Control and optimisation of irreversible processes in non-equilibrium systems

TL;DR

The thesis develops a comprehensive framework for controlling irreversible processes in non-equilibrium systems. By combining stochastic-dynamics formalisms (Fokker-Planck/Langevin) with optimal-control techniques (Pontryagin’s Maximum Principle and inverse engineering), it identifies two major directions: (i) the emergence and global stability of long-lived non-equilibrium attractors (LLNES) that drive glassy relaxation and memory effects (Mpemba, Kovacs) in molecular fluids and granular gases, including universal scaling laws and Lyapunov-type H-theorems; and (ii) the design of shortcuts between stationary states via heat-bath protocols in overdamped harmonic dynamics, yielding Bang-bang optimal controls and explicit finite/infinite heating-power limits. The work shows LLNES as a robust, universal attractor across isotropic and anisotropic confining potentials and extends to Enskog-Fokker-Planck settings with collisions. It further develops thermal brachistochrones and minimum-time connections for multidimensional oscillators and analyzes the Brownian gyrator, unveiling dimension-dependent speed limits and geometric bounds. Collectively, these results provide principled, scalable strategies to accelerate or tailor relaxation in nanoscale systems, with implications for nanos heat engines and stochastic thermodynamics.

Abstract

This thesis is devoted to the study of physical systems embedded within the field of non-equilibrium statistical mechanics. Specifically, the state of the systems of interest constitutes a stochastic process that can be externally driven by a set of controllable parameters. On the one hand, for systems in contact with a thermal bath, we have studied the emergence of strong memory effects and glassy behaviour upon varying the bath temperature, and how these are related to the existence of non-equilibrium attractors governing the dynamics. On the other hand, for overdamped harmonic systems, we have studied the problem of minimising the connection time between arbitrary stationary (either equilibrium or non-equilibrium) states, by suitably varying either the bath temperature or the stiffnesses of the potential.

Figures (44)

• Figure 1: Qualitative sketch of the Mpemba memory effect, following the thermal approach. The hotter sample A, with initial temperature $T_{i,A}$, is further away from the stationary state at the common bath temperature $T_{\text{s}}$ than the colder sample B, with initial temperature $T_{i,B} < T_{i,A}$. The thermal Mpemba effect emerges when the time evolution of the initially hotter sample (red solid line) overtakes that of the initially colder one (blue dashed).
• Figure 2: Qualitative sketch of the Kovacs effect. Specifically, we plot the time evolution of a physical observable $K$ in the top panel, when submitting the system to the two-jump protocol for the bath temperature shown in the bottom panel. The relaxation from $T_i$ to $T_1$ (dashed line) is interrupted after a waiting time $t = t_w$, when the observable $K$ has its stationary value at the final temperature $T_f$, $K(t_w) = K_{\text{s}}(T_f)$. However, $K(t)$ deviates from $K_{\text{s}}(T_f)$ in a non-monotonic way, by reaching a maximum before returning thereto. The latter shows the need of additional variables in order to completely determine the state of the system.
• Figure 3: Sketch of the hysteretic behaviour for a physical system described in terms of the average energy $E(T)$. Left panel corresponds to the hysteresis cycle: a cooling protocol (blue line) followed by a reheating one (red line)---while the right panel shows the behaviour of the apparent heat capacity $dE/dT$ along the cycle with the same colour code. The black, dashed line in the left panel corresponds to the equilibrium curve $E(T) = E_{\text{eq}}(T)$. The purple, dashed line on the right panel accounts for the temperature $T_g$ where $dE/dT$ reaches a maximum along the heating protocol.
• Figure 4: Time evolution of the scaled kinetic temperature (left panels) and the excess kurtosis (right panels). Three different values of the initial scaled temperature $\theta_i$ are considered: $2$, $10$ and $100$. Additional parameters employed are $d=2$, $\gamma = 0.1$ and $\xi = 1$. Symbols (red triangles) correspond to DSMC data, while curves correspond to the numerical integration of Eqs. \ref{['ch2_eq:first-sonine-edos']} for the first Sonine approximation (dashed lines) and Eqs. \ref{['ch2_eq:second-sonine-eqs']} for the extended Sonine approximation (solid lines).
• Figure 5: Time evolution of the scaled kinetic temperature $\theta$ under the first Sonine approximation for $\theta_i = 2$. Additional parameters employed are $d=2$, $\gamma = 0.1$ and $\xi = 1$. The solid line corresponds to the numerical integration of Eqs. \ref{['ch2_eq:first-sonine-edos']}, while the dashed and dotted ones account for the regular perturbative solution from Eq. \ref{['ch2_eq:regular-perturbative-O2-sol-temp']} and the multiple-scale solution from Eq. \ref{['ch2_eq:relaxation-exponential']}, respectively. Within the inset, we have plotted both solid and dotted lines for different values of $\theta_i$ in logarithmic scale. Specifically, from top to bottom: $\theta_i = 5$ (red), $2$ (blue) and $1.5$ (purple).