Functional integration is a mathematical technique used in physics to calculate the probabilities of various outcomes of a physical system, based on its possible paths or configurations. It involves integrating over all possible configurations of a system, taking into account the probability amplitudes associated with each configuration, and summing them up to obtain the overall probability of the system.
A simple example of functional integration is the path integral formulation of quantum mechanics. In this approach, the probability amplitude of a particle moving from one point to another is obtained by summing over all possible paths that the particle could take, with each path weighted by a phase factor proportional to its action. The functional integral is then defined as the limit of a sum of such weighted probabilities, as the number of intermediate points in the path is taken to infinity.
Another example of functional integration is in statistical mechanics, where it is used to calculate the partition function of a system, which determines its thermodynamic properties such as its energy and entropy. The partition function is obtained by summing over all possible states of the system, with each state weighted by its energy and a Boltzmann factor proportional to its entropy.