Microcanonical vs. canonical ensembles

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Homework Help Overview

The discussion revolves around the comparison between microcanonical and canonical ensembles in statistical mechanics, particularly in calculating thermodynamic variables such as energy and entropy. The original poster is exploring how the choice of ensemble affects the calculation of the partition function and the resulting thermodynamic properties.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the compatibility of results obtained from different ensembles, questioning whether the choice of ensemble impacts the thermodynamic variables. Some participants express confusion over differing expressions for entropy derived from each ensemble and seek clarification on the conditions under which results align.

Discussion Status

The discussion is active, with participants sharing insights about the relationship between the two ensembles. Some have provided guidance on the compatibility of results, while others are probing deeper into the conditions necessary for equivalence in energy values across ensembles. Multiple interpretations and lines of reasoning are being explored.

Contextual Notes

Participants are considering the implications of energy exchange between subsystems and the fixed nature of energy in microcanonical ensembles versus the variable energy in canonical ensembles. There is an emphasis on understanding the foundational principles that govern these statistical mechanics frameworks.

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Homework Statement


Suppose we have a system of particles and we wish to calculate the thermodynamic variables (e.g. energy, entropy, pressure, etc.)
Will the result depend on whether we consider the microcanonical or the canonical ensemble?
I want to calculate the partition function to get the TD variables, but the way in which we need to calculate the partition function depends on the ensemble that we consider.


Homework Equations





The Attempt at a Solution

 
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The result will be consistent.
 
Great! Thanks.
 
Does it matter which ensemble I use? micro/canoncal?
 
No -- use which ever one you are more comfortable with.
 
When I do the same problem with either ensemble, I get very different looking expressions for, say, the entropy. I guess what you're saying is that if I do the correct manipulations I should always be able to cast one expression in the form of the other?
 
Yep --- they are completely compatible. After all, they are both just special cases of the general maximum entropy principle.
 
Great. Thanks
 
A further Question

Hi,

I agree with what you have said about the values being the same. I have calculated the mean square deviation of the energy in terms of Cv, using the Boltzmann Distribution, and I am trying to figure out what criterion is needed to to have the same energy values for both canonical & microcanonical ensembles. Any thoughts?
 
  • #10
Microcanonical is used when no energy or particles are is exchanged between subsystems or with the outside (no heat transfer through container walls, for instance). Canonical allows energy exchange but the number of particles must remain constant. Grand canonical also allows for change in particles (diffusion, etc.).
 
  • #11
The two distribtions are substantially different. The microcanonical distribution deals with a single, fixed system energy, while the canonical distribution does not. However, a partition of the microcanonical system can yield canonical sub-systems -- what conditions would need to hold for this to be true?

Also, think about whether a canonical system can be composed of a set of microcanonical systems. Think about entropy in this context.
Regards,
Reilly Atkinson
 

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