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![## Two-State Model in Statistical Mechanics
### Problem 16.5
Consider the two-state model:
(a) Does the two-state model consist of distinguishable or indistinguishable particles?
(b) What is the probability that particles 1 and 2 will both be in the down position?
(c) What is the probability that exactly two particles of the \( N \) will both be in the down position, regardless of their index?
### Understanding the Two-State System
#### Figure 3.9
An illustration of the classic two-state system is provided. Each particle \( i \) can exist in either:
- A low-energy state with \( \epsilon_i = 0 \)
- A high-energy state with \( \epsilon_i = b \)
Particles are independent of each other, such that the total energy is given by:
\[ E = \sum_{i} \epsilon_i \]
In the diagram, particles are represented as circles, with lines indicating the possible energy states.
#### Figure 3.10
The diagram shows several distinct microstates of the two-state model with the same total energy, \( E = 1b \). Three different microstates are depicted:
- **Microstate 1**: Shows one particle in the high-energy state.
- **Microstate 2**: Displays a different arrangement of particles still resulting in the same total energy.
- **Microstate 3**: Another configuration achieving the same energy level.
Each microstate represents a possible configuration of the particles, illustrating the concept of degeneracy in energy states.](https://content.bartleby.com/qna-images/question/f20d637b-899d-43f6-881e-0d55dde2b4c2/7efb81d7-aa95-4c26-bf14-d9e60a6ff2e0/n96xsdj_processed.png)
Transcribed Image Text:## Two-State Model in Statistical Mechanics
### Problem 16.5
Consider the two-state model:
(a) Does the two-state model consist of distinguishable or indistinguishable particles?
(b) What is the probability that particles 1 and 2 will both be in the down position?
(c) What is the probability that exactly two particles of the \( N \) will both be in the down position, regardless of their index?
### Understanding the Two-State System
#### Figure 3.9
An illustration of the classic two-state system is provided. Each particle \( i \) can exist in either:
- A low-energy state with \( \epsilon_i = 0 \)
- A high-energy state with \( \epsilon_i = b \)
Particles are independent of each other, such that the total energy is given by:
\[ E = \sum_{i} \epsilon_i \]
In the diagram, particles are represented as circles, with lines indicating the possible energy states.
#### Figure 3.10
The diagram shows several distinct microstates of the two-state model with the same total energy, \( E = 1b \). Three different microstates are depicted:
- **Microstate 1**: Shows one particle in the high-energy state.
- **Microstate 2**: Displays a different arrangement of particles still resulting in the same total energy.
- **Microstate 3**: Another configuration achieving the same energy level.
Each microstate represents a possible configuration of the particles, illustrating the concept of degeneracy in energy states.
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