Point in Phase Space and a Microstate

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Discussion Overview

The discussion revolves around the concept of phase space in statistical mechanics, specifically the relationship between points in phase space and microstates of a system. Participants explore the implications of distinguishability and indistinguishability of particles in classical and quantum contexts.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that a point in phase space corresponds to a specific microstate of a system, but another clarifies that this is only true for a single classical particle.
  • Another participant questions whether, in a system with a large number of particles, a point in phase space represents the coordinates and momenta of a single particle, proposing that a microstate may be a 'superposition' of all points in phase space.
  • A clarification is made regarding distinguishable versus indistinguishable particles, stating that for distinguishable particles, each point in the 6N-dimensional phase space corresponds to a microstate, while for indistinguishable particles, exchanging labels of two particles leads to the same microstate.
  • One participant connects the discussion to quantum mechanics, suggesting that treating particles as indistinguishable is crucial to avoid overcounting microstates, which affects macroscopic properties like entropy.
  • Another participant adds that for classical indistinguishable particles, an additional factor is needed to account for overcounting, while quantum particles must adhere to the Pauli exclusion principle for fermions.

Areas of Agreement / Disagreement

Participants express differing views on the implications of distinguishability and indistinguishability in classical and quantum statistical mechanics. The discussion remains unresolved regarding the full implications of these concepts.

Contextual Notes

Participants acknowledge the complexity of the definitions and implications surrounding microstates and phase space, particularly in relation to the treatment of particles in classical versus quantum mechanics.

SteveDC
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Currently learning about Statistical Mechanics and just wanted to check my understanding. Am I right in saying that a point in phase space is just a specific microstate of the system?
 
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Only if the system consists of a single classical particle with no internal structure.
 
Okay, if then the system has large number 'N' of particles, does a point in phase space correspond to the coordinates and momenta of a single particle in that system and a microstate is a 'superposition' (not sure if that's the right word!) of all the points in phase space for that particular microstate?
 
I realize that I was not clear at all in my previous answer.

For a system with N particles, there is a difference depending on whether the particle are distinguishable or not. If they are distinguishable, then each point in the 6N-dimensional (in 3D) phase space is a microstate. If the particles are indistinguishable, different points in phase space that correspond to exchanging the labels of two of the particles will correspond to the same microsate.
 
Ah I see, thanks. Is this what leads to the Quantum treatment of statistical mechanics? Since I'm assuming the particles are in fact indistinguishable based on QM and the classical approach of assuming they're distinguishable would lead to an overcounting of microstates, which in turn affects macroscopic properties like entropy.
 
With classical particles, making them indistinguishable often only requires an additional factor to remove the overcounting of identical states. For quantum particles, you have an additional condition due to the Pauli principle, which for fermions means that no two can be in the same state.
 
Interesting - THanks again.
 

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