About distinguishable and indistinguishable

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In summary, distinguishability and indistinguishability play a role in determining relative probabilities of events in a "bag of possible discrete events". This is important in classical and quantum regimes, as our method of numbering or naming particles can introduce redundancies and affect the overall normalization and probabilities. In classical physics, this can be seen with the example of Tsars of Russia, while in quantum physics, it relates to the distinct positioning of particles.
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KFC
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I learned fundamental statistical physics some times ago. But so far I still don't understand how distinguishabilty and indistinguishabilty affect statistic. Could any please explain it to me? BTW, why we have to study distinguishabilty or indistinguishabilty in classical and quantum regime?
 
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Distinguishable vs. indistinguishable plays a role when you have an a priori rule to give relative probabilities to different events (the most simple one being a uniform distribution, that is, they all have the same probability, but it can be different, such as exp(- E / k T) or something).

That is, you have a "bag of possible discrete events", and you know that the *relative* probabilities are given by a property of each event (such as its "energy"). In other words, there is an overall normalization constant to be found.

Well, "distinguishable" versus "indistinguishable" comes down to saying what is a non-redundant way of enumerating all the events in the bag ; in other words, what is a "naming scheme" that doesn't point several times to the "same item" in the bag.

Consider the following: consider your bag to be certain Tsars of Russia. Let us say that we want to assign a "relative probability" to them proportional to the duration of their reign.

Now, I am going to keep this list short, but imagine that we have
{Ivan IV (37 years), Catharina the Great (34 years), Peter I (39 years), Ivan the Terrible (37 years), Feodor III (6 years), Peter the Great (39 years)}.

We could calculate, say, the normalization and the probabilities for each of them. We could calculate the probability of having a "the Great" Tsar. But we would make a mistake, because in fact:
Ivan IV is the same person as Ivan the Terrible and
Peter I is the same person as Peter the Great.

So we simply had a redundant naming scheme, and that messed up when we were adding the probabilities or normalizing the set.


We have the same with "distinguishable or indistinguishable" particles.

If we have the list
{particle 1 at position 1 and particle 2 at position 2 ; particle 1 at position 3 and particle 2 at position 2 ; particle 1 at position 2 and particle 2 at position 1 }

we have a list of possible "events" (physical states).

Now, if particle 1 and particle 2 are distinguishable, then that list is all right. But if particle 1 is indistinguishable from particle 2, then that's not true: the first and the last state are descriptions of the same state which is:

A particle at position 1 and A particle at position 2.

Our "numbering" of particles introduced a redundant naming scheme. So if we count them each individually, we make a mistake.
 
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Distinguishability and indistinguishability play a crucial role in statistical physics, especially in the classical and quantum regimes. Distinguishability refers to the ability to differentiate between individual particles or objects, while indistinguishability refers to the inability to differentiate between them.

In classical statistical physics, distinguishability is important because it allows us to track and analyze the behavior of individual particles. This is crucial for understanding macroscopic systems, as the behavior of the system is determined by the collective behavior of its constituent particles.

In the quantum regime, however, particles can exhibit indistinguishable behavior, meaning that they cannot be tracked or differentiated at the individual level. This is due to the fundamental principles of quantum mechanics, such as the uncertainty principle and the Pauli exclusion principle. In this case, we must consider the behavior of the system as a whole, rather than individual particles.

The study of distinguishability and indistinguishability is important because it allows us to accurately describe and predict the behavior of physical systems. In classical systems, distinguishability is a fundamental concept, while in quantum systems, indistinguishability plays a crucial role in understanding the behavior of particles and their interactions.

Furthermore, the concept of distinguishability and indistinguishability has practical applications, such as in quantum computing and cryptography, where the ability to control and manipulate the distinguishability of particles is key.

In summary, the study of distinguishability and indistinguishability is essential for understanding the behavior of physical systems, both in the classical and quantum regimes. It allows us to accurately describe and predict the behavior of particles and has practical applications in various fields of science and technology.
 

Related to About distinguishable and indistinguishable

What is the difference between distinguishable and indistinguishable?

Distinguishable refers to things that are unique and can be easily told apart, while indistinguishable refers to things that are very similar or identical and cannot be easily told apart.

How are distinguishable and indistinguishable related to each other?

Distinguishable and indistinguishable are two opposite concepts that are used to describe the level of similarity between things. They are related in the sense that they are both used to categorize and differentiate between different objects or concepts.

What are some examples of distinguishable and indistinguishable things?

Examples of distinguishable things include different types of fruit, colors, and shapes. Examples of indistinguishable things include identical twins, molecules, and atoms.

Why is it important to understand the concept of distinguishable and indistinguishable?

Understanding the difference between distinguishable and indistinguishable is important for many scientific fields, such as chemistry, physics, and biology. It allows scientists to accurately describe and classify objects and phenomena, which is crucial in conducting research and making scientific discoveries.

How can one determine if two things are distinguishable or indistinguishable?

The level of distinguishability between two things can be determined by observing their characteristics and comparing them. If the two things have unique and distinct properties, they are distinguishable. If they have very similar or identical properties, they are indistinguishable.

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