Getting structure data from a partition function?

In summary, the partition function is a fundamental concept in classical statistical physics, serving as a way to average through a domain in the phase space. It is not a mathematical transformation and cannot be used to reconstruct details about microscopic quantities. It only provides information about macroscopic quantities, such as the potential energy of the system.
  • #1
maajdl
Gold Member
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Hello,

From wikipedia, this is the partition function for a "classical continuous system":

1563798282228.png


This is the pillar of classical statistical physics, but it can be seen as a mere kind of "mathematical transform" .
It can be used even without thinking to statistics or temperature.
If we focus only on the potential energy part of this integral, then

H = V(q)

is a function of q and the "positions of all other particles" of the system.

I question myself:
Would the full knowledge of Z(β) contain the full information about the "other particles".
And therefore, could the knowledge of Z(β) be traced back (inverted) to the positions of the atoms?

I hope this question doesn't look too fancyful.
I find it interresting because it would cast geometrical data in a 1-variable function Z(β) !

Thanks for your suggestions!

Michel
 
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  • #2
Certainly no. Partition function is not a mathematical transformation (like Fourier) which has an inverse. It's more like averaging with weight given by the Hamiltonian. If you know the average, you cannot reconstruct the numbers that were averaged.

Moreover, partition function doesn't describe one particular configuration of the system. It is averaging through a domain in the phase space. Consider for example an ensemble of identical particles that have momenta but otherwise do not interact. Then the Hamiltonian has only kinetic part

##H = \sum\limits_i \frac{p_i^2}{2\, m}##

Then integral over dq gives just a trivial constant. Partition function does not serve to reconstruct details about microscopic quantities, only macroscopic ones.
 

Related to Getting structure data from a partition function?

1. What is a partition function?

A partition function is a mathematical function used in statistical mechanics to describe the distribution of energy states in a physical system. It is used to calculate the probability of a system being in a particular energy state at a given temperature.

2. How is structure data obtained from a partition function?

Structure data is obtained from a partition function by using the Boltzmann distribution, which relates the probability of a system being in a particular energy state to the energy of that state. This allows for the calculation of the most probable energy state and the corresponding structure of the system.

3. What types of systems can be analyzed using partition functions?

Partition functions can be used to analyze a wide range of systems, including molecules, atoms, and solids. They are also used in the study of biological systems, such as proteins and nucleic acids.

4. How accurate are the results obtained from a partition function?

The accuracy of the results obtained from a partition function depends on the complexity of the system being analyzed and the level of approximation used in the calculations. In general, partition functions provide a good estimate of the most probable energy state and structure of a system.

5. Can partition functions be used to predict the behavior of a system?

Yes, partition functions can be used to predict the behavior of a system at different temperatures. By calculating the partition function at different temperatures, it is possible to determine how the system's energy states and structures will change as the temperature changes.

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