Microcanonical vs. canonical ensembles

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SUMMARY

The discussion centers on the comparison between microcanonical and canonical ensembles in statistical mechanics. Participants confirm that while both ensembles yield consistent thermodynamic variables, the expressions for quantities like entropy may appear different. The microcanonical ensemble is applicable when no energy or particles are exchanged, while the canonical ensemble allows for energy exchange with a constant number of particles. The compatibility of both ensembles is emphasized, as they are derived from the general maximum entropy principle.

PREREQUISITES
  • Understanding of statistical mechanics concepts, specifically microcanonical and canonical ensembles.
  • Familiarity with thermodynamic variables such as energy, entropy, and pressure.
  • Knowledge of the Boltzmann Distribution and its application in calculating thermodynamic properties.
  • Basic grasp of the maximum entropy principle in statistical mechanics.
NEXT STEPS
  • Explore the derivation of the partition function for both microcanonical and canonical ensembles.
  • Study the implications of energy exchange in canonical ensembles versus fixed energy in microcanonical ensembles.
  • Investigate the conditions under which a microcanonical system can yield canonical subsystems.
  • Examine the role of entropy in the context of both ensembles and their compatibility.
USEFUL FOR

This discussion is beneficial for students and researchers in physics, particularly those focusing on statistical mechanics, thermodynamics, and the mathematical foundations of ensemble theory.

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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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