The expression e^-\beta E(x) is commonly used in statistical mechanics to describe the probability of a system being in a particular state x with energy E(x) . Here, \beta is a parameter defined as \frac1kT , where k is the Boltzmann constant and T is the absolute temperature. This formula helps in understanding how temperature influences the likelihood of a system occupying various energy states. In this context, e^-\beta E(x) represents the Boltzmann factor, which quantifies the relative probability of different energy states. As the energy E(x) increases, the value of e^-\beta E(x) decreases, indicating that higher energy states are less likely to be occupied at a given temperature. This relationship is fundamental in fields like thermodynamics and quantum mechanics.