## Two-State Model in Statistical Mechanics### Problem 16.5Consider 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 3.9An 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 3.10The 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.

(a) Given: The microstates are as follows: Introduction: Indistinguishable particles are particles that cannot be distinguished from one another, even in principle. Two-level systems, that is systems with essentially only two energy levels are important kinds of systems, as, at low enough temperatures, only the two lowest energy levels will be involved. Explanation: These particles are indistinguishable from one another because we can at most count the number of particles in each energy level but cannot distinguish which particle is in which microstate. (b) Calculation: Since the particles cannot be distinguished labelling of particles is not possible. (c)Calculation: Write the expression of probability. p=E2E Here, E2 is the energy of each microstate and E is the maximum energy. Calculate the total energy of each microstate. E=3b Calculate the value of E2. E2=b Calculate the value of probability by substituting the values of E2 and E. p=b3bp=13p=0.33 Thus, the value of probability is 0.33.

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E = b E = 0 Figure 3.9. An illustration of the classic two-state system. Each particle i can exist in eithe low-energy state with e; = 0 or a high-energy one with e; = b. Particles are independent c other such that the total energy is given by E = %3D %3D

E = b E = 0 Figure 3.9. An illustration of the classic two-state system. Each particle i can exist in eithe low-energy state with e; = 0 or a high-energy one with e; = b. Particles are independent c other such that the total energy is given by E = %3D %3D