Anomalous Cooling and Overcooling of Active Colloids

The phenomenon that a system at a hot temperature cools faster than at a warm temperature, referred to as the Mpemba effect, has recently been realized for trapped colloids. Here, we investigate the cooling and heating process of a self-propelled active colloid using numerical simulations and theoretical calculations with a model that can be directly tested in experiments. Upon cooling, activity induces a Mpemba effect and the active particle transiently escapes an effective temperature description. At the end of the cooling process the notion of temperature is recovered and the system can exhibit even smaller temperatures than its final temperature, a surprising phenomenon which we refer to as activity-induced overcooling.

Figure 1

(a) Cooling and heating scenarios, where each colored arrow represents a cooling or heating process. The arrows’ length displays the time that the system needs to cool down (left) or heat up (right). (b) Asymmetric potential with walls $U(x)$ as a function of space that the active colloid (blue circle) is subjected to. (c) Occupation probability $\pi (x)$ of an active (black line) and passive (green line) colloid in the external potential shown in (b). (d) An overcooling scenario, where during the cooling process the system escapes an effective temperature description (black represents the regime without a temperature definition) and then goes to lower temperatures than the final temperature. The color code of the arrow represents the temperature.

Figure 2

(a)–(c) Probability distributions of active colloids. A hot (red dotted line, $T_{\mathrm{hot}}=2500T_{\mathrm{cold}}$), a warm (blue dashed line, $T_{\mathrm{warm}}=50T_{\mathrm{cold}}$), and cold (black solid line) colloid is shown. (a) Initial steady state distributions at $t=0$. A temperature quench is applied at this time. (b) Distributions at $t=3\times {10}^{-3}{\tau }_D$. The hot colloid has relaxed close to $T_{\mathrm{cold}}$. (c) Distributions at $t=50\times {10}^{-3}{\tau }_D$. The warm colloid has also relaxed close to $T_{\mathrm{cold}}$. (d) Relaxation dynamics of the “distance” $\mathcal{D}(t)$ measured by the partitioning between probability distributions and the steady state distribution of a cold colloid for an initially hot, warm, and cold colloid. [Parameters: $F_0/(\gamma v_D)=1.5\times {10}^5$, $F_B/(\gamma v_D)=24$, $x_{\min }/l=-1/3$, $x_{\max }/l=2/3$, $v_0/v_D=120$, ${\tau }_p/{\tau }_D=2\times {10}^{-4}$.].

Figure 3

(a) Cooling time as a function of initial temperature $T$ to final temperature $T_{\mathrm{cold}}$ for passive (gray circles) and active (orange diamonds, $v_0/v_D=3.9\times {10}^3$, ${\tau }_p/{\tau }_D=5\times {10}^{-4}$) particles. (b) Cooling state diagram as a function of self-propulsion and persistence showing normal cooling and the active Mpemba effect, where the black crosses show theoretical transition points. [Parameters (a)-(b): $F_0/(\gamma v_D)=1.5\times {10}^5$, $F_B/(\gamma v_D)=24$, $x_{\min }/l=-1/2$, $x_{\max }/l=1/2$.] (c) Heating time as a function of initial temperatures $T$ to final temperature $T_{\mathrm{hot}}$ for active (green triangles, $v_0/v_D=9$, ${\tau }_p/{\tau }_D=1.2$) and passive (purple pentagons) particles. (d) Heating state diagram for varying self-propulsion and persistence, where the black crosses show theoretical transition points. [Parameters (c)–(d): $F_0/(\gamma v_D)=450$, $F_B/(\gamma v_D)=0.054$, $x_{\min }/l=-2/9$, $x_{\max }/l=7/9$.].

Figure 4

(a) Effective temperature $T_{\mathrm{eff}}$ as a function of time for an active (purple solid line, $v_0/v_D=150$, ${\tau }_p/{\tau }_D=2.7\times {10}^{-4}$) and passive (grey dashed line) particle. The initial temperature corresponds to the hot particle in Fig. 2. At intermediate times, the effective temperature description breaks down. Inset: Effective temperature $T_{\mathrm{eff}}$ as a function of time extracted from our theoretical approach [Inset parameters: $F_B/(\gamma v_D)=4.4$, $x_{\min }/l=-0.36$, $x_{\max }/l=0.63$, $v_0/v_D=27.5$, ${\tau }_p/{\tau }_D=1.6\times {10}^{-3}$]. (b)–(c) Probability distributions after time $t$ of the cooling process (orange dash-dotted line), effective distribution ${\pi }_{T_{\mathrm{eff}}}$ (solid blue line) and final steady state (green dotted line). (b) shows a typical state at the beginning of the cooling process and (c) a state where the system is overcooled. (d) State diagram showing overcooling (purple diamonds) and monotonic cooling (grey circles) for varying self-propulsion and persistence. (e) Reduced lag diffusion as a function of time for different lag times (color code, $v_0/v_D=150$, ${\tau }_p/{\tau }_D=2.7\times {10}^{-4}$). [Parameters: $F_0/(\gamma v_D)=1.5\times {10}^5$, $F_B/(\gamma v_D)=15$, $x_{\min }/l=-1/3$, $x_{\max }/l=2/3$.].