Energy storage in steady states under cyclic local energy input

We study periodic steady states of a lattice system under external cyclic energy supply using simulation. We consider different protocols for cyclic energy supply and examine the energy storage. Under the same energy flux, we found that the stored energy depends on the details of the supply, period, and amplitude of the supply. Further, we introduce an adiabatic wall as an internal constraint into the lattice and examine the stored energy with respect to different positions of the internal constrain. We found that the stored energy for constrained systems is larger than its unconstrained counterpart. We also observe that the system stores more energy through large and rare energy delivery, comparing to small and frequent delivery.

Figure 1

Scheme of an unconstrained system. A two-dimensional open lattice system, denoted in white in the center, is surrounded by a heat bath at temperature $T_0$ denoted in blue (gray). External energy ${\sigma }_E$ is supplied into the system. Energy currents are denoted as $J_E$.

Figure 2

Scheme of the checkerboard update of the lattice system. The system of interest is inside the thick black lines and is surrounded by two layers of spins representing the heat bath. Black and gray dots of the system denote two group of spins during the checkerboard update. For a system of size $L\times L$, $x,y$ ranges from 0 to $L+3$.

Figure 3

Schemes of the three geometries $A_{1,2,3}$ used for the energy supply. Area of each geometry is shaded in red (light gray). (a) $A_1$ spans the whole system. (b) $A_2$ is square of size $10\times 10$ at the center of the system. (c) $A_3$ is a stripe of size $L\times 4$ in the middle of the system.

Figure 4

Averaged column temperature profiles under energy supply protocol $J_U(A_3,100,1000)$. The whole period is divided into time intervals of 200 MC steps over which the temperature profiles are averaged. The $x$ coordinate of the stripe subjected to the energy supply lies between two vertical dashed lines. Temperature is measured in units of $J/k_{\mathrm{B}}$.

Figure 5

Contour plots of averaged temperature profiles $\overline{T}(\vec{r},\tau )$ under different geometries shown in Fig. 3. Corresponding protocols and averaged fluxes for energy supply are denoted under each graph. Temperature is measured in units of $J/k_{\mathrm{B}}$. The white area in the center of (b) denotes temperature higher than the shown scale.

Figure 6

(a), (c), and (e) Column temperature profiles $\overline{T}(x,\tau )$ under averaged constant energy flux $\overline{J_U}$ for geometries $A_1,A_2$, and $A_3$, respectively. The value of $\overline{J_U}$ per spin is indicated over each panel. The specific values of the $(E_0,\tau )$ pair are denoted for each curve. Temperature is measured in units of $J/k_{\mathrm{B}}$. (b), (d), and (f) Energy storage $\mathrm{\Delta }U$ per spin under constant $\overline{J_U}$ for different geometries.

Figure 7

Scheme of a constrained system. The horizontal line denotes the constraint, which is an adiabatic wall. Energy flux is nonzero in each subsystem.

Figure 8

Row temperature profiles $\overline{T}(y,\tau )$ and the stored energy under $A_1$ geometry, energy supply $\overline{J_U}(A_1,E_0,\tau )=\overline{J_U}(A_1,20,10000)=0.002\times L^2$. Temperature is measured in units of $J/k_{\mathrm{B}}$.

Figure 9

Row temperature profiles $\overline{T}(y,\tau )$ and the stored energy under $A_3$ geometry. Averaged total flux is fixed at $\overline{J_U}(A_3,E_0,\tau )=0.002\times L^2$. Panels (a)–(d) show results for $\overline{T}\left(y\right)$ against $y$ for four pairs of $(E_0,\tau )$. Corresponding energy amplitude and the period are indicated above each graph. Positions of the constraint are denoted with dashed lines. Panel (e) shows the stored energy for different pairs of $(E_0,\tau )$. Horizontal lines correspond to the stored energy without constraint. Temperature is measured in units of $J/k_{\mathrm{B}}$.