Step 1: Understanding the Boltzmann distribution.
In thermodynamics, the Boltzmann distribution quantifies the probability \( P(E) \) that a system's particle possesses energy \( E \):
\[P(E) = \frac{e^{-\frac{E}{kT}}}{Z}\]
where:
- \( E \) represents the particle's energy,
- \( k \) is the Boltzmann constant,
- \( T \) denotes the system's temperature,
- \( Z \) is the partition function, which ensures the probability distribution is normalized.
The Boltzmann distribution demonstrates an exponential decrease in the probability of a particle having a given energy as that energy increases.
Step 2: Calculating the probability of energy exceeding \( E_0 \).
The objective is to determine the probability that a particle's energy surpasses \( E_0 \). This is equivalent to the complement of the probability that the particle's energy is less than or equal to \( E_0 \):
\[P(E > E_0) = 1 - P(E \leq E_0)\]
Based on the Boltzmann distribution, the probability of a particle having energy less than or equal to \( E_0 \) is computed as:
\[P(E \leq E_0) = \int_0^{E_0} \frac{e^{-\frac{E}{kT}}}{Z} dE\]
The integral's solution yields the probability of energy being less than \( E_0 \). Consequently, the probability of energy exceeding \( E_0 \) is:
\[P(E > E_0) = e^{-\frac{E_0}{kT}}\]
Step 3: Conclusion.
Therefore, according to the Boltzmann distribution, the probability that a particle will have energy greater than \( E_0 \) is:
\[P(E > E_0) = e^{-\frac{E_0}{kT}}\]
Answer: The probability is \( e^{-\frac{E_0}{kT}} \).