Refined bounds on energy harvesting from anisotropic fluctuations

We now explain the inherently geometric nature of the problem. Firstly, dissipation can be expressed as

where $\alpha(t)=\{(r(t),\theta(t),z(t)):\,t\in[0,t_{f}]\}$ denotes a trajectory on the $\Sigma$-space and $\left\lVert\cdot\right\rVert^{2}_{g}$ denotes the square norm of a vector with respect to the metric

Note that when $z$ remains constant, the metric in [5] is recovered. By the Cauchy-Schwarz inequality, the minimal dissipative heat for any trajectory is given by

where $\ell$ is the length of the path with respect to $g$ on the $\Sigma$-space, which coincides with $\mathcal{Q}_{\rm diss}$, and hence, with the $2$-Wasserstein length of the cycle of thermodynamic states [9]. The minimum is attained when $\left\lVert\dot{\alpha}(t)\right\rVert_{g}$ remains constant along the path.

Second, the quasi-static heat can be written as a weighted surface integral over a domain $\mathcal{D}$, precisely as shown in [5]. Note however, that $\mathcal{D}$ is no longer the domain encircled by a path drawn on some two-dimensional submanifold in the $\Sigma$-space, but instead, it is the area enclosed by the projection of the cycle onto a plane that corresponds to a constant value for $z$.

Following [5], by means of Stokes’ theorem, quasi-static heat from (7) gives

where the $\pm$ sign depends on the direction chosen, and

is an area integral with respect to the Riemannian canonical $2$-form of the metric $g$ and the work density function

that results in $\mathcal{A}_{h}$ being independent of $z$.

Thus, the problem to maximize work extraction $\mathcal{W}_{\text{out}}$ along a cycle, namely,

for a given $\mu=\frac{\gamma}{k_{B}\Delta Tt_{f}}$, that can be interpreted as a Lagrange multiplier, amounts to maximizing the area integral $\mathcal{A}_{h}$ enclosed by a cycle of fixed length $\ell$, as in [5]. Since $\mu$ acts as a penalty on the length of the cycle, it is clear that larger values of $\mu$ lead to smaller values for the optimal length $\ell$. On the other hand, efficiency (see (5)) can also be expressed in geometric terms as

with the problem to maximize efficiency along a cycle of arbitrary length turning into a search for an isoperimetric-like inequality in the space of thermodynamics states. That is, maximizing efficiency amounts to seeking

We remark that, due to the exponential term $e^{z/2}$ in the expression for the metric $g$, increasingly negative values of $z$ (and thus, increasingly negative entropy) result in a vanishingly small dissipation. However, such a tight confinement requires an arbitrarily strong potential. To ensure physically meaningful conditions, we specify a starting value for $z$ along the cycle, which amounts to specifying the entropy at that point.

To gain intuition on the shape of the optimal curves, we perform a local analysis, valid when $\ell\to 0$. To this end, we consider cycles of infinitesimally small length around an operating point $(r_{0},\theta_{0},z_{0}),$ which are optimal for large enough values of the parameter $\mu$.

Since the work density (11) is proportional to $\sin(\theta)$ and $g$ is independent of $\theta$, the choice $\theta_{0}=\tfrac{\pi}{2}$ maximizes $\mathcal{A}_{h}$ locally. Without loss of generality we also fix $z_{0}=0$ (corresponding to $\det(\Sigma)=1$) and consider closed paths $\{(r(t),\theta(t),z(t)\mid t\in[0,t_{f}]\}$ with

for $r_{0}$ to be determined with $\epsilon>0$ assumed small. Up to $o(\epsilon^{2})$, the quasi-static and dissipative heat are

where $c_{0}=\cosh(r_{0})$ and $s_{0}=\sinh(r_{0})$. Note that $\mathcal{Q}^{\epsilon}_{\rm qs}$ can be written as an area integral as before, or as the line integral above.

We now consider maximizing work extracted, namely,

over a small cycle about $(r_{0},\theta_{0}=\frac{\pi}{2},z_{0}=0)$. The Euler-Lagrange equation (first order necessary condition for optimality) for the functional in (14) with respect to the $z$-coordinate gives

Integrating over time gives $\dot{z}_{1}=-2\frac{s_{0}}{c_{0}}\dot{r}_{1}+\rm const.$ This constant however must be equal to zero on periodic orbits. Substituting $\dot{z}_{1}=-2\frac{s_{0}}{c_{0}}\dot{r}_{1}$ into (13), we obtain that

is quadratic in the $2$-dimensional velocity vector $(\dot{r},\dot{\theta})$ for the metric (cf. (8))

As observed in the geometric analysis section, instead of Problem (14) we may instead consider the problem to maximize $\mathcal{Q}^{\epsilon}_{\rm qs}$ subject to $\mathcal{Q}^{\epsilon}_{\rm diss}$ in (16) being specified. To this end, cf. (9), we set

Our problem now becomes

where the constraint (17) has been included in the Lagrangian $\mathcal{L}$ with Lagrange multiplier $\lambda$. The corresponding Euler-Lagrange equations are

giving

with $\omega=\tfrac{2\pi}{t_{f}}=\tfrac{s_{0}}{2\lambda c_{0}}$. Since we are interested in the complete (closed) orbit, we may take $B=0$. Then, $A=\tfrac{\ell_{\epsilon}\sqrt{c_{0}}}{2\pi s_{0}}$ from (17), and we obtain the equations of an ellipse

for $t\in[0,t_{f}]$.

Up to $\epsilon^{2}$, the quasi-static heat and dissipation are

respectively, giving

Thus, the efficiency can be expressed as

Here, the subscript $3\rm d$ refers to the fact that optimization takes place in the $3$-dimensional parameter space, of coordinates $(r,\theta,z)$. The efficiency is seen to be greater than that obtained over cycles of constant entropy ($z=\rm constant$), when it was found that $\eta^{\epsilon}_{2\rm d}=1-4\pi\mu\,\frac{c_{0}^{2}}{s_{0}}$[5]. It was further shown in [5] that $\eta_{2\rm d}^{\epsilon}=1-8\pi\mu$ is maximal (achieved for $r_{0}=\text{asinh}(1)$). Instead, $\eta^{\epsilon}_{3\rm d}$ increases as $r_{0}\to\infty$ towards the limit $1-4\pi\mu$, see Figure 2.

Equation (19) implies an inherent speed limit, in that for

it is impossible to extract work ($\eta_{3\rm d}^{\epsilon}<0$). A corresponding speed limit for isentropic cycles obtained in [5], $t_{f}<\tfrac{4\pi\gamma c^{2}_{0}}{s_{0}k_{B}\Delta T}$, differs by a factor of $c_{0}$.

For isentropic cycles, it was inferred in [5] that the efficiency - and thus the ratio of the weighted area and the $2$-Wasserstein length of the cycle squared - was maximized as $\ell\to 0$. In the present case of cycles where the entropy is allowed to vary, it appears that the same is true, and thus we presume the efficiency to be bounded by $\eta\,\leq 1-4\pi\,\frac{t_{c}}{t_{f}}$, where $t_{c}=\mu t_{f}$ is a characteristic time, pointing towards a more general isoperimetric inequality and speed limit. Numerical experiments appear to back this hypothesis but a formal proof is lacking.

Both limits $\ell\to 0$ (with $t_{f}$ fixed) and $t_{f}\to\infty$ (with $\ell$ fixed) represent quasi-static operation, for which the cycle is traversed arbitrarily slowly, leading to vanishing dissipation. Remarkably, these two scenarios are distinctly different. The limit of arbitrarily slow operation is achieved by lengthening the time to complete the cycle in one case, and by shrinking the path to be traversed in the other. This makes the efficiency different in the two limits. Specifically, as $t_{f}\to\infty$, $\eta\to 1$, as the process becomes quasi-static in the traditional sense. In the other case, as $\ell\to 0$ (with a finite $t_{f}$), dissipation vanishes at the same rate as the quasi-static work, leading to a negative contribution to the efficiency and $\eta\to 1-8\pi\mu<1.$

We now consider general cycles of finite lengths $\ell$, that are in correspondence with values of $\mu$. Since the Riemannian metric $g$ can no longer be assumed constant, closed-form expressions for the optimal cycles are not feasible, and the cycles are constructed numerically. Specifically, we solve (12) numerically by fixing the length $\ell$ of the cycle, interpreting $\mu$ as a Lagrange multiplier. To this end, we implement gradient descent on the space of functions $(r(t),\theta(t),z(t))$ to determine cycles that maximize work extraction for different lengths.

The discrepancy between a numerically obtained optimal cycle and its local approximation is negligible for small $\ell$ (corresponding to large $\mu$), as highlighted in Figure 3. Note that the coordinates $x=r\cos\theta$ and $y=r\sin\theta$ have been used as axes (in accordance with their portrayal in Figure 1). As depicted in the figure, the approximation is precisely the intersection of a cone with an elliptic cylinder (18); the equation of the cone $z_{1}=-2\tanh(r_{0})r_{1}+\,\rm const.$ follows from (15). Due to the scale of the figure (range of $y$-values far from the origin), the slice of the cone appears as planar.

For larger values of $\ell$, optimal cycles are depicted in Figure 4. The dashed curves on the left plot outline the local approximations of optimal cycles. These are in surprisingly good agreement with the exact numerical solutions (solid curves). The subfigure on the right displays the side-view of optimal cycles, lying on the generatrix of the cone with slope $-2\tanh(r_{0})\simeq-2$ (since here, $r_{0}$ is large).

The thermodynamic efficiency of heat engines is maximal in the quasi-static limit ($t_{f}\to\infty$), a regime with vanishing power output. Indeed, quantifying the power that an engine is capable of has motivated this and earlier studies. In the regime where power is maximal, it is also of interest to quantify efficiency.

In the present context, for a specified thermodynamic cycle (hence, with $\mathcal{A}_{h},\ell$ given), the power output

is maximized for $t_{f}=\frac{2\gamma\ell^{2}}{k_{B}\Delta T\mathcal{A}_{h}}$, so that power and efficiency (defined in (5)) become

These depend on $\mathcal{A}_{h},\ell$ and apply to all cycles (traversed in constant speed on the Wasserstein manifold, as explained earlier). The efficiency at maximum power $\eta^{*}$ matches the universal linear-response bound of $1/2$ [10, 11].