Quantum Scale, Statistical Physics

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SUMMARY

The discussion focuses on the probabilistic analysis of quantum statistics, specifically comparing Bose-Einstein and Fermi-Dirac statistics. It highlights that for Bose-Einstein statistics, the permutation of identical particles is calculated using the formula involving the factorial of the number of particles (Ni!) and the factorial of the number of blocks (Gi-1) due to the allowance of multiple particles per block. In contrast, Fermi-Dirac statistics restrict occupancy to one particle per block, leading to a different permutation approach. The question raised pertains to the absence of consideration for empty blocks in the Bose-Einstein case.

PREREQUISITES
  • Understanding of quantum statistics, specifically Bose-Einstein and Fermi-Dirac distributions.
  • Familiarity with combinatorial mathematics, particularly permutations and factorials.
  • Knowledge of quantum mechanics concepts, including particle degeneracy (Gi).
  • Basic grasp of statistical mechanics principles.
NEXT STEPS
  • Explore the mathematical derivation of Bose-Einstein statistics and its implications.
  • Study Fermi-Dirac statistics in detail, focusing on occupancy restrictions.
  • Investigate the role of degeneracy in quantum systems and its effect on statistical distributions.
  • Learn about advanced combinatorial techniques in statistical physics.
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Students and researchers in physics, particularly those specializing in quantum mechanics and statistical physics, as well as educators seeking to deepen their understanding of quantum statistical distributions.

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So the probabilistic studying of Quantum statistics of Böse-Einstein suggests that we permutate the number of objects (objects, are the blocks (Gi-1), and the particles in the boxes Ni) and divide those by Ni! And (Gi-1)! Because these particles are identical. But the case of Fermi-Dirac, where only one particle is allowed in a block(unlike the previous case of B.E where many particles are allowed per block), probabilistic studying of it suggests that we permutate an empty block by a block containing one particle.

My question is why didn't our probabilistic study in the first case suggests also the permutation with an empty block.

Don't hesistate to ask if the question is not clear enough.Thanks.
 
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Gi here is the degeneracy of each block.
 

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