In various branches of physics, there arises the problem of minimizing a functional F[q(t)] that depends on a t-dependent angular variable q(t), in the presence of a random force f(t). A paradigmatic model for this class of phenomena is provided by balancing an inverted pendulum subjected to a time-dependent horizontal force f(t). The problem is to find an exceptional never-falling trajectory (NFT) q(t) that remains in the upper half-plane for all t. In mathematics, this problem is known as Whitney' s problem and the existence of the solution has been proved for any f(t). The resulting q(t) is a non-Markovian functional of the force f(t), that prohibits the naïve use of the Fokker-Planck evolutionary equation.

We have shown that the solution to the Whitney problem, considered over the entire time axis, is unique. This raises the question of describing the statistical properties of this NFT when averaged over an ensemble of random forces. For the white-noise random force, we construct a complete field-theoretical description of the NFT statistics, based in the Parisi-Sourlas supersymmetric representation. After disorder averaging, the action remains local in time that allows reducing the arising functional integral to an effective supersymmetric quantum mechanics. We show that statistics of the NFT is determined by the zero mode of the emerging Hamiltonian. Employing the BRST symmetry, we reduce it to the Fokker-Planch form. The joint probability distribution function of the angle q(t) and its velocity dq(t)/dt is expressed via the Poisson bracket of the zero mode, which is found analytically in the limits of weak and strong noise. We also determine the Lyapunov exponent, describing the divergence of nearby solutions around the NFT. Finally, we discuss the intuitive way of understanding the supersymmetric wave function and relate it to physical observables.

Biography

Prof. Mikhail A. Skvortsov is a Chief Researcher at the Landau Institute for Theoretical Physics, Russian Academy of Sciences. He earned his academic degrees from the Moscow Institute of Physics and Technology (MIPT), where he also holds a professorial appointment. His research focuses on condensed matter theory, particularly the physics of disordered superconductors, mesoscopic physics, and quantum transport.

Inviter: Chu-Shun Tian