Shortcuts of Freely Relaxing Systems Using Equilibrium Physical Observables

Many systems, when initially placed far from equilibrium, exhibit surprising behavior in their attempt to equilibrate. Striking examples are the Mpemba effect and the cooling-heating asymmetry. These anomalous behaviors can be exploited to shorten the time needed to cool down (or heat up) a system. Though, a strategy to design these effects in mesoscopic systems is missing. We bring forward a description that allows us to formulate such strategies, and, along the way, makes natural these paradoxical behaviors. In particular, we study the evolution of macroscopic physical observables of systems freely relaxing under the influence of one or two instantaneous thermal quenches. The two crucial ingredients in our approach are timescale separation and a nonmonotonic temperature evolution of an important state function. We argue that both are generic features near a first-order transition. Our theory is exemplified with the one-dimensional Ising model in a magnetic field using analytic results and numerical experiments.

Figure 1

Staging the anomalous effects. (a) The Mpemba effect. (b) Preheating for faster cooling. (c) Asymmetry of heating and cooling processes.

Figure 2

Validation of $(M_{\mathrm{st}}^2{)}^{\perp }$ as a proxy for the slowest decaying observable ${\mathcal{O}}_2^b$. (a) ${\chi }_{\mathrm{st}}$ as computed for $N\to \infty$ vs $k_{B}T/|J|$, see Eq. (10), has a maximum at $T_b^{*}\approx 4.15$. The curves for $N=8$ and $N\to \infty$ are hardly distinguishable at this scale. (b) For $N=8$, the cosine of the angle between $({\mathcal{M}}_{\mathrm{st}}^2{)}^{\perp }$ and ${\mathcal{O}}_2^b$ is close to 1 for a wide range of bath temperatures. The colored dots in (a) and (b) indicate the three temperatures that we mostly use to demonstrate unconventional OD.

Figure 3

Mpemba effect. Evolution of ${\mathrm{\Delta }}_t[\mathcal{A}]$ [cf. Eq. (11)] for the observables $\mathcal{A}=\mathcal{E}$ (a) and $\mathcal{A}={\mathcal{C}}_1$ (b). We show the results for $N=8$ (blue), $N=12$ (red), and $N=32$ (purple, with a lighter shade representing the error bars of the MC data). The time at which ${\mathrm{\Delta }}_t[\mathcal{A}]$ changes sign is marked by a dot.

Figure 4

Preheating strategy for faster cooling. Evolution of ${\delta }_t[\mathcal{A}]$ [cf. Eq. (12)] for $\mathcal{A}={\mathcal{M}}_{\mathrm{st}}^2$ (a), and $\mathcal{A}={\mathcal{C}}_1$ (b). Colors red and purple (respectively, blue and green) denote that the protocol includes (respectively, does not include) an initial quench at temperature $T_q$. We show data for $N=8$ (dashed lines), $N=12$ (solid lines), and $N=32$ (solid lines with a lighter shade representing the error bars of the MC data). Both panels show the expected speedup for the preheating protocol.

Figure 5

(A)symmetry in heating and cooling. Evolution of ${\delta }_t[{\mathcal{M}}_u]$ [cf. Eq. (12)] for cooling (blue and green) and heating (red and purple) processes. We show data for $N=8$ (dashed lines), $N=12$ (solid lines), and $N=32$ (solid lines with a lighter shade representing the errors of the MC data). The cooling and heating processes were carried out between (a) $T_A=1$ and $T_B=4.15$; and (b) $T_A=15.177$ and $T_B=4.15$. Note that in panel (b), ${\mathbb{E}}_t[{\mathcal{M}}_u]-{\mathbb{E}}^{T_b=T_B}[{\mathcal{M}}_u]$ changes sign at $t\approx 3$ for the system originally at $T_A=15.177$. It is always slower to approach $T_B=4.15$: i.e., to heat up in (a), and to cool down in (b).