Canonical Ensemble Homework: Equal Probabilities Postulate

In summary, the canonical ensemble allows for a more accurate calculation of probabilities by taking into account the energy levels and their multiplicities, rather than assuming equal probabilities for all states.
  • #1
ehrenfest
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1

Homework Statement


In the canonical ensemble, the probability that a system is in state r is given by

[tex] P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)} [/tex]

where g_i is the multiplicity of state i. This is confusing me because I thought

[tex] P_i = \frac{g_i}{\sum_i g_i} = [/tex] states consistent with i / total number of states

was always true by the equal probabilities postulate. What am I missing? Are those two expressions the same?

Homework Equations


The Attempt at a Solution

 
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  • #2
ehrenfest said:

Homework Statement


In the canonical ensemble, the probability that a system is in state r is given by

[tex] P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)} [/tex]

where g_i is the multiplicity of state i. This is confusing me because I thought

[tex] P_i = \frac{g_i}{\sum_i g_i} = [/tex] states consistent with i / total number of states

was always true by the equal probabilities postulate. What am I missing? Are those two expressions the same?


Homework Equations





The Attempt at a Solution


The basic idea of stat mech is that the configurations of different energies are not equally likely. The higher the energy of a state is, the least likely the system will be in that state. This is reflected in the Boltzmann distribution you cite at the top. Your second equation would be valid if all states (irrespective of their energy) would be equally probable.
 

Related to Canonical Ensemble Homework: Equal Probabilities Postulate

1. What is the Equal Probabilities Postulate in the Canonical Ensemble?

The Equal Probabilities Postulate states that in a canonical ensemble, all microstates that have the same energy have an equal probability of being occupied by the system.

2. How is the Equal Probabilities Postulate used in the canonical ensemble?

The Equal Probabilities Postulate is used to determine the probability of a system occupying a specific energy state in a canonical ensemble. It allows us to calculate the distribution of energies among the different microstates in the system.

3. What is the significance of the Equal Probabilities Postulate in statistical mechanics?

The Equal Probabilities Postulate is a fundamental principle in statistical mechanics that helps us understand the behavior of systems at the microscopic level. It allows us to make predictions about the distribution of energies in a system and calculate thermodynamic quantities such as entropy and free energy.

4. Can the Equal Probabilities Postulate be applied to all types of systems?

Yes, the Equal Probabilities Postulate can be applied to all types of systems, as long as they are in thermal equilibrium and have a well-defined energy. It is a general principle in statistical mechanics and is used to study a wide range of systems, including gases, liquids, and solids.

5. How does the Equal Probabilities Postulate relate to the concept of equilibrium in thermodynamics?

The Equal Probabilities Postulate is closely related to the concept of thermal equilibrium in thermodynamics. When a system is in thermal equilibrium, all possible microstates are equally likely to be occupied, as stated by the Equal Probabilities Postulate. This allows us to define the equilibrium state of a system as the one with the maximum number of microstates, or the one with the highest entropy.

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