Tipping of a classical point mass pendulum: Role of statistical fluctuations

Abstract

The behavior of a stationary inverted point mass pendulum pivoted at its lower end in a gravitational potential is studied under the influence of statistical fluctuations. It is shown using purely classical equations that the pendulum eventually tips over i.e evolves out of its initial position of unstable equilibrium, and, in a finite amount of time points down assuming a position of stable equilibrium. This `tipping time' is calculated by solving the appropriate Fokker- Planck equation in the overdamped limit. It is also shown that the asymptotic time solution for probability corresponds to the Boltzmann distribution, as expected for a system in stable equilibrium, and that the tipping time tends to infinity as the parameter corresponding to the strength of thermal fluctuations is tuned to zero, thereby defining the limit where one recovers the classical result that a stationary inverted point mass pendulum never tips over. The paper provides a unique perspective showing that phenomena like tipping that have been often attributed to quantum mechanics can be studied even in the domain of purely classical physics.