Universal trade-off between power, efficiency, and constancy in steady-state heat engines

TL;DR

This paper proves a universal trade-off in steady-state heat engines showing that achieving high power, efficiency, and constancy simultaneously is impossible, unifying recent ideas to overcome this limitation.

Contribution

It establishes a fundamental bound linking power, efficiency, and constancy in steady-state heat engines, unifying previous approaches.

Findings

Constancy constrains the power-efficiency trade-off.

Universal bound applies to quantum dot solar cells.

Bound also holds for Brownian gyrators.

Abstract

Heat engines should ideally have large power output, operate close to Carnot efficiency and show constancy, i.e., exhibit only small fluctuations in this output. For steady-state heat engines, driven by a constant temperature difference between the two heat baths, we prove that out of these three requirements only two are compatible. Constancy enters quantitatively the conventional trade-off between power and efficiency. Thus, we rationalize and unify recent suggestions for overcoming this simple trade-off. Our universal bound is illustrated for a paradigmatic model of a quantum dot solar cell and for a Brownian gyrator delivering mechanical work against an external force.

Equations

$$P\frac{\eta }{{\eta }_{\mathrm{C}}-\eta }\frac{T_c}{{\Delta }_P}\le \frac{1}{2}$$

$${\Delta }_P\equiv \underset{t\to \infty }{\lim }⟨(P(t)-P{)}^2⟩t.$$

$$\eta \le \frac{{\eta }_{\mathrm{C}}}{1+2P{T}_c/{\Delta }_P}$$

$$\frac{k_{ij}^{c,h}}{k_{ji}^{c,h}}=\exp [(E_i-E_j+b_{ij}^{c,h}{\mu }_{c,h}-f{d}_{ij})/T_{c,h}],$$

$$j_h=\sum _{i<j}(p_i{k}_{ij}^h-p_j{k}_{ji}^h)(E_j-E_i-b_{ij}^h{\mu }_h)$$

$$j_c=-\sum _{i<j}(p_i{k}_{ij}^c-p_j{k}_{ji}^c)(E_j-E_i-b_{ij}^c{\mu }_c)$$

$$j_w=j_h-j_c=P,$$

$$D_{\alpha }\equiv \underset{t\to \infty }{\lim }⟨(j_{\alpha }(t)-j_{\alpha }{)}^2⟩t/2$$

$$\sigma =j_c/T_c-j_h/T_h=j_w({\eta }_{\mathrm{C}}/\eta -1)/T_c.$$

$$\sigma D_{\alpha }\ge j_{\alpha }^2.$$

$$P\le ({\eta }_{\mathrm{C}}-\eta )\eta D_h/T_c$$

$$P\le ({\eta }_{\mathrm{C}}-\eta )\eta D_c/[(1-\eta {)}^2{T}_c],$$

$$k_{0i}^c={\Gamma }_i/[1+\exp (x_i)],k_{i0}^c={\Gamma }_i/[1+\exp (-x_i)]$$

$$k_{lr}^a={\Gamma }_a/[\exp (x_a)-1],k_{rl}^a={\Gamma }_a/[1-\exp (-x_a)],$$

$$P$$

$$j_h$$

$$j_c$$

$$D_w$$

$$D_h$$

$$D_c$$

$$\ln \frac{k_{0l}^c{k}_{lr}^h{k}_{r0}^c}{k_{0r}^c{k}_{rl}^h{k}_{l0}^c}=x_r-x_l-x_h=({\eta }_{\mathrm{C}}\Delta E-\Delta \mu )/T_c,$$

$$d\mathbf{x}/dt=\mu \left[-\partial V/\partial \mathbf{x}+{\mathbf{F}}^{\mathrm{ext}}(\mathbf{x})\right]+\mathbf{\zeta }(t),$$

$$d\stackrel{ˉ}{}w=-d\mathbf{x}\cdot {\mathbf{F}}^{\mathrm{ext}}(\mathbf{x})$$

$$d\stackrel{ˉ}{}{q}_c=d{x}_1[-\partial V(\mathbf{x})/\partial x_1+F_1^{\mathrm{ext}}(\mathbf{x})]$$

$$d\stackrel{ˉ}{}{q}_h=-d{x}_2[-\partial V(\mathbf{x})/\partial x_2+F_2^{\mathrm{ext}}(\mathbf{x})]$$

$$P=j_w=⟨d\stackrel{ˉ}{}w/dt⟩=4P_{\max }\frac{k}{k_s}\left(1-\frac{k}{k_s}\right)$$

$$\eta =j_w/j_h=2k/(c+k)\le {\eta }_C.$$

$$\frac{D_w\sigma }{j_w^2}=\frac{D_c\sigma }{j_c^2}=\frac{D_h\sigma }{j_h^2}=1+\frac{[c{\eta }_C(1-k/k_s){]}^2}{(u_1+u_2{)}^2(1-{\eta }_C)}\ge 1,$$

$$\lambda (z_1,z_2)$$

$$=j_1{z}_1+j_2{z}_2+D_{11}{z}_1^2+D_{22}{z}_2^2+2D_{12}{z}_1{z}_2+\mathcal{O}(z^3)$$

$${\mathcal{L}}_{ij}(z_1,z_2)=k_{ji}^h{e}^{d_{ji}{z}_1}+k_{ji}^c{e}^{d_{ji}{z}_2}-{\delta }_{ij}\sum _{\ell }(k_{i\ell }^c+k_{i\ell }^h)$$

$$j_1=[-k_{l0}^c{k}_{0r}^c{k}_{rl}^h+k_{0l}^c(k_{r0}^c{k}_{lr}^h+k_{rl}^c{k}_{lr}^h-k_{lr}^c{k}_{rl}^h)+k_{rl}^c{k}_{0r}^c{k}_{lr}^h-k_{0r}^c{k}_{lr}^c{k}_{rl}^h]/\mathcal{N},$$

$$j_2=[-k_{l0}^c{k}_{rl}^c{k}_{0r}^c+k_{r0}^c{k}_{0l}^c{k}_{lr}^c+(k_{0l}^c+k_{0r}^c)(k_{lr}^c{k}_{rl}^h-k_{rl}^c{k}_{lr}^h)]/\mathcal{N},$$

Full text

The efficiency $\eta$ of heat engines operating cyclically between a hot heat reservoir at temperature $T_{h}$ and a cold one at $T_{c}$ is universally bounded from above by the Carnot value $\eta_{\mathrm{C}}\equiv 1-T_{c}/T_{h}$. Moreover, it was commonly believed that reaching Carnot efficiency inevitably comes with a vanishing power $P$ since such engines require quasi-static conditions leading to an infinite cycle-time. This supposedly universal trade-off between power and efficiency has more recently been challenged by studies hinting at the possibility to come at least arbitrarily close to Carnot efficiency with finite power by a particular coupling between subsystems Allahverdyan et al. (2013) or by exploiting a working substance at a critical point Campisi and Fazio (2016). On the other hand, a general bound of the type $P\leq A(\eta_{\mathrm{C}}-\eta)$ with a system-specific amplitude $A$ has been proven for cyclic engines both in linear response Brandner et al. (2015) and beyond Shiraishi et al. (2016). These two strands of reasoning can be reconciled only if one allows for a diverging amplitude $A$ as the efficiency approaches the Carnot value.

For steady-state heat engines permanently coupled to two heat baths like in thermoelectric setups, the common argument from above invoking quasistatic conditions and hence an infinite cycle time is not directly applicable. It has usually been replaced with the idea that finite currents necessarily lead to dissipation which should spoil the option of reaching the Carnot limit at finite power. This view was challenged by a seminal paper from Benenti et al. Benenti et al. (2011), who pointed out that if time-reversal symmetry is broken, like in the presence of a magnetic field, the usual approach based on linear irreversible thermodynamics does not forbid Carnot efficiency at finite power. Subsequent studies showed with somewhat more specific assumptions that Carnot efficiency at finite power is not accessible. This holds true, e.g., for any finite number of terminals in a multiterminal set-up within linear response theory Brandner and Seifert (2013, 2015), and for an effective two-terminal device containing inelastic electron-phonon processes Yamamoto et al. (2016). However, as for cyclic engines, it has recently been pointed out that in certain limits Carnot efficiency at finite power can be reached in models of steady-state engines. Specific proposals include two-cycle engines with diverging affinities Polettini and Esposito (2017), a specially designed Feynman-Smoluchowski ratchet Lee and Park (2017), and systems with a singular transport law Shiraishi (2015); Koning and Indekeu (2016).

These observations taken together indicate that when searching for a universal trade-off it may not be enough to focus only on the two characteristics power and efficiency, and their relationship if one of them becomes maximal as it is typically done, see also Curzon and Ahlborn (1975); Van den Broeck (2005); Schmiedl and Seifert (2008); Tu (2008); Esposito et al. (2009); Nakpathomkun et al. (2010); Seifert (2011); Izumida and Okuda (2012); Golubeva and Imparato (2012); de Tomas et al. (2013); Stark et al. (2014); Whitney (2014); Jiang (2014); Raz et al. (2016); Bauer et al. (2016); Proesmans et al. (2016). As we will show in this Letter, as a crucial third quantity, power fluctuations enter quantitatively into this balance. Specifically, we will prove for a huge class of steady-state heat engines, which includes all thermodynamically consistent machines with a classical discrete set of internal states and continuous ones modeled with an overdamped Langevin dynamics, that there is a universal trade-off between three desiderata: Finite (or even large) power, an efficiency close to the Carnot value, and constancy in the sense of small fluctuations in the power output are not compatible. Specifically, the bound

[TABLE]

constrains (mean) output power, efficiency, and the power fluctuations in finite time as measured by

[TABLE]

Here, $P(t)$ is the fluctuating power after time $t$ evaluated in the steady state, for which $\langle...\rangle$ denotes averages. Since the mean output work grows linearly in time as does its variance, converting work fluctuations into power fluctuations requires the additional factor of $t$ to reach a finite limit for $\varDelta_{P}$, with which we characterize the constancy, or stability Holubec (2014), of the engine. In particular for nanoscopic heat engines, power fluctuations due to thermal noise are not negligible compared to the mean work output on relevant time scales and should therefore be taken into account for a thermodynamic description. The crucial role power fluctuations play in the above bound is complementary to their effect in the statistics of efficiency in a finite time Verley et al. (2014); Gingrich et al. (2014); Polettini et al. (2015); Proesmans and Van den Broeck (2015).

As a main first consequence of this new bound for steady-state heat engines, it is obvious that as long as the power fluctuations remain finite (and $T_{c}$ as well), approaching Carnot efficiency implies that the power has to vanish at least linearly. The explicit occurrence of $\varDelta_{P}$ as an amplitude in a putative linear relationship between power and efficiency gap to Carnot, however, offers a second option. If the fluctuations blow up, then a finite (or even diverging) power is not ruled out as $\eta\to\eta_{\mathrm{C}}$ which unifies quantitatively the various observations recalled above.

The bound (1) can be rearranged as a bound on efficiency

[TABLE]

determined by mean and fluctuations of the output power. Thus, the efficiency of any steady-state heat engine is bounded from above by this simple expression independent of the specific design of the engine. The formal similarity of (3) with a bound derived for the efficiency of molecular motors Pietzonka et al. (2016) indicates as common origin of these bounds the thermodynamic uncertainty relation Barato and Seifert (2015); Gingrich et al. (2016), which describes a universal inequality between entropy production and mean and variance of an arbitrary current.

For a proof of the universal trade-off (1), we consider an engine characterized by a set of internal states $\{i\}$ with internal energies $\{E_{i}\}$. A transition between state $i$ and $j$ takes place with a rate $k^{c}_{ij}$ if it is mediated by the contact to the cold bath and with a rate $k^{h}_{ij}$ if it is mediated by the hot bath. One of these rates can be zero which means that this particular transition always involves the other bath. For any nonzero rate, the corresponding backward rate does not vanish either, and it obeys the local detailed balance condition Seifert (2012)

[TABLE]

where we set Boltzmann’s constant to 1 throughout the Letter. Here, $E_{i,j}$ are the internal energies of the two states. If the the transition from $i$ to $j$ requires transport of an electron from the bath with temperature $T_{a}$ $(a=c,h)$ and chemical potential $\mu_{a}$ to the system, then $b_{ij}^{a}=1$. Likewise, if this transition involves the release of an electron to a bath, $b_{ij}^{a}=-1$. In both cases, the chemical potentials enter the expression in the exponent providing a contribution to the total free energy involved in such a transition. The last term in the exponent is non-zero if this transition additionally involves a step of length $d_{ij}$ against an external force $f$. Generalizations to several species of particles, further baths, or the case of rotary motion against an applied torque should be obvious.

In the steady state, where state $i$ is realized with probability $p_{i}$, this engine can be characterized by three important mean currents, the heat current

[TABLE]

from the hot bath to the engine, the heat current

[TABLE]

from the engine to the cold bath and the “work current”

[TABLE]

which, due to the first law, is the power delivered by the engine. Running the engine for a finite time $t$, each of the currents will fluctuate around these mean values with a dispersion 111The fluctuating currents $j_{\alpha}(t)$ are formally defined as $j_{h}(t)=\sum_{i<j}n_{ij}^{h}(t)(E_{j}-E_{i}-b_{ij}^{h}\mu_{h})/t$, $j_{c}(t)=-\sum_{i<j}n_{ij}^{c}(t)(E_{j}-E_{i}-b_{ij}^{c}\mu_{c})/t$, and $j_{w}(t)=\sum_{i<j}[-b_{ij}^{h}n_{ij}^{h}(t)(\mu_{h}-\mu_{c})+(n_{ij}^{h}(t)+n_{ij}^{c}(t))fd_{ij}]/t$, with $n_{ij}^{a}(t)$ being the net number of transitions between $i$ and $j$ mediated by the hot ($a=h$) or cold ($a=c$) bath up to the time $t$.

[TABLE]

where $\alpha=h,c,w$. The mean entropy production rate becomes

[TABLE]

Since this Markovian network is thermodynamically consistent, one can directly apply the thermodynamic uncertainty relation, which reads for any of the three currents Barato and Seifert (2015); Gingrich et al. (2016)

[TABLE]

Evaluating this relation for the work current, $\alpha=w$, leads, with $\varDelta_{P}=2D_{w}$, (7) and (9), to the bound (1).

Two related, but not identical, forms of this bound can be derived similarly by applying the thermodynamic uncertainty relation to either the heat current from the hot bath or the one entering the cold bath. Expressed as bound on power, they read explicitly

[TABLE]

and

[TABLE]

respectively. Obviously, in order to reach a finite value for the power as $\eta\to\eta_{\mathrm{C}}$, the fluctuations in all three currents have to diverge at least $\sim 1/(\eta_{\mathrm{C}}-\eta)$ since each of the three bounds (1,11,12) has to be respected.

For an illustration of these bounds, we consider a simple but instructive model for a solar cell introduced in Ref. Rutten et al. (2009) as shown in Fig. 1a. It consists of a two level quantum dot that can either be empty (state [math]) or contain an electron in one of the levels $E_{l}<E_{r}$ (states $l$ and $r$, respectively). Electrons are transported through the dot from a left reservoir with chemical potential $\mu_{l}$ and temperature $T_{c}$ to a right reservoir with higher chemical potential $\mu_{r}$ and the same temperature $T_{c}$. They enter the level $E_{l}$ from the left reservoir at a rate $k_{0l}^{c}$, and they jump back at a rate $k_{l0}^{c}$. Analogously, the level $E_{r}$ is connected to the right reservoir via the transition rates $k_{r0}^{c}$ and $k_{0r}^{c}$. Transitions between the two states can occur either nonradiatively through interactions with the surrounding phonon bath at temperature $T_{c}$ with rates $k_{lr}^{c}$ and $k_{rl}^{c}$ or are mediated by the black body radiation of the sun with temperature $T_{h}$ at rates $k_{lr}^{h}$ and $k_{rl}^{h}$. The transition rates to and from the two electron reservoirs ($i=l,r$) are chosen according to the Fermi-Dirac distribution as

[TABLE]

with frequencies $\varGamma_{i}$ and affinities $x_{i}=(E_{i}-\mu_{i})/T_{c}$. The transition rates between the two states in contact with the cold phonon bath or hot photon bath ($a=c,h$, respectively) are determined by the Bose-Einstein distribution

[TABLE]

with frequencies $\varGamma_{j}$ and affinities $x_{a}=(E_{r}-E_{l})/T_{a}$. All four pairs of rates satisfy the local detailed balance condition (4).

\floatsetup

[figure]style=plain,subcapbesideposition=top

In a network representation of this system (Fig. 1b), we can identify two independent cycle currents. The variables $X_{1}$ and $X_{2}$ count the net number of electrons transported from state $l$ to state $r$ via radiative and nonradiative transitions, respectively. The total number of transported electrons is thus $X_{e}=X_{1}+X_{2}$ (up to a single electron that might still be in the two-level system, which does not affect the statistics of fluctuations in the long-time limit). The joint fluctuations of these counting variables are characterized by the average currents $\left\langle X_{1,2}(t)\right\rangle=j_{1,2}t$ and their covariance $\left\langle(X_{i}(t)-j_{i}t)(X_{j}(t)-j_{j}t)\right\rangle=2D_{ij}t$ for large $t$, which can be calculated directly from the rates (13) and (14) 222See Supplemental Material (including Refs. Koza (1999); Baiesi et al. (2009); Touchette (2017)) below..

For the mean power, mean heat current from the hot reservoir, and mean heat current into the cold reservoir, one obtains

[TABLE]

respectively, with $\varDelta E\equiv E_{r}-E_{l}$ and $\varDelta\mu\equiv\mu_{r}-\mu_{l}$. For the corresponding dispersion coefficients one gets

[TABLE]

In the “strong coupling” limit of negligible nonradiative transitions ($\varGamma_{c}\to 0$ and hence $X_{2},j_{2},D_{22}$, and $D_{12}\to 0$), the efficiency of the photoelectric device becomes $\eta=\varDelta\mu/\varDelta E$. By gradually increasing the chemical potential difference $\varDelta\mu$ with fixed $\varDelta E$, this ratio can approach $\eta_{\mathrm{C}}$ from below. In Fig. 1c, this limit is realized by increasing the chemical potential $\mu_{r}$ via a scaling parameter $\varOmega$ while keeping $\mu_{l}$, the energies and the temperatures fixed, as detailed in the caption. We show that finite power can be achieved in this limit by increasing the rates $\varGamma_{l}$, $\varGamma_{r}$ and $\varGamma_{h}$ while reducing $\varGamma_{c}$ 333A similar scaling has been used to achieve Carnot efficiency at divergent power output in Ref. Polettini and Esposito (2017).. Crucially, as predicted by relation (1), the fluctuations of the output power diverge in such a scenario.

The uncertainty relation (10) becomes tight in the linear response limit for unicyclic networks Barato and Seifert (2015). Hence, for strong coupling and vanishing cycle affinity

[TABLE]

as realized for a large scaling parameter $\varOmega$ in Fig. 1c, all three bounds on the power output (1,11,12) saturate. Beyond the linear response and strong-coupling limit, the bounds can become weaker. As explored in the Supplemental Material Note (2), depending on the values of the transition frequencies, each of the three bounds can become the strongest one demonstrating that these are independent bounds.

Since the thermodynamic uncertainty relation (10) has also been proven for overdamped Langevin dynamics Polettini et al. (2016); Gingrich et al. (2017), the resulting trade-offs apply as well to steady-state heat engines modeled on such a continuous state space. For an analytically solvable illustration, we consider the so-called Brownian gyrator Filliger and Reimann (2007), which has recently been realized experimentally in an electronic Chiang et al. (2017) and in a colloidal system Argun et al. (2017). The setup, shown in Fig. 2a, consists of a point particle with mobility $\mu$ in two dimensions and an anisotropic harmonic potential $V(\boldsymbol{x})=(u_{1}x_{1}^{2}+u_{2}x_{2}^{2})/2+cx_{1}x_{2}$ with $u_{1,2}>0$ and $0<c<\sqrt{u_{1}u_{2}}$. Its motion obeys the Langevin equation

[TABLE]

where the noise term has mean $\left\langle\boldsymbol{\zeta}(t)\right\rangle=0$ and correlations $\left\langle\zeta_{i}(t)\zeta_{j}(t^{\prime})\right\rangle=2\mu T_{i}\delta_{ij}\delta(t-t^{\prime})$, i.e., the two components of the fluctuating force are associated with heat reservoirs at two different temperatures $T_{i}$. We choose $T_{1}\equiv T_{c}<T_{2}\equiv T_{h}$. In the established case with $\boldsymbol{F}^{\mathrm{ext}}(\boldsymbol{x})=0$, the coupling $c\neq 0$ in the potential mediates a steady transfer of heat from the hot to the cold reservoir, that is accompanied by a persistent directed gyration of the particle Filliger and Reimann (2007). In order to extract mechanical work from this gyration, we load the engine with a nonconservative external counter force $\boldsymbol{F}^{\mathrm{ext}}(\boldsymbol{x})$. As the simplest, linear realization of such a force, we choose $\boldsymbol{F}^{\mathrm{ext}}(\boldsymbol{x})=(F^{\mathrm{ext}}_{1},F^{\mathrm{ext}}_{2})^{T}=k\,(x_{2},-x_{1})^{T}$ with a parameter $k$. Following the rules of stochastic thermodynamics Seifert (2012), along an individual trajectory we identify the work

[TABLE]

performed against the external force, the heat

[TABLE]

dissipated into the cold heat bath, and the heat

[TABLE]

extracted from the hot heat bath. The corresponding integrated currents follow from these differentials through the Stratonovich integration scheme.

The average mechanical power can be calculated as Note (2)

[TABLE]

with the stall-parameter $k_{s}\equiv c\eta_{C}/(2-\eta_{C})$ and the maximal power $P_{\mathrm{max}}\equiv\mu k_{s}^{2}(T_{c}+T_{h})/(2(u_{1}+u_{2}))$. The system operates as a heat engine delivering mechanical work for $0<k<k_{s}$. Its efficiency is then given by

[TABLE]

The diffusion coefficients $D_{w,c,h}$ of the three currents can be calculated analytically as well. It turns out that the uncertainty relation (10) becomes the same for each current,

[TABLE]

with the entropy production rate $\sigma$ as defined in Eq. (9). The resulting bounds on the power and efficiency are shown in Fig. 2b. These bounds become strong when the ratio $D_{w}\sigma/j_{w}^{2}$ in Eq. (28) is close to one, as it is the case for moderate temperature differences or small couplings $c$ with respect to the parameters $u_{1,2}$. For $k$ close to $k_{s}$, the bounds on both power and efficiency become tight in linear order in $(k_{s}-k)$ independently of all other parameters. Because of a tight coupling between heat and work currents, this regime corresponds to a weak driving of the system within the linear response regime, for which the uncertainty relation can generally be saturated. ”For $k\to k_{s}$, as the efficiency approaches the Carnot limit, the power vanishes. Since the constancy of the power remains finite, this typical behavior of a heat engine can be interpreted as a consequence of the universal relation (1). The unusual case of finite power close to Carnot efficiency can yet be obtained by scaling the mobility $\mu$, which comes at the cost of diverging power fluctuations Note (2).

In summary, we have derived a bound providing a universal trade-off between power, efficiency and constancy, i.e., fluctuations in the power output. A finite (or even diverging) power as Carnot efficiency is approached necessarily requires the latter to diverge. The three versions of the bound hold beyond the linear response regime of a small temperature difference between the two heat reservoirs. We have derived these results for steady-state engines described by a thermodynamically consistent Markovian dynamics on a discrete state space and for overdamped Langevin dynamics. They also apply to apparently non-Markovian heat engines, provided that there is a sufficiently fine-grained level of description on which the system obeys a Markovian stochastic dynamics, however complex this description might be. The generalization to underdamped Langevin dynamics might be somewhat more subtle.

One might expect that similar results could be derived for cyclic, i.e., periodically driven heat engines Schmiedl and Seifert (2008) that can be experimentally realized for colloidal systems Blickle and Bechinger (2012); Martínez et al. (2016) and with active baths Krishnamurthy et al. (2016). In particular, a certain formal analogy of (1) with the bound derived in Shiraishi et al. (2016) for cyclic engines is striking. However, the analysis of a periodically driven Brownian clock in Barato and Seifert (2016) shows that the steady-state bound (10) relating entropy production, mean current, and dispersion cannot naively be extended to periodically driven systems. Therefore, it remains an exciting open question whether there are periodically driven heat engines that beat the, for steady-state engines universal, bound (1).

We have focused on the question whether Carnot efficiency can be reached with finite power. More generally, one can ask for any given class of machines whether the power at the maximal efficiency, which may be less than Carnot (or less than 1 for isothermal machines), can be bounded or shown to vanish. Exploring this issue using the techniques introduced here will be left to future work. Finally, it will be interesting to investigate whether and how constancy enters bounds for genuine quantum heat engines that exploit coherences—see Brandner et al. (2017) and references therein.

Supplemental Material: Universal trade-off between power, efficiency,

and constancy in steady-state heat engines

I Quantum dot solar cell

The joint fluctuations of the counting variables $X_{1}$ and $X_{2}$ give rise to the scaled cumulant generating function

[TABLE]

with the mean currents $j_{1,2}=\left\langle X_{1,2}\right\rangle/t$ and the symmetric diffusion matrix $D$. Technically, this function follows as the maximal eigenvalue of the matrix

[TABLE]

with $i,j\in\{0,l,r\}$, $d_{lr}=-d_{rl}=1$ and $d_{ij}=0$ otherwise. The cumulants in the expansion (S.1) can then be calculated using the standard methods explained in Refs. Koza (1999); Baiesi et al. (2009) as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

with

[TABLE]

The cumulants of heat and work then follow from the expressions Eqs. (15)-(20) in the main text.

The tightness of the resulting bounds on power (1), (11) and (12) are explored in Fig. 3 as a function of the two model parameters $\varGamma_{c}$ and $\varGamma_{h}$. Each of the three bounds can become the tightest for some combination of parameters, showing that the bounds are independent for this model. The ratio to the actual power becomes small for small $\varGamma_{c}$, when nonradiative transitions between the two levels are suppressed, such that a tight coupling between heat and work is established.

II Brownian gyrator

The Langevin equation for the Brownian gyrator, (22) in the main text, can be written as

[TABLE]

with the non-symmetric matrix

[TABLE]

subsuming the potential force and the non-conservative, external force, the Einstein relation $\boldsymbol{D}\equiv\mu\operatorname{diag}(T_{c},T_{h})$, and a two-dimensional standard Wiener process $\boldsymbol{W}(t)$. In the following, we re-scale time such that $\mu=1$. The differentials for work and heat, Eqs. (23)-(25), are identified as

[TABLE]

with the matrix

[TABLE]

and

[TABLE]

where we choose $\boldsymbol{Q}=\operatorname{diag}(1,0)$ for the heat $q_{c}$ dissipated into the cold reservoir and $\boldsymbol{Q}=\operatorname{diag}(0,-1)$ for the heat $q_{h}$ extracted from the hot reservoir. We employ the Stratonovich convention for stochastic integrals throughout, e.g., for the calculation of the integrated work and heat from above differentials. Sample trajectories for the integrated heat and work are shown in Fig. 4.

Work and heat are connected through the first law

[TABLE]

moreover, we find the relation

[TABLE]

Since the right hand sides of Eqs. (S.14) and (S.15) are total differentials of functions that do not scale with time, we conclude that the fluctuations of the heat currents and the work are mutually dependent on large time scales. It is therefore possible to focus on the current $q_{c}(t)$ only (setting $\boldsymbol{Q}=\operatorname{diag}(1,0)$ in the following), with mean $j_{c}$ and diffusion coefficient $D_{c}$. The mean and diffusion of the current $q_{h}(t)$ then follow readily as $j_{h}=(a_{21}/a_{12})j_{c}$ and $D_{h}=(a_{21}/a_{12})^{2}D_{c}$, respectively, and likewise for the work $j_{w}=(a_{21}/a_{12}-1)j_{c}$ and $D_{w}=(a_{12}/a_{21}-1)^{2}D_{c}$. These considerations yield the efficiency

[TABLE]

(Eq. (27) in the main text) and the equality

[TABLE]

c.f. Eq.(28).

The fluctuations of the heat $q_{c}(t)$ can be derived from the scaled cumulant generating function

[TABLE]

with a real parameter $z$. Employing standard methods, as recently pedagogically reviewed in Touchette (2017), $\lambda(z)$ follows as the largest eigenvalue of the tilted adjoint Fokker-Planck operator

[TABLE]

with $\boldsymbol{\nabla}\equiv\partial/\partial\boldsymbol{x}$. For $z=0$, the largest eigenvalue is $\lambda(0)=0$ with the corresponding eigenfunction $g(\boldsymbol{x},z=0)=1$. For general $z$, we use the Gaussian ansatz $g(\boldsymbol{x},z)=\exp[-\boldsymbol{x}^{T}\boldsymbol{G}(z)\boldsymbol{x}/2]$ with a symmetric matrix $\boldsymbol{G}(z)$ for the eigenfunction, leading to

[TABLE]

where we suppress the argument of $\boldsymbol{G}(z)$ here and in the following for readability. Comparing the coefficients of the quadratic form then yields

[TABLE]

and

[TABLE]

where we denote the symmetrization of a matrix $\boldsymbol{X}$ as $(\boldsymbol{X})_{s}\equiv(\boldsymbol{X}+\boldsymbol{X}^{T})/2$. Eq. (S.22) presents three conditions for the three independent coefficients of $\boldsymbol{G}$. The Gaussian ansatz thus yields indeed an eigenfunction of $\mathcal{L}^{\dagger}(z)$. However, since Eq. (S.22) is nonlinear in $\boldsymbol{G}$, it is hard to find a general solution for $\boldsymbol{G}$ for arbitrary $z$. Nonetheless, Eqs. (S.22) and (S.21) can be reduced to a set of linear equations by expanding $\lambda(z)$ and $\boldsymbol{G}(z)$ as Taylor series (S.18) and

[TABLE]

respectively. The first order in $z$ yields

[TABLE]

with $\boldsymbol{G}_{1}$ determined by

[TABLE]

and the second order in $z$ yields

[TABLE]

with $\boldsymbol{G}_{2}$ determined by

[TABLE]

Solving these linear systems of equations, we finally obtain

[TABLE]

and

[TABLE]

Restoring the original units and notation (S.10), we obtain the expressions (26) for the power and the explicit result Eq. (28) for the uncertainty relation. In particular, for the constancy of power we obtain

[TABLE]

with the stall-parameter $k_{s}=c\eta_{C}/(2-\eta_{C})$. The efficiency (S.16) reaches the Carnot limit for $k\to k_{s}$. With all other parameters kept fixed, the constancy remains finite in this limit, such that the bound (1) demands vanishing power as visible in Fig. 2b of the main text. Yet, finite power can be achieved in this limit if the mobility $\mu$ is scaled up, which comes at the cost of diverging power fluctuations (S.30). This behavior, which is akin to what we have explored for the quantum dot solar cell in the main text, is illustrated in Fig. 5.

Our bounds on efficiency and power become the tighter, the closer $D_{w}\sigma/j_{w}^{2}$ is to unity. An upper bound on this ratio that is specific for the present model of the Brownian gyrator and $0<k<k_{s}$ follows from

[TABLE]

and is approached for $k\to 0$, coupling $c\to\sqrt{u_{1}u_{2}}$ (larger coupling leads to an unstable potential), and $u_{1}=u_{2}$.