Logical arrow of time, 11

The initial microstates should be averaged, because it forms an ensemble for the initial macrostate.

Note that a macrostate is actually one particular microstate, not a collection of microstates; it is just that we don’t know which particular microstate.

But how come the final possible states should be summed over, not be averaged?

— Me@2013-08-13 05:16 PM

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a macrostate = (a microstate in) a set of macroscopically-indistinguishable microstates

— Me@2022-01-09 07:43 AM

Note that, by definition, two macroscopically-indistinguishable microstates will never separate into two distinct macrostates.

— Me@2022-04-14 05:55 PM

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The initial macrostate is with probability one, because it is already known. So the summation of the probabilities of all possible mutually exclusive initial microstates that are corresponding to that initial macrostate is one, such as

\displaystyle{P(I_1) + P(I_2) = 1}

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By definition, the final macrostate is not known yet. Each possible final macrostate is not with probability one.

The probability of getting a particular final macrostate from that initial macrostate is the summation of the probabilities of all possible mutually exclusive final microstates that are corresponding to that final macrostate.

\displaystyle{P(F_1~\text{or}~F_2) = P(F_1) + P(F_2)}

\displaystyle{P(I\to F) = \frac{1}{N_{\text{initial}}} \sum_{ij} P(I_i \to F_j)}

— Me@2022-04-13 01:09 PM

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The only assumptions I made are those about the addition of probabilities of assumptions and their effects – and these logical rules are fundamentally asymmetric when it comes to the role of the assumptions and their consequences. This logical arrow of time can’t be removed from any reasoning about a world that depends on time – time only copies the logical relationship of implication. And this logical arrow of time is the source of the thermodynamic arrow of time as well.

— edited Feb 2, 2011 at 15:23

— answered Jan 14, 2011 at 11:42

— Luboš Motl

— Calculation of the cross section

— Physics StackExchange

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2022.04.14 Thursday (c) All rights reserved by ACHK

Remove time from physics

motohagiography on May 14, 2018 | next [–]

I once saw a fridge magnet that said “time is natures way of making sure everything doesn’t happen all at once,” and it’s stuck with me.

The concept of time not being “real,” can be useful as an exercise for modelling problems where to fully explore the problem space, you need to decouple your solutions from needing them to occur in an order or sequence.

From an engineering perspective, “removing” time means you can model problems abstractly by stepping back from a problem and asking, what are all possible states of the mechanism, then which ones are we implementing, and finally, in what order. This is different from the relatively stochastic approach most people take of “given X, what is the necessary next step to get to desired endstate.”

More simply, as a tool, time helps us apprehend the states of a system by reducing the scope of our perception of them to sets of serial, ordered phenomena.

Whether it is “real,” or an artifact of our perception is sort of immaterial when you can choose to reason about things with it, or without it. A friend once joked that math is what you get when you remove time from physics.

I look forward to the author’s new book.

— Hacker News

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2022.04.10 Sunday ACHK

Logical arrow of time, 6.4.3

Logical arrow of time, 6.1.3

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The source of the macroscopic time asymmetry, aka the second law of thermodynamics, is the difference between prediction and retrodiction.

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In a prediction, the deduction direction is the same as the physical/observer time direction.

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In a retrodiction, the deduction direction is opposite to the physical/observer time direction.

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In other words:

In a prediction, the meta-time direction is the same as the object-time direction.

In a retrodiction, the meta-time direction is opposite to the object-time direction.

— Me@2022-02-18 06:52:27 AM

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— guess —

If a retrodiction is done by a time-opposite observer, he will see the entropy increasing. For him, he is really making a prediction.

— guess

— Me@2013-10-25 3:33 AM

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How to create a time-inverted observer?

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Just invert the retrodiction direction.

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Retrodiction to a backward-time observer is just equivalent to retrodiction-for-backward-time to a forward-time observer.

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However, retrodiction-for-backward-time is just prediction.

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In other words, retrodiction to a backward-time observer is equivalent to a prediction for a normal time direction observer.

That’s why

— guess —

If a retrodiction is done by a time-opposite observer, he will see the entropy increasing. For him, he is really making a prediction.

— guess

— Me@2013-10-25 3:33 AM

— Me@2022-02-18 06:37:59 AM

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2022.02.18 Friday (c) All rights reserved by ACHK

EPR paradox, 11.3

Black hole information paradox, 2.2.3

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It shouldn’t be so surprising that unitarity survives completely while causality doesn’t. After all, the basic postulates of quantum mechanics, including unitarity, the probabilistic interpretation of the amplitudes, and the linearity of the operators representing observables, seem to be universally necessary to describe physics of any system that agrees with the basic insights of the quantum revolution.

On the other hand, geometry has been downgraded into an effective, approximate, emergent aspect of reality. The metric tensor is just one among many fields in our effective field theories including gravity.

— Black hole information puzzle

— Lubos Motl

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identical particles

~ some particles are identical, except having different positions

~ some particle trajectories are indistinguishable

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trajectory indistinguishability

~ particle identity is an approximate concept

~ causality is an approximation

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spacetime is defined by causality

~ so spacetime is also an approximation

— Me@2022-02-11 12:47:14 AM

— Me@2022-02-13 03:38:35 PM

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2022.02.13 Sunday (c) All rights reserved by ACHK

程式員頭腦 15

數學教育 7

這段改編自 2010 年 4 月 24 日的對話。

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But while you don’t literally need math for most kinds of hacking, in the sense of knowing 1001 tricks for differentiating formulas, math is very much worth studying for its own sake. It’s a valuable source of metaphors for almost any kind of work.[3] I wish I’d studied more math in college for that reason.

[3] Eric Raymond says the best metaphors for hackers are in set theory, combinatorics, and graph theory.

— Undergraduation

— March 2005

— Paul Graham

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(安:另外,他提的另一個,有關學習數學的要點是,即使假設你在大學中,學到的數學,在日常生活中沒有用,單單是為獲取,那些嶄新的元素概念本身,就已經能夠令你有超能力;令你有一些,常人沒有的思考工具、比喻語言。)

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那和我之前,叫你學 programming(電腦編程),意思是一樣的。

那本 programming 教科書《SICP》(Structure and Interpretation of Computer Programs),竟然教曉我,時間的定義和數字的定義等。

— Me@2022-02-13 10:46:08 AM

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2022.02.13 Sunday (c) All rights reserved by ACHK

EPR paradox, 11.2

Black hole information paradox, 2.2.2

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physical definition

~ define the microscopic events in terms of observable physical phenomena such as the change of readings of the measuring device

~ define unobservable events in terms of observable events

— Me@2022-01-31 08:33:01 AM

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superposition

~ lack of the existence of measuring device to provide the physical definitions for the (difference between) microscopic events

— Me@2022-01-31 08:33:01 AM

— Me@2022-02-12 10:22:09 AM

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In the EPR experiment, how come the two always correlate if there are no definite states before the measurements?

When you actually know the results of your experiment, it does affect your expectations of the faraway results if there are correlations – and correlations are almost always there iff the two subsystems have interacted or been in contact in the past). — Lubos Motl

Microscopically, there is no time, in the sense that all the (past and future) quantum states have one-one correspondences. All results are deterministic. No causality violation required nor allowed. — Me@2016-10-14 07:55:48 PM

This is called quantum determinism, which may or may not be correct. But quantum determinism, even if true, is not necessary for explaining the EPR experiment, if we understand that:

1. Superposition is mathematical, not physical.

2. “Wave function collapse” is mathematical, not physical. It just means that we have to replace the wave function with another if we replace the system with another.

The system before and after the detectors activated should be regarded as two distinct systems. In other words, when you activate the detectors, you have actually replaced a system-without-detectors with a system-with-detectors.

“Wave function collapse” replaces the pure state wave function with a mixed state wave function. In other words, it replaces the pure state of superposition with a mixed state of eigenstates. In other other words, it replaces quantum probability with classical probability.

Before opening the box, the cat is not in a superposition state. Instead, it is in a mixed state.

The uncertainty is classical probability, which is due to lack of detailed knowledge, not quantum probability, which is due to lack of definition (in terms of physical phenomena difference).

— Me@2022-01-29 10:38:19 PM

— Me@2022-02-12 10:28:57 AM

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2022.02.12 Saturday (c) All rights reserved by ACHK

Entropy at the Beginning of Time, 1.2

Logical arrow of time, 10.2.2

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If at the beginning, the universe had a high entropy, it was at a macrostate corresponding to many indistinguishable microstates.

That description is self-contradictory, because “two macroscopically-indistinguishable microstates” is meaningful only if they were once macroscopically distinguishable before.

That is not possible for the state(s) at the beginning of the universe, because at that moment, there was no “before”.

So it is meaningless to label the universe’s beginning macrostate as “a state corresponding to many indistinguishable microstates”.

Instead, we should label the universe’s beginning state as “a state corresponding to one single microstate”.

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For example, assume that the universe was at the macrostate \displaystyle{A} at the beginning; and the \displaystyle{A} is corresponding to two macroscopically-indistinguishable microstates \displaystyle{a_1} and \displaystyle{a_2}.

Although microstates \displaystyle{a_1} and \displaystyle{a_2} are macroscopically-indistinguishable, we can still label them as “two” microstates, because they have 2 different histories — history paths that are macroscopically distinguishable.

However, for the beginning of the universe, there was no history. So it is meaningless to label the state as “a macrostate with two (or more) possible microstates”.

So we should label that state not only as one single macrostate but also as one single microstate.

In other words, that state’s entropy value should be defined to be zero.

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If in some special situation, it is better to label the universe’s beginning state as “a state with non-zero entropy”, that state will still have the smallest possible entropy of the universe throughout history.

So it is not possible for the universe to have “a high entropy” at the beginning.

— Me@2022-01-08 02:38 PM

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2022.01.09 Sunday (c) All rights reserved by ACHK

Entropy at the Beginning of Time, 1.1

Logical arrow of time, 10.2.1

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Two distinguishable macrostates can both evolve into one indistinguishable macrostate.

— Me@2013-08-11 11:08 AM

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Note that, tautologically, any system can be at only one single macrostate at any particular time.

So the statement actually means that it is possible for two identical systems at different macrostates evolve into the same later macrostate.

— Me@2022-01-08 03:12 PM

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But the opposite is not possible. Two indistinguishable macrostates is actually, by definition, one macrostate. It cannot evolve into two distinguishable macrostates.

One single macrostate is logically impossible to be corresponding to two different possible later macrostates.

— Me@2022-01-08 01:29 PM

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If the beginning universe state had a high entropy, by definition, it was at a macroscopic state with many possible macroscopically-indistinguishable microstates.

However, if it is really the state of the universe at the beginning, it is, by definition, a single microstate, because “different microstates” is meaningful only if they were once distinguishable.

— Me@2013-08-11 01:42 PM

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a macrostate = a set of macroscopically-indistinguishable microstates

— Me@2022-01-09 07:43 AM

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The meaning of “entropy increases” is that state \displaystyle{S_1} and state \displaystyle{S_2} both evolve into state \displaystyle{S_3}.

But for the beginning of the universe, there were no multiple possible macrostates that the beginning state could be evolved from.

— Me@2013-08-11 01:44 PM

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2022.01.09 Sunday (c) All rights reserved by ACHK

時光起源

The Origin of Time

這段改編自 2021 年 12 月 5 日的對話。

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昨日的那段影片說,個別的粒子,其實都是質量零;以光速行走,感受不到時間。

但是,把一堆粒子圈成一件東西的話,那件東西整體而言,就會有質量;就會低於光速行走;就會感受到時間。

變相來說,時間的來源就是,你將一堆粒子組成一個物件、一個身份、一個自我時,那個自我就會,感受到時間。

— Me@2022-01-04 12:23:25 PM

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To form an object (an observer), the component particles cannot all always move at light speed in the same direction, for that would prevent the object as a whole from feeling time.

Anything moving at the speed of light cannot feel the passage of time. If a set of particles all moves at light speed in the same direction all the time, they cannot feel time either as individuals or as a whole; so they cannot form an “object”.

To form an object (an observer), the component particles need to interact. So some component particles need to move in other directions sometimes.

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An object requires an internal structure to exist and evolve. The component particles need to interact in order to evolve as a single identity. So different particles need to move in different directions sometimes. As a result, the component particles as a whole, aka “the object”, will move slower than light.

— Me@2021-12-08 08:09:17 AM

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2022.01.05 Wednesday (c) All rights reserved by ACHK

Visualizing higher dimensions, 2

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

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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

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2021.08.23 Monday (c) All rights reserved by ACHK

Logical arrow of time, 9.4

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

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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

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

— Me@2017-02-12 07:37:54 PM

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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

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2021.08.07 Saturday (c) All rights reserved by ACHK

Logical arrow of time, 9.2

To confirm or disconfirm a prediction, you cannot check record; you can only observe the system evolving.

To confirm or disconfirm a retrodiction, you can only check record (or the logical consequence of that retrodiction); you cannot observe that past event directly.

— Me@2013-08-10 08:00 PM

— Me@2021-05-03 12:28 PM

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2021.05.03 Monday (c) All rights reserved by ACHK

Modification of the cumulative concept of time

The past is part of the present, in the sense that part of the present is the same as the past, or is based on something of the past. For example, “I was 10 years old in the past” is the necessary condition for “I am 20 years old in the present”.

However, not all of the past is still part of the present, because some of the past is already lost. In other words, some of the past is not stored in the present.

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You need physical macroscopic law (induction) in order to get the cumulative property of time.

Microscopically, due to the unitarity of the Schrödinger equation, any past quantum state has a one-one correspondence to a quantum state at any particular future moment. So any future quantum state already has all the information needed to deduce its past state. In this sense, there is no time microscopically.

Time is not cumulative microscopically, because the past is not part of the future.

We can also say that time is 100% cumulative microscopically, because the past is all of the future; instead of being part of any future quantum state, the past quantum state already has all the (equivalent) information of a quantum state at any particular future moment.

— Me@2013-08-10 07:45 PM

— Me@2021-04-21 04:24 PM

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2021.04.21 Wednesday (c) All rights reserved by ACHK

Causal diamonds in time travel

Quantum mechanics is a set of rules that allows an observer to predict, explain, and/or verify observations (and especially their mutual relationships) that he has access to.

No observer can detect inconsistencies within the causal diamonds. However, inconsistencies between “stories” as told by different observers with different causal diamonds are allowed (and mildly encouraged) in general (as long as there is no observer who could incorporate all the data needed to see an inconsistency).

— Raphael Bousso is right about firewalls

— Lubos Motl

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There is no “god’s eye view” in physics.

— Me@2021-04-17 03:12:58 PM

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Macroscopic time is with respect to an observer. Actually, physics is with respect to an observer.

In the real universe, any observer’s observations must be consistent. When two observers compare their observations, their results must be consistent, because the comparison itself is an observation of an observer.

Time travel in the absolute sense is logically impossible. Let’s assume that it is logically possible.

If a time travel story follows the principle of “an observer’s observations must be consistent”, each character in that story must see a consistent timeline, even if different characters’ timelines may be inconsistent. That is fine as long as such inconsistent observers never meet to compare their results.

If two of such observers choose to meet to compare their results, the action to “meet to compare” itself will render the results consistent. It is similar to the resolution of the twin paradox in special relativity.

There is no “god’s eye view” in physics. Every physical event must be described with respect to an observer. Every physical event, even if the event is “to compare observation results”, must be described with respect to an observer.

— Me@2017-05-10 07:45:36 AM

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2021.04.17 Saturday (c) All rights reserved by ACHK

Time travel, 3.2

時光機 3.2 | Time is other things, 4 | Cumulative concept of time, 9.2

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If you keep deleting your blog posts one by one, your blog effectively keeps becoming its past. However, its environment is still cumulative. In other words, the environment is not becoming its past.

If the environment and the blog interchange labels, the blog’s environment is going to the past, while the blog is not.

— Me@2013-08-10 07:41 PM

— Me@2021-04-12 06:02 PM

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2021.04.12 Monday (c) All rights reserved by ACHK