Gibbsian Ensemble: Kerson & Huang Explained

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SUMMARY

The discussion focuses on the Gibbsian Ensemble as presented in Kerson and Huang's "Statistical Mechanics." The key point is the representation of the system's state in phase space, where the density function \(\rho(p,q,t) = d^{3N}p d^{3N}q\) indicates the number of representative points in an infinitesimal volume. The confusion arises regarding the integration of this density over allowed momenta and positions, leading to an infinite result when considering an infinite number of systems. The conclusion emphasizes that the volume of integration must be finite or integrable to avoid contradictions.

PREREQUISITES
  • Understanding of Gibbsian Ensembles in statistical mechanics
  • Familiarity with phase space concepts
  • Knowledge of integration techniques in mathematical physics
  • Basic principles of statistical mechanics as outlined in Kerson and Huang's textbook
NEXT STEPS
  • Study the mathematical foundations of phase space in statistical mechanics
  • Explore the implications of infinite systems in thermodynamics
  • Review integration techniques relevant to statistical mechanics
  • Examine alternative treatments of Gibbsian Ensembles in other statistical mechanics literature
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Students and researchers in statistical mechanics, particularly those using Kerson and Huang's textbook, as well as anyone seeking to deepen their understanding of Gibbsian Ensembles and phase space integration.

emob2p
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Hi,
I'm taking a course in Stat Mach using Kerson and Huang's Statistical Mechanics book. I am quite confused with their treatment of a Gibbsian Ensemble. They say imagine an infinite copies of the same system whose state can be represented by a point in phase space. Then [tex]\rho (p,q,t) = d^{3N}p d^{3N}q[/tex] is the number of representative points contained in the infinitesimal volume. So if we integrate this over allowed p's and q's, we should get infinity because we started out w/ an infinite number of total systems. Can this be correct?
 
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Not if the volume of integration is finite or infinite but integrable.
 

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