Nonholonomic constraints at finite temperature

Abstract

We investigate the behavior of dynamical systems with nonholonomic constraints when coupled to a thermal bath, focusing on the paradigmatic case of the Chaplygin sleigh. A straightforward Langevin-type approach—obtained by naively adding stochastic and dissipative terms to the equations of motion—predicts a regime in which useful work can be extracted, violating the second law of thermodynamics. To resolve this paradox, we resort to a physically motivated implementation of the nonholonomic constraint as the limiting case of a viscous interaction. However, at finite temperature, fluctuation-dissipation relations imply that the viscous force has to be complemented with stochastic forces acting at the contact. We show that their incorporation restores compliance with the second law. Therefore, our results place fundamental limits on the physical realizability of idealized nonholonomic constraints.