Eigenstate thermalization hypothesis (in progress)

 Why statistical mechanics works for classical physics and why it should not work for quantum physics

Source: Wikipedia

The assumption of statistical physics in a many body (10^23 particles) isolated classical system with given fixed energy is that every micro-state is equally probable. Why should this be the case? Because a system with 10^23 particles is highly likely to be chaotic. And a chaotic system will in general spend equal time in equal areas of its phase space. Hence, the expectation value of an observable will be the average of the expectation value corresponding to each of the micro-state.

$$\bar{A}_{\mathrm{classical}} = \frac{1}{N}\sum_{i=1}^N A_i$$

But in the quantum case, there appears to be a problem. Even for a many body quantum system, if we go in the energy eigenbasis, then time evolution of the state just rotates the phases in the energy eigenbasis. That is, if the system is prepared an eigenstate, it will stay there.

Hence, long time expectation value of a general observable depends upon the initial state of the system. This does NOT happen in the classical case. See the following calculation.

$$|\psi(0)\rangle=\sum _\alpha c_\alpha |E_\alpha \rangle$$

$$\langle \hat{A}  \rangle_t \equiv \langle \Psi(t) | \hat{A} |\Psi(t) \rangle = \sum_{\alpha, \beta} c_{\alpha}^{*} c_{\beta} A_{\alpha \beta} e^{-i \left ( E_{\beta} - E_{\alpha} \right )t / \hbar }$$

The time average is then,

$$\langle \hat{A}  \rangle_t \overset{t\to\infty}{\approx} \sum_{\alpha} \vert c_{\alpha}\vert^2 A_{\alpha \alpha}$$

This is surprising because on one hand, the expectation value of general observable depends on the initial state of the system and on the other hand, we intuitively know that a sub system of this many body state is going to thermalize in finite time for a large class of initial state, and hence so will thermalize the expectation values of the local operators on this subsystem.

How to resolve this paradox?

 Cognitive Biases for Scientists (in progress)

We all suffer with Cognitive Biases. What is Cognitive Bias? One typical example is how we stick to and defend an information that we received earlier in life, no matter what it may be. Therefore, we have people born in a religious environment incapable of seeing the illogicality of their faith, and under adequate provocation, also ready to kill or get killed for their faith.

Such cognitive bias can prove fatal for a scientist. A scientist is in search of truth. Cognitive biases will make them fool themselves into believing something that is actually not true. But nature cannot be fooled, and the said scientist would waste all of their exsistence because of cognitive biases. We want to avoid that.

Why is Cognitive Biases a matter for absolute concern for a scientist, more so than a common person. I mean, the opposite can be argued. Science is a deductive discipline, a logical framework with rigour and objectivity? What room for subjectivity and let alone a Cognitive Bias is for a scientist? A scientist should be least worried about cognitive biases than common people!

If only this picture I have painted was true. The argument can be summarized in a single quote.

“It is by logic that we prove, but by intuition that we discover.”

― Henri Poincaré

And when intuition is at play, congnitive biases follow. Hence the necessity to guard oneself from being fooled by one's own mind. Because, as Feynman said, "you must not fool yourself and you are the easiest person to fool.".

Before we dive into a list of cognitive biases, it is important to note that there does not exist an exhaustive list of cogitive biases, in the same way as there does not exist an exhaustive list of nutritients required for a human being. Scientists know about a certain number of nutrients, but they might not know about many other ones. Moreover, we as human beings are not static machines. We are evolving. We do not have a strict list of prescriptions of nutrients. The body dynamically reacts to chemicals that it is exposed to, and this reaction also evolves through time.

What I am trying to say that, be watchful for cognitive lapses that perhaps nobody has heard of. How will you do that. Stay intellectually honest. Don't fool your own self.

Can Quantum-Mechanical Description of Physical Reality be Considered Complete ?

 The Podolsk's argument

Consider a pair of particles, A and B, having entangled positions and momentum, separated by a distance. We can hence determine either the position or momentum of B by measuring the corresponding quantity of A. But note that in doing so, we could NOT have disturbed B whatsoever, because of special relativity.

Since the value of position or momentum of B can be determined without disturbing it, a concept of reality can be associated with these physical quantities. But quantum mechanics denies to attribute a definite position AND momentum to the particles prior to their measurement, and also claims that both these quantities do not coexist to arbitrary precision. Hence Quantum mechanics must be incomplete.

Bohr's reply to  The Podolsk's argument

Position and momentum are complementary quantities, and determining one destroys anything that can be said about the other. That is, these measurements are NOT non disturbing. Measuring the position of A DOES disturb B. There is a nonlocal influence in a way that no superluminal signaling is still valid.

Einstein's argument

The choice of measuring the position OR momentum of A collapses the wavefunction of B. But locality demands that such influence should not be possible. Hence, wavefunction is not true description of the 'real' state of the system, because it is not possible to influence the 'real' state of a system nonlocally.

My take

By measuring one of the EPR pair, we do end up disturbing the other particle. Nonlocal influences exist in quantum mechanics. These influences are such that a useful superluminal signal cannot be sent. Hence, measuring position of A instantaneously disturbs the state of B (in A's rest frame) in the same manner as measuring the position of B would disturb the state of B. Hence the reality claim based on non disturbance is invalid.

Einstein is wrong because when choice of measurement on particle A steers particle B's state, such superluminal influence does not transfer any information. Hence special relativity is not violated.

Einstein believed that the real state of a particle cannot be influenced by choices made at a distance. Hence wavefunction cannot be in one-to-one correspondence with the real states.

If by real state he meant all the information that can be extracted from the system, then this cannot indeed be influenced by the choices made at a distance. BUT, wavefunction gives a probabilistic description of a system, and hence can be disturbed from a distance without revealing superluminal information.

The superluminal signals are censored, inaccessible.

C* algebra

 algebra is a vector space  over a field  with a multiplication. The multiplication must be distributive and, for every  and  must satisfy

f(xy)=(fx)y=x(fy).

Note- by multiplication defined on teh evector space, we mean multiplication of two vectors to yield a number (or a vector maybe, depending on the case).

Examples of algebras include the algebra of real numbers, vectors and matrices, tensors, complex numbers, and quaternions.

PC hacks

For people who find the sound only in left earphone irritating, you can try the following- On Android, go to setting, accessibility, mono audio. Exact path may vary depending on Android version, but it is there in recent ones. On ipados 13, go to setting, accessibility, audio/visual, mono audio

Ubuntu on windows

sudo apt-get install gcc //To update ubuntu

GNU plot on windows

p 'plot.txt' using 1:2 //to get a 2d plot from text file in which first column is used as 1st axis and 2nd column 2nd.

splot 'plot.txt' using 1:2:3 //surface plot with x, y , z data. gnuplot> set hidden3d //To get nice surface plot without points but actual surface. gnuplot> set dgrid3d 50,50 qnorm 2 gnuplot> splot 'data_file' with lines

Generate pdf thumbnail

Install genuine adobe acrobat reader. Go to edit, preferences, general, thumbnail. Also, go to control panel, set 'view by' on upper right side to large icons, find file explorer options, go to view, then set show thumbnail, and not icon. apply. If the files don't load, open acrobat, open file, go to pdf files and do large icon view. Thumbnails will generate. IF THE THUMBNAILS ARE WRONG, CLEAR CACHE. Search for disk cleanup, choose c drive, and clean only the thumbnail caches.

Download youtube videos

(but only when your life depends upon it. Downloading youtube videos is unethical, and this blog does not endorse it)

Use: youtube-dl

Download youtube-dl from github.

in cmd line, go to the youtube-dl directory. ALWAYS go to this folder to execute programs. type youtube-dl then give a space, and paste the link. youtube-dl -U for update To use options for download, just leave a space after youtube-dl and type the code for the option, then leave a space, and then paste the link. press shift enter.

Speed up a youtube video by an amount larger than 2x.

press ctrl + shift +j paste following command, with x replaced with teh speed you want. press enter document.getElementsByTagName("video")[0].playbackRate = x press ctrl + shift +j

Most economical export method from goodnotes

is flatened, with background. Handwriting recognision makes no difference.
If you make it editable, for pure notes, the file size becomes 10 times! Using blank background actually increases the file size.

gmail mail bulk delete code.

Your Gmail occupies space. If you want to free up space in bulk, search the following on gmail.

has:attachment larger:10mb

OR

Size:5m

Summary: 'What is Life' by Schrodinger

Ch1: The classical Physicist's Approach to the subject

Why are organisms so large as compared to atoms?

Because of emergence of laws from the background heat as a result of large number of atoms. For example, motion of  each molecule in a gas is random, but diffusion emerges as an approximate law due to the principle of large numbers. Author points out that even if we know the number density of a gas, given a volume that should have N molecules, the actual number fluctuates by an amount $\sqrt{N}$.

Author suggests that all that we know about physics is approximate in this sense, as the pattern emerges out of large number of atoms that we have access to.

Author notes how because of the background heat, if life was susceptible to influence from individual atomic interactions, we would not be capable of organised thought.

Solution to the game 'Among us'

Solution to the game 'Among us'

Today I played a very interesting murder mystery game on android called Among us.

Suppose you are stuck in a house, probably due to storm, with some people you know. Suddenly, all of you realize that there is a murderer among you, bent on killing everyone. You discover one murder, and then two... Everyone starts suspecting everyone else. Nobody can be trusted. What will you do?

There have been many movies and novels on this plot. 'Murder on the Orient Express' and 'And then there were none' are some works that I can recall of Agatha Christie on these lines. 'Johnny Gaddaar is a Bollywood movie on similar plot.

Among us lets you live through such a situation.

There are more than 4 players (typically 10) on a spaceship. They realize that there is an impostor among them, bent on killing everyone. The crew needs to successfully perform some tasks on the ship in order to win the game. On discovery of a dead colleague or gaining new clues, the crew can also call for an emergency meeting and vote to determine an impostor and kick him out of the ship, hence killing him. It is after the suspect is killed that it is disclosed whether he was actually the impostor or not. If they successfully kill the impostor, they win. If not, the game continues and their numbers dwindle. If the impostor manages to kill the second last of the crew, he wins.

If the impostor and the crew play a perfect game, can one force its victory on the other? Yes. Spoilers alert. You are about to learn the solution.

The crew can always defeat the impostor. All they need to do is split in groups of two and keep an eye on their partner while performing the tasks. If the impostor kills his partner, the rest of the crew will kick him out of the ship.

The developers of the game need to change the rules of the game to prevent this loophole.