Why Cannot I Factorize the Partition Function?

This is due to the fact that the number of particles in a grand canonical ensemble is not fixed, allowing for different permutations of particles to contribute to the overall partition function. In summary, the difference between quantum and classical statistics is that in quantum statistics, the partition function cannot be written as the product of single-particle partition functions due to the symmetrization and antisymmetrization requirements for bosons and fermions, respectively. This can be further explored in any book on quantum mechanics, as it is a fundamental concept in the field.
  • #1
Frank Einstein
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Hi everibody, the other day in a stadistical physics lesson we were studyng Fermi Dirac and Bose Einstein stadistics and comparing it to the classical Maxwell Boltzmann's.

We learned that in the quantum stadistics for indistinguishable particles the partition function of the whole system couldn't be written as the multiplication of the partition function of all the single particles of the system as we used to do when we were working in the classical limit.

So, can please anybody tell me if there is a physical reason why I can't write the partition formulal for Bose Einstein's and Fermi Dirac's stadistic as ∏(zn)?

Thanks for reading.
 
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  • #2
The wavefunction has to be symmetric under exchange of two identical bosons, and antisymmetric under exchange of two identical fermions - otherwise you cannot reproduce the observed statistics. With a factorized partition function this does not work.
 
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  • #3
Thank you very much for your anwser.
By the way, can you point me to a book where whis is treated?
 
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  • #4
This should be covered in every book about quantum mechanics.
 
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However, while it is not possible to factorize the canonical partition function, it is possible to factorize the grand canonical partition function.
 
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FAQ: Why Cannot I Factorize the Partition Function?

What is the partition function?

The partition function is a mathematical concept used in statistical mechanics to describe the distribution of energy and states among particles in a system. It is denoted by the letter Z and is a function of temperature, volume, and the number of particles in the system.

Why is it difficult to factorize the partition function?

The partition function is difficult to factorize because it is a complex mathematical expression that involves summations over all possible energy states of the particles in a system. It also includes exponential functions, making it challenging to manipulate and simplify algebraically.

Why is the partition function important?

The partition function is essential because it allows us to calculate thermodynamic properties of a system, such as the internal energy, entropy, and free energy. It also provides a link between microscopic and macroscopic properties of a system, allowing us to understand and predict its behavior.

Are there any approximations that can be made to simplify the partition function?

Yes, there are several approximations that can be made to simplify the partition function, such as the classical approximation, the high-temperature approximation, and the low-density approximation. These approximations are valid under certain conditions and can make the partition function more manageable to work with.

Can the partition function be solved exactly for all systems?

No, the partition function cannot be solved exactly for all systems. In some cases, it may be possible to calculate it analytically, but in most cases, it requires numerical methods to approximate its value. Additionally, for complex systems with many particles and energy states, the partition function may be impossible to calculate exactly.

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