What is the physical meaning of the partition function in statistical physics? Ask Question. Asked 6 years, 2 months ago. Active 3 years, 10 months ago. Viewed 31k times. Improve this question. If yes, what doesn't satisfy you about "it encodes how the probabilities are partitioned among the different microstates"? Add a comment. Active Oldest Votes. Improve this answer. Because partition function in literature is generally defined like this.
Zent A. Zent 21 4 4 bronze badges. It is a dimensionless quantity and provides the most convenient way for linking the microscopic property of individual molecules such as their discrete energy level moment of inertia and dipole moment with the macroscopic property such as molar heat, entropy and polarisation.
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Learn more. What is the physical significance of molecular partition function? Ask Question. Asked 4 years, 4 months ago. Active 3 years, 4 months ago. Viewed 4k times. Improve this question. Ronny Ronny 81 1 1 silver badge 4 4 bronze badges.
Add a comment. Firstly, let us consider what goes into it. The partition function is a function of the temperature T and the microstate energies E 1 , E 2 , E 3 , etc.
The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles.
This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability P j that the system occupies microstate j is. This is the well-known Boltzmann factor. For a detailed derivation of this result, see canonical ensemble.
The partition function thus plays the role of a normalizing constant note that it does not depend on j , ensuring that the probabilities add up to one:.
This is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies.
The letter Z stands for the German word Zustandssumme , "sum over states". In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value, or ensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities:. This provides us with a trick for calculating the expected values of many microscopic quantities.
This is analogous to the source field method used in the path integral formulation of quantum field theory. In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system.
These results can be derived using the method of the previous section and the various thermodynamic relations. Suppose a system is subdivided into N sub-systems with negligible interaction energy. However, there is a well-known exception to this rule.
If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a N! N factorial :. This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox.
A specific example of the partition function, expressed in terms of the mathematical formalism of measure theory, is presented in the article on the Potts model. The grand canonical partition function, although conceptually more involved, simplifies the calculation of the physics of quantum systems. The grand canonical partition function for an ideal quantum gas is written:. For bosons , the occupation numbers can take any integer values as long as their sum is equal to N.
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