Thermal relaxation asymmetry persists under inertial effects

Abstract

We algebraically prove the asymmetry in thermal relaxation in phase space in the entire range from overdamped dynamics to underdamped dynamics. We show that for the same setup as for overdamped dynamics, even in the more general case of phase-space relaxation, i.e., underdamped dynamics, far-from-equilibrium heating is faster than cooling. Upon isolating the relevant relaxational contribution to the entropy production, we find that the asymmetry persist for underdamped dynamics that are linearly driven out of equilibrium. The coupling of positions and velocities emerging in this generalization further underscores, in a striking manner, the intricate dynamics of such thermal relaxation processes that do not pass through local equilibria. Investigating the overdamped limit, our generalized approach reveals, interestingly, that an excess free energy contribution from the velocity degrees of freedom does not trivially vanish in the overdamped limit, but is instead affected by the precise interpretation of temperature quenches in overdamped systems.

Figures (4)

• Figure 1: (a) Position component of three trajectories (generated via Euler algorithm with $\rmd t=0.005$) and (b) EFE for dynamics in Eq. \ref{['underdamped SDE x 1d']} with $a=m=\gamma=k_BT=1$ and TE temperatures $k_BT_c=0.2$ and $k_BT_h=2.86$. (c) Trajectory of total length $t=1000$ ($\rmd t=0.002$) with time running from dark to bright and (d) EFE for dynamics in Eq. \ref{['underdamped SDE x 2d']} with $a=m=\gamma=k_BT=1$, $\omega=0.9$ and TE temperatures $k_BT_c=0.2$ and $k_BT_h=2.86$. The black line in (b) and (d) denotes $\Delta\mathcal{D}(t) \equiv \mathcal{D}_h(t)-\mathcal{D}_c(t)$ and is scaled up by a factor 3 for improved visibility.
• Figure 2: Trajectory from Fig. \ref{['fig:trajectories_and_EFE']}c shown for several $r$- and $v$-components.
• Figure 3: EFE for the process from Eq. \ref{['underdamped SDE x 1d']} with decreasing values of $m$ and $a=\gamma=k_BT=1$ and TE temperatures $k_BT_c=0.2$ and $k_BT_h=2.86$. Note that (c) and (d) show the same curves with a different left limit on the $x$-axis.
• Figure 4: (a) EFE for the process from Eq. \ref{['underdamped SDE x 1d']} with $a=m=\gamma=k_BT=1$ and TE temperatures $k_BT_c=0.2$ and $k_BT_h=2.86$ alongside projections on the (b) $r$- and (c) $v$-component. (d) Comparison of the curves from (a) (solid lines in (d)) with the sum of the same-color lines in (b) and (c) (dashed lines in (d)) reveal that not all of the two-dimensional relaxation is contained in the one-dimensional marginalizations. The difference is due to off-diagonal terms in $\Sigma(t)$ that occur via Eq. \ref{['solution dynamic Lyapunov']}.