Microscopic thermal machines using run-and-tumble particles

Abstract

Microscopic thermal machines that are of dimensions of around few hundred nanometres have been the subject of intense study over the last two decades. Recently, it has been shown that the efficiency of such thermal engines can be enhanced by using active Ornstein–Uhlenbeck particles (AOUP). In this work, we numerically study the behaviour of tiny engines and refrigerators that use an active run-and-tumble particle as the working system. We find that the results for the engine mode are in sharp contrast with those of engines using AOUP, thus showing that the nature of activity has a strong influence on the qualitative behaviours of thermal machines for non-equilibrium cycles. The efficiency of an engine using a run-and-tumble particle is found to be smaller in general than a passive microscopic engine. However, when the applied protocol is time-reversed, the resulting microscopic refrigerator can have a much higher coefficient of performance under these conditions. The effect of variation of different parameters of the coefficient of performance has been explored. A non-monotonic variation of coefficient of performance with active force has been found.

Appendices

Appendix A. Cosine distribution

To generate the distribution, we have used the technique of inverse transform sampling (ITS) [42]. The cumulative density function (CDF) for $P(\eta )$ is given by

$$\begin{aligned} F(\eta )&= \frac{1}{4}\int _{-\pi }^\eta \cos \left( \frac{\eta '}{2}\right) ~\mathrm{d}\eta ' = \frac{1}{2}\left[ 1+\sin \left( \frac{\eta }{2}\right) \right] . \end{aligned}$$

(A.1)

Its inverse is given by

$$\begin{aligned} \eta&= 2\sin ^{-1}[2F(\eta )-1]. \end{aligned}$$

(A.2)

Following the method of ITS, we define the random variable in the $i{\mathrm {th}}$ iteration to be

$$\begin{aligned} \eta _i&= 2\sin ^{-1}(2u_i-1), \end{aligned}$$

(A.3)

where $u_i$ is a uniform random number in the range [0,1]. Figure 12 shows that the distribution of $\eta _i$ indeed follows the desired form.

Fig. 12

Distribution of tumble angle $\eta$.

Appendix B. Variance of RT particle in the presence of thermal noise

We follow the analytical technique implemented in [25]. Consider the equation

$$\begin{aligned} m\dot{v}&= -\gamma v + (\sqrt{D})\xi (t) \end{aligned}$$

(B.1)

which is interrupted by tumbles. In the absence of tumble events, the velocity correlation is given by

$$\begin{aligned} \langle v_x(t_1)v_x(t_2)\rangle&= \langle v_y(t_1)v_y(t_2)\rangle = \frac{D}{2m\gamma }\mathrm{e}^{-\gamma |t_1-t_2|/m}. \end{aligned}$$

(B.2)

The variance in the absence of tumble events is therefore given by

$$\begin{aligned} \langle x^2(t)\rangle&= \int _0^t \mathrm{d}t_1 \int _0^t \mathrm{d}t_2 \left<v_x(t_1)v_x(t_2)\right> = \langle y^2(t)\rangle . \end{aligned}$$

(B.3)

The variance in the absence of tumble events is given by

$$\begin{aligned} \sigma (t)&\equiv \langle x^2(t)\rangle + \langle y^2(t)\rangle = \frac{2D}{\gamma ^2}t \nonumber \\&\quad - \frac{2Dm}{\gamma ^3}\left( 1-\mathrm{e}^{-\gamma t/m}\right) . \end{aligned}$$

(B.4)

Next, let us include tumble events. If the probability of a run duration $\tau _r$ is given by $P(\tau _r)$, then the probability that no tumble has taken place in time t is given by

$$\begin{aligned} Q(t)&= 1-\int _0^t P(t')\mathrm{d}t', \end{aligned}$$

(B.5)

so that the modified variance in the presence of tumble events is

$$\begin{aligned} \langle r^2(t) \rangle&= Q(t)\sigma (t) + \int _0^t P(t')[\sigma (t')\nonumber \\&\quad +\langle r^2(t-t') \rangle ]\mathrm{d}t'. \end{aligned}$$

(B.6)

The first term on the RHS is the contribution from runs that are uninterrupted by tumble events, the second term is that of the first run and the third is due to subsequent evolution. If we define the function

$$\begin{aligned} f(t)&= Q(t)\sigma (t) + \int _0^t P(t')\sigma (t')\mathrm{d}t', \end{aligned}$$

(B.7)

then performing Laplace transform on both sides, we get

$$\begin{aligned} {\tilde{\sigma }}(s)&= \frac{{\tilde{f}}(s)}{1-{\tilde{P}}(s)}. \end{aligned}$$

(B.8)

Finally, performing the inverse Laplace transform, we obtain eq. (9).

Appendix C. Velocity correlations

Intuitively, we would expect a partial reorientation to correspond to a higher activity compared to the total reorientation, as the former imparts a stronger persistence in the motion of the particle. In order to verify this, we have plotted the velocity correlations along x-direction $\langle v_x(0)v_x(t)\rangle$ obtained for a particular set of parameters (as mentioned in the figure caption) in figure 13 as a function of time t. Figure 13a compares the velocity correlations for 1d and 2d RT particles, both for total and partial reorientations. As expected, the correlations are higher for partial reorientation in each case. Figure 13b compares the velocity correlations of the 1d and 2d particles undergoing partial reorientations, with those of the corresponding passive particles. The correlations are found to be generally weaker for the RT particles, due the degradation in the correlations caused by tumble events.

Fig. 13

(a) Velocity correlations $\langle v_x(0)v_x(t)\rangle$ as a function of time for a 1d and a 2d RT particle in the case of total and partial reorientations. The parameters used are: $m=0.2,~\tau _p=0.1,~F=0.1,~D=0.01,~k_0=0$ and (b) velocity correlations for 1d and 2d passive and active particles with partial reorientation. Parameters used are same as above.

Appendix D. Brief description of engine using an active Ornstein–Uhlenbeck particle

We have compared our results with those of [4], where the overdamped AOUP has been used. Such a particle follows a Langevin equation where, in addition to the thermal noise, there is an exponentially correlated noise that leads to persistence in its motion. The equations of motion for the AOUP are given as follows for the expansion and compression steps respectively:

$$\begin{aligned} \gamma \dot{x}&= -k_\mathrm{exp}(t)x + (\sqrt{D_h})\xi (t), \, 0\le t<\tau /2;\nonumber \\ \gamma \dot{x}&= -k_\mathrm{com}(t)x + (\sqrt{D_c})\xi (t) + (\sqrt{D_\eta /\tau _\eta })\eta (t), \nonumber \\& \tau /2\le t <\tau ; \nonumber \\ \tau _\eta {\dot{\eta }}&= -\eta + (\sqrt{2\tau _\eta })\xi _\eta (t). \end{aligned}$$

(D.1)

Here, $\tau _\eta$ is the decay time of the exponentially correlated noise $\eta$:

$$\begin{aligned} \langle \eta (t)\eta (t') \rangle = \mathrm{e}^{-|t-t'|/\tau _\eta }. \end{aligned}$$

(D.2)

The constant $D_\eta$ determines the strength of the noise (for a fixed value of $\tau _\eta$), which we call the active noise strength. The functions $k_\mathrm{exp}(t)$ and $k_\mathrm{com}(t)$, as well as the constants $D_h$ and $D_c$, are the same as given in §2.1.