Thermal equilibrium control by frequent bang-bang modulation

In this paper, we investigate the non-Markovian heat transfer between a weakly damped harmonic oscillator (system) and a thermal bath. When the system is initially in a thermal state and not correlated with the environment, the mean energy of the system always first increases, then oscillates, and finally reaches equilibrium with the bath, no matter what the initial temperature of the system is. Moreover, the heat transfer between the system and the bath can be controlled by fast bang-bang modulation. This modulation does work on the system, and temporarily inverts the direction of heat flow. In this case, the common sense that heat always transfers from hot to cold does not hold any more. At the long time scale, a new dynamic equilibrium is established between the system and the bath. At this equilibrium, the energy of the system can be either higher or lower than its normal equilibrium value. A comprehensive analysis of the relationship between the dynamic equilibrium and the parameters of the modulation as well as the environment is presented.

Figure 1

$\kappa \left(\tau \right)$ and $\mu \left(\tau \right)$ vs $\tau$ according to Eq. (15). The parameters are set by: $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, $\gamma =0.1{\omega }_0$, and ${\beta }_T{\omega }_0=0.02$. Both $\Omega =1.5{\omega }_0$ (solid line) and $\Omega =2.0{\omega }_0$ (dash line) are presented for comparison.

Figure 2

Dynamics of $\Lambda \left(t\right)$ and $\Delta \left(t\right)$ according to Eqs. (10, 12), respectively. The parameters are set as: $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, $\gamma =0.1{\omega }_0$, $\Omega =1.5{\omega }_0$, and ${\beta }_T{\omega }_0=0.02$.

Figure 3

(Color online) Mean-energy evolution of the system. The parameters are set by: $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, $\Omega =1.5{\omega }_0$, and ${\beta }_T{\omega }_0=0.02$. In figure (a), $\gamma =0.1{\omega }_0$, and the initial temperature of the system is as the same as the one of the environment $\left({\beta }_0\equiv \hslash /\left(k_B{T}_0\right)={\beta }_T\right)$. In figures (b) and (c), $\gamma =0.3{\omega }_0$. In figure (b), different initial temperature of the system is presented, where blue solid line refers to ${\beta }_0=0.022$ (initial colder), green dash line refers to ${\beta }_0=0.02$, and dot red line refers to ${\beta }_0=0.018$ (initial hotter). The oscillation in figure (b) (the square region) is enlarged and presented in figure (c). The equilibrium value [Eq. (25)] is shown in figure (c) by the purple dot horizontal line. The square region in (c) is enlarged in the inset, in which we can see that the mean system energy fall below the equilibrium value even when the system is initially hotter than the environment (red dot line).

Figure 4

A sketch of the bang-bang strategy. An analytical description is given in Eq. (29).

Figure 5

Dynamics of ${\Delta }^{\prime }\left(t\right)$ and ${\Lambda }^{\prime }\left(t\right)$. The parameters are set by $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, $\gamma ={\omega }_0/10$, $\Omega =3{\omega }_0/2$, $ϵ=5\pi /2$, and ${\beta }_T{\omega }_0=0.02$. The time separation between every two consecutive squares is $0.05/{\omega }_0$. In figures (a) and (b), we set ${\tau }_d=0.8$ and ${\tau }_u=0.2$, and therefore $\delta \phi =ϵ{\omega }_0{\tau }_u=\pi /2$. In figures (c) and (d), we set ${\tau }_d=0.8$ and ${\tau }_u=0.6$, and thus $\delta \phi =ϵ{\omega }_0{\tau }_u=3\pi /2$. When $\delta \phi =3\pi /2$, we notice that ${\Delta }^{\prime }\left(t\right)$ instantaneously becomes negative when the modulation is on. However, when the modulation is turned off, it turns positive as predicted in Table .

Figure 6

(Color online) Anomalous heating and cooling. The parameters are set by $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, $\gamma ={\omega }_0/10$, $\Omega =3{\omega }_0/2$, $ϵ=5\pi /2$, ${\tau }_d=0.8/{\omega }_0$, and ${\beta }_T{\omega }_0=0.02$. In figure (a), the initial temperature of the system is as the same as the one of the environment $\left({\beta }_0{\omega }_0={\beta }_T{\omega }_0=0.02\right)$. When ${\tau }_u=0.2/{\omega }_0$ (solid blue line), $\delta \phi =ϵ{\omega }_0{\tau }_u=\pi /2$. So anomalous cooling takes place and the system is cooled down. When ${\tau }_u=0.6$ (dot red line), anomalous heating happens and the system is heated up. The case without BB is plotted (dash green line) for comparison. In figure (b), the system is initially hotter than the environment $\left({\beta }_0{\omega }_0=0.01\right)$. With the help of anomalous cooling (solid blue line), cooling is much more efficient than in the normal case (dash green line).

Figure 7

Net work done on the system vs $t_{-}$ [see Eq. (45)]. The parameters are set by $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, $\gamma ={\omega }_0/10$, $\Omega =3{\omega }_0/2$, $ϵ=5\pi /2$, ${\tau }_d=0.8/{\omega }_0$, and ${\beta }_T{\omega }_0=0.02$. For the heating case (square), ${\tau }_u=0.6/{\omega }_0$ and $\delta \phi =3\pi /2$; while for the cooling case (triangle), ${\tau }_u=0.2/{\omega }_0$ and $\delta \phi =\pi /2$.

Figure 8

(Color online) The relation between $\overline{H}$ [Eq. (48)] and the parameters of BB (Fig. 4): $ϵ$, ${\tau }_u$, and ${\tau }_d$. The parameters are set by $\hslash =1$, $\alpha =1$, $m_0=1$, ${\omega }_0=1$, ${\beta }_T{\omega }_0=0.02$, and $\gamma /{\omega }_0=0.1$. Two cases with $\Omega =1.5{\omega }_0$ (blue solid line or blue square) and $\Omega =2.0{\omega }_0$ (red dash line or red triangle) are presented for comparison. It is obvious that $\Omega$ has litter influence on cooling equilibrium while the heating equilibrium is significantly affected. In figure (a), $\overline{H}$ oscillates as $ϵ$ increases. Here we set ${\tau }_d=0.8$ and ${\tau }_u=0.2$. Especially, when $ϵ{\omega }_0{\tau }_u$ is around $3\pi /2$, no dynamic equilibrium exists and the system is persistently heated up. In figure (b), where ${\tau }_d=0.8/{\omega }_0$ and $ϵ=5\pi /2$, $\overline{H}$ oscillates as ${\tau }_u$ increases. In figure (c), $\overline{H}$ gradually approaches $H_E$ with the increase in ${\tau }_d$. We set ${\tau }_u=0.4/{\omega }_0$, and $ϵ=5\pi /4$ for the colder case [two bottom lines which overlap each other in figure (c)] and $ϵ=15\pi /4$ for the hotter case (two upper lines).