趣，但是，Notepad++ 卻是日常生活也有用。

原初辛苦上山的目的是，傳說中，山頂上的寶藏。但是，上到山頂後，才發現最珍貴的寶藏，反而是沿途找到的那些。

— Me@2021-08-16 04:31:19 PM

1986, 2

Posted on August 13, 2021 by rpflee

— Me@2021-08-13 04:52:08 PM

Ex 1.21 The dumbbell, 3.2

Posted on August 10, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

~~~

[guess]

…

$\displaystyle{ \begin{aligned} m_0 \ddot y_0 &= F \sin \theta \\ m_0 \ddot x_0 &= F \cos \theta \\ m_1 \ddot y_1 &= -F \sin \theta \\ m_1 \ddot x_1 &= -F \cos \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_{cm} &= \frac{m_0 y_0 + m_1 y_1}{m_0 + m_1} \\ x_{cm} &= \frac{m_0 x_0 + m_1 x_1}{m_0 + m_1} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \ddot y_{cm} &= \frac{F \sin \theta - F \sin \theta}{m_0 + m_1} = 0 \\ \ddot x_{cm} &= \frac{F \cos \theta - F \cos \theta}{m_0 + m_1} = 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_0 &= y_{cm} - \frac{m_1}{M} c(t) \sin \theta \\ x_0 &= x_{cm} - \frac{m_1}{M} c(t) \cos \theta \\ y_1 &= y_{cm} + \frac{m_0}{M} c(t) \sin \theta \\ x_1 &= x_{cm} + \frac{m_0}{M} c(t) \cos \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} x_1 - x_0 &= \frac{m_0}{M} c(t) \cos \theta + \frac{m_1}{M} c(t) \cos \theta \\ &= c(t) \cos \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \dot x_1 - \dot x_0 &= \dot c(t) \cos \theta - c(t) \dot \theta \sin \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \ddot x_1 - \ddot x_0 &= \ddot c(t) \cos \theta - \dot c(t) \dot \theta \sin \theta - \dot c(t) \dot \theta \sin \theta - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}}$

$\displaystyle{ \begin{aligned} y_1 - y_0 &= c(t) \sin \theta \\ \dot y_1 - \dot y_0 &= \dot c(t) \sin \theta + c(t) \dot \theta \cos \theta \\ \ddot y_1 - \ddot y_0 &= \ddot c(t) \sin \theta + \dot c(t) \dot \theta \cos \theta + \dot c(t) \dot \theta \cos \theta + c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} m_0 \ddot y_0 &= F \sin \theta \\ m_0 \ddot x_0 &= F \cos \theta \\ m_1 \ddot y_1 &= -F \sin \theta \\ m_1 \ddot x_1 &= -F \cos \theta \\ \end{aligned}}$

When $\displaystyle{\dot c(t) = 0}$ and $\displaystyle{\ddot c(t) = 0}$ ,

$\displaystyle{ \begin{aligned} \ddot x_1 - \ddot x_0 &= - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta \\ - \left( \frac{1}{m_1} + \frac{1}{m_1} \right) F \cos \theta &= - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \ddot y_1 - \ddot y_0 &= ... \\ - \left( \frac{1}{m_1} + \frac{1}{m_0} \right) F \sin \theta &= c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \tan \theta &= \frac{ c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta }{- c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta} \\ &= \frac{ \ddot \theta - \dot \theta^2 \tan \theta }{- \ddot \theta \tan \theta - \dot \theta^2} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \tan \theta \left( - \ddot \theta \tan \theta - \dot \theta^2 \right) &= \ddot \theta - \dot \theta^2 \tan \theta \\ &... \\ 0 &= \ddot \theta (1 + \tan^2 \theta) \\ \ddot \theta &= 0 \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} - \left( \frac{1}{m_1} + \frac{1}{m_0} \right) F \sin \theta &= c(t) \ddot \theta \cos \theta - c(t) \dot \theta^2 \sin \theta \\ - \left( \frac{1}{m_1} + \frac{1}{m_1} \right) F \cos \theta &= - c(t) \ddot \theta \sin \theta - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}}$

Let $\displaystyle{\frac{1}{\mu} = \left( \frac{1}{m_1} + \frac{1}{m_1} \right)}$ and since $\displaystyle{\ddot \theta = 0}$ ,

$\displaystyle{ \begin{aligned} - \frac{1}{\mu} F \sin \theta &= - c(t) \dot \theta^2 \sin \theta \\ - \frac{1}{\mu} F \cos \theta &= - c(t) \dot \theta^2 \cos \theta\\ \end{aligned}}$

Since $\displaystyle{\sin \theta}$ and $\displaystyle{\cos \theta}$ cannot be both zero at the same time,

$\displaystyle{ \begin{aligned} - \frac{1}{\mu} F &= - c(t) \dot \theta^2 \\ \end{aligned}}$

Put $\displaystyle{c(t) = l}$ ,

$\displaystyle{ \begin{aligned} \frac{1}{\mu} F &= l \dot \theta^2 \\ \dot \theta^2 &= \frac{1}{l \mu} F \\ \end{aligned}}$

[guess]

— Me@2021-08-08 05:41:21 PM

Logical arrow of time, 9.4

Posted on August 7, 2021 by rpflee

The second law of thermodynamics’ derivation (Ludwig Boltzmann’s H-theorem) is with respect to an observer.

How does an observer keep losing microscopic information about a system?

— Me@2017-02-12 07:37:54 PM

This drew the objection from Loschmidt that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism: something must be wrong (Loschmidt’s paradox).

The resolution (1895) of this paradox is that the velocities of two particles after a collision are no longer truly uncorrelated. By asserting that it was acceptable to ignore these correlations in the population at times after the initial time, Boltzmann had introduced an element of time asymmetry through the formalism of his calculation.

— Wikipedia on Molecular chaos

Physical entropy’s value is with respect to an observer.

— Me@2017-02-12 07:37:54 PM

This “paradox” can be explained by carefully considering the definition of entropy. In particular, as concisely explained by Edwin Thompson Jaynes, definitions of entropy are arbitrary.

As a central example in Jaynes’ paper points out, one can develop a theory that treats two gases as similar even if those gases may in reality be distinguished through sufficiently detailed measurement. As long as we do not perform these detailed measurements, the theory will have no internal inconsistencies. (In other words, it does not matter that we call gases A and B by the same name if we have not yet discovered that they are distinct.) If our theory calls gases A and B the same, then entropy does not change when we mix them. If our theory calls gases A and B different, then entropy does increase when they are mixed. This insight suggests that the ideas of “thermodynamic state” and of “entropy” are somewhat subjective.

— Wikipedia on The mixing paradox

— Wikipedia on Gibbs paradox

Graduate school

Posted on August 6, 2021 by rpflee

Turn my life into an ideal graduate school. Be my own professor.

— Me@2011.05.26

Doing real research, doing real teaching.

— Me@2021-08-06 05:03:16 PM

To realize is to realize, 1.3

Posted on July 27, 2021 by rpflee

搜神記 2

The ultimate self-fulfilling prophecies:

1. free will or not

2. god or no god

3. afterlife or not

4. future spouse exists or not

Why self-fulfilling?

2. god or no god

Whether “god” exists or not depends on your definition of the word “god”.

In some definitions, god does not exist, because of the definitions’ self-contradictory nature. For example, god is good but he wants you to suffer for no reason.

In some definitions, god is possible to exist. For example, a god is any being that has a higher level of consciousness than a human being.

We can say that a human being’s consciousness is higher than a dog’s, in the sense that a human can understand things that a dog cannot. For example, most dogs do not understand what a computer is.

Similarly, a dog’s consciousness is higher than an ant’s. An ant’s consciousness is higher than a tree’s. A tree’s consciousness is higher than a rock’s.

In the opposite direction, it is highly possible that in this universe, there are beings that have a higher level of consciousness than human beings. It is highly unlikely that human beings are the highest beings.

Even within the human being species itself, different people can have different levels of consciousness. Even within a single person’s lifetime, one can be at different levels at different ages.

In some definitions, god can exist. You become that god.

In those cases, whether god exists or not depends on whether YOU are willing to become that god, taking up his responsibilities.

— Me@2021-07-26 05:49:53 PM

1995, 2

Posted on June 9, 2021 by rpflee

— 1995

物理一個字：力學第一步

Posted on June 7, 2021 by rpflee

This is my first created video.

Please click, like, share, and subscribe!

— Me@2021-06-07 05:00:34 PM

Physics Town

continues

Author Archives: rpflee

Logical arrow of time, 10

Ultimate Freedom, 2

改變因果鏈起點

1990s, 5.2

Ex 1.21 The dumbbell, 3.3

Meta numbers

Year 2000 Reloaded

相聚零刻 2.1

Pier, 2

2.10 A spacetime orbifold in two dimensions, 3

Visualizing higher dimensions, 2

Broken

上山尋寶, 2

1986, 2

Ex 1.21 The dumbbell, 3.2

Logical arrow of time, 9.4

Graduate school

To realize is to realize, 1.3

1995, 2

物理一個字：力學第一步