Partition Function of a System with One State at Zero Energy and Another State at Energy E

The partition function is a fundamental concept in statistical mechanics that describes the distribution of energy among different states of a system. It is denoted by the symbol Z and is defined as the sum of the Boltzmann factors for all possible states of the system.

Partition Function (Z)
The partition function is given by the equation:

Z = Σ e^(-βE)

where Σ is the sum over all possible states of the system, β = 1/(kT) is the inverse temperature, E is the energy of each state, and k is the Boltzmann constant.

System with One State at Zero Energy and Another State at Energy E
In this particular system, there are two states: one state at zero energy and another state at energy E. Let's denote these states as S1 and S2, respectively.

Calculating the Partition Function
To calculate the partition function for this system, we need to evaluate the Boltzmann factors for each state and sum them up.

1. State S1 (Zero Energy):
- The energy of state S1 is zero, so E1 = 0.
- The Boltzmann factor for state S1 is e^(-βE1) = e^(-β*0) = 1.

2. State S2 (Energy E):
- The energy of state S2 is E2 = E.
- The Boltzmann factor for state S2 is e^(-βE2) = e^(-β*E).

3. Partition Function:
- The partition function Z is the sum of the Boltzmann factors for both states: Z = e^(-β*0) + e^(-β*E) = 1 + e^(-βE).

Explanation
The partition function of a system with one state at zero energy and another state at energy E is given by Z = 1 + e^(-βE), where β = 1/(kT) is the inverse temperature. The partition function represents the sum of the Boltzmann factors for each state, which determines the distribution of energy among the states of the system. In this case, the state with zero energy has a Boltzmann factor of 1, while the state with energy E has a Boltzmann factor of e^(-βE). The partition function provides a measure of the probability of each state being occupied by the system at a given temperature.