Isolated System of Distinguishable Particles with Two Energy Levels
In an isolated system of N distinguishable particles, each particle can occupy one of two energy levels, E1 and E2. This scenario is commonly encountered in physics and statistical mechanics when studying the behavior of particles in a closed system.

Energy Levels E1 and E2
- The energy levels E1 and E2 represent two possible states that a particle can occupy in the system.
- The energy difference between E1 and E2 is denoted as ΔE = E2 - E1.

Particle Distribution
- The particles in the system are distinguishable, meaning each particle has a unique identity.
- Due to the distinguishability of particles, the total number of ways to distribute N particles among the two energy levels is given by the binomial coefficient formula, C(N, n) = N! / [n!(N-n)!], where n is the number of particles in one energy level.

Statistical Mechanics Analysis
- In statistical mechanics, the system's behavior can be analyzed using the principles of thermodynamics and probability.
- The distribution of particles among energy levels follows the Boltzmann distribution, which describes the probability of a particle occupying a specific energy level based on its energy and temperature.

Conclusion
In summary, an isolated system of N distinguishable particles with two energy levels, E1 and E2, can be analyzed using statistical mechanics principles. The distribution of particles among energy levels is determined by the distinguishability of particles and the energy difference between the two levels. By applying statistical mechanics concepts, one can understand the behavior of such systems in terms of thermodynamic properties and probability.