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.

Figures (6)

• Figure 1: Geometry of the Chaplygin sleigh. The center of mass is at point $M:(X,Y)$. The standard NHC at $P$ requires the velocity $u$ perpendicular to $PM$ to be zero. The inset shows the way in which a "sail" (in red) is included, to exchange energy with the particles of a thermal bath.
• Figure 2: Phase portrait of the Chaplygin sleigh.
• Figure 3: Average velocity in numerical solutions of Eqs. (\ref{['2l']}) and (\ref{['2f']}) (points), and analytical solution [Eq. (\ref{['vanal']})] (continuous line). Parameters are $I/ma^2=1$, $\widetilde{T}\equiv mk_BT/a^2\lambda_\perp^2$ as indicated. We see how the typical velocity increases as $\lambda_\parallel/\lambda_\perp$ is reduced, and diverges as $(\lambda_\parallel/\lambda_\perp)^{-1}$ (dotted line has slope $-1$).
• Figure 4: Evolution of the translational kinetic energy $E_c\equiv mv^2/2$ from numerical simulations of Eqs. (\ref{['2l']}) and (\ref{['2f']}), with $\lambda_\parallel=0$ for two different values of the dimensionless temperature $\widetilde{T}\equiv mk_BT/a^2\lambda_\perp^2$, as indicated (see Appendix \ref{['AppC']} for numerical details). The initial condition is $v=\omega=0$, and $I/ma^2=1$. Thin continuous lines are the results for five individual realizations, for each temperature. Thick continuous line is the result of averaging over 1000 realizations. Dotted lines show the asymptotic analytic form from Eq. (\ref{['analytic']}).
• Figure 5: (a) Time evolution of $v(t)$ from single realizations of Eqs. (\ref{['uf']}) and (\ref{['uf2']}) (initial condition $u=v=\omega=0$), at different values of $T_\Lambda$, and with $\Lambda/\lambda_\perp=100$, $I/ma^2=1$, and $mk_BT/a^2\lambda_\perp^2=1$. (b) Asymptotic values of $\langle v\rangle$ as a function of $T/T_\Lambda$, and for different values of $\Lambda/\lambda_\perp$. Continuous straight line corresponds to $m\langle v\rangle/a\lambda_\perp=T/T_\Lambda-1$, and it is a guide to the eye.