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 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.
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.
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.
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.
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.