Bounds on current fluctuations in periodically driven systems

We consider a Markov jump process with finite number of states Ω. The space of states is written as {1, 2, ..., Ω}. The transition rate from state i to state j at time t is denoted by w_ij(t). Since we are interested in periodically driven systems, these transition rates have a period τ, i.e., w_ij(t) = w_ij(t + τ). Furthermore, we assume that if ${w}_{{ij}}(t)\ne 0$ then ${w}_{{ji}}(t)\ne 0$.

The master equation that governs the time-evolution of _Pi(t), the probability to be in state i at time t, reads

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{P}_{i}(t)=\displaystyle \sum _{j\ne i}[{P}_{j}(t){w}_{{ji}}(t)-{P}_{i}(t){w}_{{ij}}(t)].\end{eqnarray} \tag{ 1 }$$

In the long time limit, _Pi(t) tends to an invariant time-periodic distribution π_i(t) = π_i(t + τ). An important quantity in this paper is the average elementary current

$$\begin{eqnarray}&&{{ \mathcal J }}_{{ij}}(t)\equiv {\pi }_{i}(t){w}_{{ij}}(t)-{\pi }_{j}(t){w}_{{ji}}(t).\end{eqnarray} \tag{ 2 }$$

Fluctuations can be analyzed if we consider stochastic variables that are defined as functionals of a stochastic trajectory ${({a}_{t})}_{0\leqslant t\leqslant m\tau }$, where mτ is the final time and m is an integer number. This trajectory is a sequence of jumps and waiting times. If a jump takes place at time t, the state of the system before and after the jump is denoted by ${a}_{t}^{-}$ and ${a}_{t}^{+}$, respectively. Two basic fluctuating quantities are

$$\begin{eqnarray}&&{\rho }_{i}^{(m)}(t)\equiv \displaystyle \frac{1}{m}\displaystyle \sum _{k=0}^{m-1}{\delta }_{{a}_{\tau k+t},i}\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&{C}_{{ij}}^{(m)}(t)\equiv \displaystyle \frac{1}{{m}{\rm{d}}{t}}\displaystyle \sum _{k=0}^{m-1}\left(\displaystyle \sum _{t^{\prime} \in [t,t+{\rm{d}}{t}]}{\delta }_{{a}_{\tau k+t^{\prime} }^{-},i}{\delta }_{{a}_{\tau k+t^{\prime} }^{+},j}\right),\end{eqnarray} \tag{ 4 }$$

where dt is an infinitesimal time interval and t ∈ [0, τ]. The empirical density ${\rho }_{i}^{(m)}(t)$ counts the fraction of periods with the system in state i at time t. The empirical flow ${C}_{{ij}}^{(m)}(t)$ counts the number of jumps per period from i to j at time t. Even though both quantities are functionals of the stochastic trajectory, to simplify notation, we do not keep the explicit dependence on ${({a}_{t})}_{0\leqslant t\leqslant m\tau }$. The fluctuating empirical current from state i to state j is given by

$$\begin{eqnarray}&&{J}_{{ij}}^{(m)}(t)={C}_{{ij}}^{(m)}(t)-{C}_{{ji}}^{(m)}(t).\end{eqnarray} \tag{ 5 }$$

The average in equation (2) is

$$\begin{eqnarray}&&{{ \mathcal J }}_{{ij}}(t)=\mathop{\mathrm{lim}}\limits_{m\to \infty }\langle {J}_{{ij}}^{(m)}(t)\rangle ,\end{eqnarray} \tag{ 6 }$$

where the brackets denote an average over stochastic trajectories.

A generic current ${J}_{\alpha }^{(m)}$ is defined by its periodic increments α_ij(t), which are anti-symmetric, i.e., α_ij(t) = − α_ji(t), as

$$\begin{eqnarray}&&{J}_{\alpha }^{(m)}\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}(t){J}_{{ij}}^{(m)}(t),\end{eqnarray} \tag{ 7 }$$

where ${\sum }_{i\lt j}$ represents a sum over all pairs of states (i, j) with $i\lt j$ and with non-zero transition rates. The current in equation (7) can also be written in the form

$$\begin{eqnarray}&&{J}_{\alpha }^{(m)}=\displaystyle \frac{1}{m\tau }\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{s\in [0,m\tau ]:}{{a}_{s}^{-}\ne {a}_{s}^{+}}}{\alpha }_{{a}_{s}^{-}{a}_{s}^{+}}(s).\end{eqnarray} \tag{ 8 }$$

In stochastic thermodynamics, physical observables such as heat fluxes and particle fluxes are expressed as currents ${J}_{\alpha }^{(m)}$. The average rate associated with ${J}_{\alpha }^{(m)}$ in the limit $m\to \infty$ reads

$$\begin{eqnarray}&&{{ \mathcal J }}_{\alpha }\equiv \mathop{\mathrm{lim}}\limits_{m\to \infty }\langle {J}_{\alpha }^{(m)}\rangle =\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}(t){{ \mathcal J }}_{{ij}}(t).\end{eqnarray} \tag{ 9 }$$

Furthermore, the diffusion coefficient associated with ${J}_{\alpha }^{(m)}$ is defined as

$$\begin{eqnarray}&&{D}_{\alpha }\equiv \mathop{\mathrm{lim}}\limits_{m\to \infty }m\tau \displaystyle \frac{\langle {({J}_{\alpha }^{(m)}-{{ \mathcal J }}_{\alpha })}^{2}\rangle }{2}.\end{eqnarray} \tag{ 10 }$$

An important current in stochastic thermodynamics is the entropy increase of the environment [2], which corresponds to the increments ${\alpha }_{{ij}}(t)=\mathrm{ln}\tfrac{{w}_{{ij}}(t)}{{w}_{{ji}}(t)}$. The average rate of entropy production is then given by

$$\begin{eqnarray}&&\sigma \equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\mathrm{ln}\displaystyle \frac{{w}_{{ij}}(t)}{{w}_{{ji}}(t)}{{ \mathcal J }}_{{ij}}(t)=\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\mathrm{ln}\displaystyle \frac{{\pi }_{i}(t){w}_{{ij}}(t)}{{\pi }_{j}(t){w}_{{ji}}(t)}{{ \mathcal J }}_{{ij}}(t).\end{eqnarray} \tag{ 11 }$$

The second equality follows from π_i(t) = π_i(t + τ) and from equation (1), which leads to ${\partial }_{t}{\pi }_{i}+{\sum }_{j\ne i}{{ \mathcal J }}_{{ij}}=0$.

The rate function from large deviation theory quantifies exponentially rare events in the long time limit [16–19]. It is defined through the relation

$$\begin{eqnarray}&&{\rm{Prob}}({J}_{\alpha }^{(m)}\approx x)\sim \exp [-m\tau {I}_{\alpha }(x)],\end{eqnarray} \tag{ 12 }$$

where the symbol ∼means asymptotic equality in the limit $m\to \infty$ and ${J}_{\alpha }^{(m)}\approx x$ means that ${J}_{\alpha }^{(m)}$ lies in an infinitesimal interval around x. Our main result is a parabola that bounds ${I}_{\alpha }(x)$, which is a convex function, from above. This parabola depends on an average rate. For the known parabolic bound for stationary states from [14, 15], this rate is the average rate of entropy production σ in equation (11). In our bound for periodically driven systems, this rate is, in general, different from σ.

Current fluctuations can also be characterized by the scaled cumulant generating function

$$\begin{eqnarray}&&{\lambda }_{\alpha }(z)\equiv \mathop{\mathrm{lim}}\limits_{m\to \infty }\displaystyle \frac{1}{m\tau }\mathrm{ln}\langle \exp (m\tau {J}_{\alpha }^{(m)}z)\rangle ,\end{eqnarray} \tag{ 13 }$$

where z is a real number. The cumulants associated with ${J}_{\alpha }^{(m)}$ can be obtained as derivatives of λ_α(z) at z = 0. The scaled current generating function λ_α(z) is a Legendre–Fenchel transform of the rate function I_α(x), i.e.

$$\begin{eqnarray}&&{\lambda }_{\alpha }(z)\equiv \mathop{\sup }\limits_{x}\{{xz}-{I}_{\alpha }(x)\}.\end{eqnarray} \tag{ 14 }$$

If a parabola bounds ${I}_{\alpha }(x)$ from above then a corresponding parabola, which can be determined from equation (14), bounds λ_α(z) from below. For illustrations of our results we perform calculations of λ_α(z) using known methods [40, 50].

We also consider the case of an external protocol that is stochastic [40, 49, 50]. In order to mimic a deterministic periodic protocol, this stochastic protocol is cyclic and has N states. The transition rate from state i to state j with the external protocol in state n = 0, 1, ..., N − 1 is denoted by w_ijⁿ. The transition rate for a change in the external protocol from state n to state $n+1\,\mathrm{mod}\,N$ is γ, whereas the transition rate for the reversed transition is 0. Consider a deterministic periodic protocol characterized by the rates w_ij(t) and the period τ. If the rates of the model with a stochastic protocol are ${w}_{{ij}}^{n}={w}_{{ij}}(t=n\tau /N)$ and γ = N/τ, then in the limit of $N\to \infty$, current fluctuations for the stochastic protocol become equal to current fluctuations for the deterministic protocol [50]. Hence, the deterministic protocol corresponds to an asymptotic limit of a stochastic protocol. We point out that we do not consider the cost of the external protocol [51].

In the appendix, we derive bounds on current fluctuations for the case of a stochastic protocol. These derivations are similar to the derivation in section 4 for a deterministic periodic protocol. An advantage of models with a stochastic protocol is that they are Markov processes with time-independent transition rates, which can simplify the exact evaluation of the scaled cumulant generating function in equation (13), as explained in [40]. Whereas the expressions in the main text are for the case of a deterministic protocol, the expressions for a stochastic protocol can be obtained from these expressions for a deterministic protocol by making the substitution ${\tau }^{-1}{\int }_{0}^{\tau }{\rm{d}}{t}\to {N}^{-1}{\sum }_{n}$, as explained in the appendix.

2.4.1. Colloidal particle driven by a time-periodic field

The first model in figure 1(a) is a biased random walk on a ring with Ω states driven by a time-periodic force $F(t)\equiv {F}_{0}\cos (2\pi t/\tau )$. A physical realization of this model is a charged colloid on a ring subjected to a time-periodic electrical field. We set Boltzmann constant k_B and the temperature T to k_BT = 1 throughout. The transition rate for a jump in the clockwise direction is ${k}_{+}(t)\equiv k{{\rm{e}}}^{F(t)/{\rm{\Omega }}}$ and the reversed transition rate is ${k}_{-}(t)\equiv k$. These transition rates satisfy the generalized detailed balance relation [2]. The current we consider is the net number of jumps in the clockwise direction per unit time. For this model, the scaled cumulant generating function in equation (13) can be calculated exactly [50].

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2.4.2. Molecular pump

The other two models are driven by a stochastic protocol. The model illustrated in figure 1(b) is a molecular pump with Ω = 3. This model has been introduced in [40]. The external protocol changes energies and energy barriers between states, which can lead to net rotation in the ring with three states. The number of states of the external protocol is N = 3. The states of the external protocol are denoted by 0, 1, 2, which correspond, respectively, to the top left circle, the top right circle and the bottom circle in figure 1(b). In this model, the energies and energy barriers are rotated in the clockwise direction by one step if a jump (with rate γ ) that changes the state of the protocol takes place. The energies are denoted by E₁, E₂, and E₃, whereas the energy barriers are denoted by B₁, B₂, and B₃. The internal transition rates are given by

$$\begin{eqnarray}&&{w}_{{ij}}^{n}={{\rm{e}}}^{{E}_{i-n}-{B}_{j-n}},\end{eqnarray} \tag{ 15 }$$

for j = i + 1, and

$$\begin{eqnarray}&&{w}_{{ij}}^{n}={{\rm{e}}}^{{E}_{i-n}-{B}_{i-n}},\end{eqnarray} \tag{ 16 }$$

for j = i − 1, where we assume periodic boundary conditions. An important property of molecular pumps is that the thermodynamic force is zero for any state n of the external protocol. This physical condition is manifested in the following restriction on the transition rates

$$\begin{eqnarray}&&\displaystyle \frac{{w}_{12}^{n}{w}_{23}^{n}{w}_{31}^{n}}{{w}_{21}^{n}{w}_{32}^{n}{w}_{13}^{n}}=1.\end{eqnarray} \tag{ 17 }$$

The current we consider is the net number of jumps in the clockwise direction per unit time. The scaled cumulant generating function in equation (13) associated with this current can be calculated from the eigenvalue of a modified generator, as shown in [40].

2.4.3. Enzymatic reaction with stochastic substrate concentrations

The model illustrated in figure 1(c) is a model with Ω = 4 and two independent thermodynamic forces F₁ⁿ and F₂ⁿ, which depend on the state of the external protocol n. This model can be interpreted as a enzyme that can consume two different substrates and produces one product [9]. The two enzymatic cycles are $E+{S}_{1}\to {{ES}}_{1}\to {EP}\to E+P$ and $E+{S}_{2}\to {{ES}}_{2}\to {EP}\to E+P$, where E is the enzyme, P is the product, S₁ is one substrate, and S₂ is another substrate. State 1 corresponds to the free enzyme E, state 2 corresponds to ES₁, state 3 corresponds to ES₂, and state 4 corresponds to EP. The external control of the concentrations of the substrates S₁ and S₂ generate thermodynamic forces that depend on n. The number of states of the external protocol is N = 2. The generalized detailed balance relation for this model reads

$$\begin{eqnarray}&&{F}_{1}^{n}=\mathrm{ln}\displaystyle \frac{{w}_{12}^{n}{w}_{24}^{n}{w}_{41}^{n}}{{w}_{21}^{n}{w}_{42}^{n}{w}_{14}^{n}}\qquad {F}_{2}^{n}=\mathrm{ln}\displaystyle \frac{{w}_{13}^{n}{w}_{34}^{n}{w}_{41}^{n}}{{w}_{31}^{n}{w}_{43}^{n}{w}_{14}^{n}}.\end{eqnarray} \tag{ 18 }$$

The thermodynamic forces change between two values of the same modulus and different sign stochastically, i.e., F₁ⁿ is given by ${F}_{1}^{0}={F}_{1}$ and ${F}_{1}^{1}=-{F}_{1}$, whereas F₂ⁿ is given by ${F}_{2}^{0}={F}_{2}$ and ${F}_{2}^{1}=-{F}_{2}$.

The transition rate for a change of the external protocol is γ. The transitions rates are set to ${w}_{12}^{n}=k{{\rm{e}}}^{{F}_{1}^{n}/2},{w}_{13}^{n}=k{{\rm{e}}}^{{F}_{2}^{n}/2},{w}_{14}^{n}=k$, ${w}_{21}^{n}=k,{w}_{24}^{n}=k{{\rm{e}}}^{{F}_{1}^{n}/2},{w}_{31}^{n}=k,{w}_{34}^{n}=k{{\rm{e}}}^{{F}_{2}^{n}/2}$, ${w}_{41}^{n}=k,{w}_{42}^{n}=k$, and ${w}_{43}^{n}=k$. The current we consider is the elementary current from state 1 to state 2, which corresponds to the net number of S₁ molecules that have been consumed per unit time. As is the case of the previous model, the scaled cumulant generating function in equation (13) can be calculated with the method explained in [40].

The parabolic bound on the rate function is

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\sigma }^{* }}{4{{ \mathcal J }}_{\alpha }^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2},\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray}&&{\sigma }^{* }\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }\displaystyle \sum _{i\lt j}\displaystyle \frac{{({\bar{{ \mathcal J }}}_{{ij}})}^{2}}{{{ \mathcal J }}_{{ij}}(t)}\mathrm{ln}\displaystyle \frac{{\pi }_{i}(t){w}_{{ij}}(t)}{{\pi }_{j}(t){w}_{{ji}}(t)}{\rm{d}}{t},\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&{\bar{{ \mathcal J }}}_{{ij}}\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{{ \mathcal J }}_{{ij}}(t){\rm{d}}{t}.\end{eqnarray} \tag{ 21 }$$

The inequality ${\sigma }^{* }\geqslant 0$ comes from the fact that for fixed t every term in the sum ${\sum }_{i\lt j}$ in equation (20) is not negative. In general, the average rate ${\sigma }^{* }$ is different from the thermodynamic rate of entropy production σ in equation (11). Furthermore, there is no simple inequality relating both quantities, as illustrated in figure 2(b).

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For the case of time-independent transition rates ${w}_{{ij}}(t)={w}_{{ij}},{\sigma }^{* }=\sigma$ and the bound (19) becomes

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{\sigma }{4{{ \mathcal J }}_{\alpha }^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2}.\end{eqnarray} \tag{ 22 }$$

This bound is the known parabolic bound for time-independent transition rates proved in [15]. Hence, equation (19) constitutes a generalization of this parabolic bound to periodically driven systems.

In terms of the scaled cumulant generating function, the bound in equation (19) is written as

$$\begin{eqnarray}&&{\lambda }_{\alpha }(z)\geqslant z{{ \mathcal J }}_{\alpha }(1+{{ \mathcal J }}_{\alpha }z/{\sigma }^{* }),\end{eqnarray} \tag{ 23 }$$

where we used equation (14). The universality of our result is illustrated in figure 2(a). There we compare the function

$$\begin{eqnarray}&&{\tilde{\lambda }}_{\alpha }(\tilde{z})\equiv {\lambda }_{\alpha }(z)/{\sigma }^{* }={\lambda }_{\alpha }(\tilde{z}{\sigma }^{* }/{{ \mathcal J }}_{\alpha })/{\sigma }^{* }\geqslant \tilde{z}(1+\tilde{z}),\end{eqnarray} \tag{ 24 }$$

where $\tilde{z}\equiv z{{ \mathcal J }}_{\alpha }/{\sigma }^{* }$, for the models in figure 1, with the lower bound $\tilde{z}(1+\tilde{z})$. This bound, or the bound in equation (19), is a particular case of two bounds, one derived in section 4.1 and the other derived in section 4.5.

Taking the second derivative of I_α(x) at $x={{ \mathcal J }}_{\alpha }$, we obtain the diffusion coefficient D_α defined in equation (10) as

$$\begin{eqnarray}&&I^{\prime\prime} ({{ \mathcal J }}_{\alpha })=\displaystyle \frac{1}{2{D}_{\alpha }}.\end{eqnarray} \tag{ 25 }$$

The inequality in equation (19) and the fact that this inequality is saturated at $x={{ \mathcal J }}_{\alpha }$, leads to the following bound on D_α,

$$\begin{eqnarray}&&{D}_{\alpha }\geqslant \displaystyle \frac{{{ \mathcal J }}_{\alpha }^{2}}{{\sigma }^{* }}.\end{eqnarray} \tag{ 26 }$$

In section 4.3, we derive a local quadratic bound on ${I}_{\alpha }(x)$, which is valid for x close to the average ${{ \mathcal J }}_{\alpha }$. This local bound together with equation (25), gives a tighter bound on D_α that reads

$$\begin{eqnarray}&&{D}_{\alpha }\geqslant \displaystyle \frac{{{ \mathcal J }}_{\alpha }^{2}}{\tilde{\sigma }}\geqslant \displaystyle \frac{{{ \mathcal J }}_{\alpha }^{2}}{{\sigma }^{* }},\end{eqnarray} \tag{ 27 }$$

where

$$\begin{eqnarray}&&\tilde{\sigma }\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\displaystyle \frac{2{({\bar{{ \mathcal J }}}_{{ij}})}^{2}}{{\pi }_{i}(t){w}_{{ij}}(t)+{\pi }_{j}(t){w}_{{ji}}(t)}.\end{eqnarray} \tag{ 28 }$$

The second inequality in equation (27) is a consequence of ${\sigma }^{* }\geqslant \tilde{\sigma }$, which follows from the inequality

$$\begin{eqnarray}&&(a-b)\mathrm{ln}\displaystyle \frac{a}{b}\geqslant \displaystyle \frac{2{(a-b)}^{2}}{a+b},\end{eqnarray} \tag{ 29 }$$

where a and b are positive. An inequality similar to ${\sigma }^{* }\geqslant \tilde{\sigma }$ has been considered in [52]. We point out that there is no general inequality between the entropy production σ and the rate $\tilde{\sigma }$, as illustrated in figure 2(b).

Rearranging the terms in equation (27), we write the following universal trade-off relation between speed and precision for periodically driven systems,

$$\begin{eqnarray}&&{{ \mathcal F }}_{\alpha }^{-1}{{ \mathcal J }}_{\alpha }\leqslant \displaystyle \frac{\tilde{\sigma }}{2}\leqslant \displaystyle \frac{{\sigma }^{* }}{2},\end{eqnarray} \tag{ 30 }$$

where ${{ \mathcal F }}_{\alpha }\equiv 2{D}_{\alpha }/{{ \mathcal J }}_{\alpha }$ is the Fano factor. The Fano factor characterizes the precision associated with ${J}_{\alpha }^{(m)}$, whereas ${{ \mathcal J }}_{\alpha }$ quantifies the speed. In periodically driven systems, a current with small fluctuations, as characterized by a small Fano factor ${{ \mathcal F }}_{\alpha }$, can only be as fast as $\tilde{\sigma }{{ \mathcal F }}_{\alpha }/2$.

This trade-off relation is a generalization of the thermodynamic uncertainty relation to periodically driven systems. In particular, for the case of time-independent transition rates w_ij(t) = w_ij, inequality (30) implies the thermodynamic uncertainty relation ${{ \mathcal F }}_{\alpha }^{-1}{{ \mathcal J }}_{\alpha }\leqslant \sigma /2$, since ${\sigma }^{* }=\sigma$ for this case. Furthermore, the inequality ${{ \mathcal F }}_{\alpha }^{-1}{{ \mathcal J }}_{\alpha }\leqslant \tilde{\sigma }/2$, for time-independent transition rates, provides an even tighter bound than the thermodynamic uncertainty relation.

This result is relevant for the mapping between an artificial molecular pump and a system driven by a fixed thermodynamic force such as a biological molecular motor, which is modelled with time-independent transition rates that lead to a nonequilibrium stationary state, proposed in [42]. With this mapping, one can construct a molecular pump that mimicks a stationary state and vice-versa, in the sense that both the average rate of entropy production and the average elementary currents between a pair of states are conserved. However, a mapping of a molecular pump onto a stationary state that also preserves fluctuations is not always possible, since a molecular pump may not fulfill the relation ${{ \mathcal J }}_{\alpha }^{2}/(2{D}_{\alpha }\sigma )\leqslant 1/2$, as shown in [40], whereas a system that reaches a nonequilibrium stationary state must fulfill this relation.

Our trade-off relations do not imply the generalization of the thermodynamic uncertainty relation from [29] for the case of periodic protocols that are symmetric, i.e., $w(\tau /2+{\rm{\Delta }}t)=w(\tau /2-{\rm{\Delta }}t)$, where $0\leqslant {\rm{\Delta }}t\leqslant \tau /2$. The trade-off relation from this reference involves the thermodynamic entropy production σ and for symmetric protocols the rate σ is, in general, different from the rates ${\sigma }^{* }$ and $\tilde{\sigma }$.

In figure 3(a), we show plots of ${R}_{\alpha }\equiv {{ \mathcal J }}_{\alpha }^{2}/(2{D}_{\alpha }{\sigma }^{* })\leqslant 1/2$ as a function of the rate γ, which quantifies the speed of the protocol, for the models illustrated in figure 1(b) and in figure 1(c). For the first model, which is a molecular pump, we find that this bound is saturated if the transitions of the protocol are much slower than the internal transition rates associated with changes of the state of the system. For this model, in this limit the bound is saturated independent of the values of the energies and energy barriers. However, for the second model the bound is not saturated in this limit.

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In figure 3(b), we show plots of ${R}_{\alpha }\equiv {{ \mathcal J }}_{\alpha }^{2}/(2{D}_{\alpha }{\sigma }^{* })\leqslant 1/2$ for the model illustrated in figure 1(c). The quantity in the horizontal axis is the thermodynamic force F₁. For this model, the bound is saturated for F₁ small and the other thermodynamic force F₂ = 0. This saturation of the bound is similar to the saturation of the bound for stationary states known as thermodynamic uncertainty relation, which happens in the linear response regime [9].

Let us comment on the rate ${\sigma }^{* }$ that we have introduced here. Its physical interpretation is that ${\sigma }^{* }$, and not the rate of entropy production σ, provides a bound on the whole spectrum of fluctuations for any current (with time-independent increments) in a generic periodically driven system arbitrarily far from equilibrium. In terms of the trade-off relation from equation (30), ${\sigma }^{* }$ (and also $\tilde{\sigma }$) provides a limit on how precise and fast a thermodynamic current can be. The rate of entropy production σ quantifies the energetic cost of sustaining the operation of the nonequilibrium system. Interestingly, for time-independent transition rates corresponding to a system driven by a fixed thermodynamic force, ${\sigma }^{* }=\sigma$ is a rate that has both physical properties, i.e., it bounds current fluctuations and quantifies energetic cost.

The rate ${\sigma }^{* }$ of the original periodically driven system can be interpreted as the rate of entropy production associated with the stationary state of an auxiliary Markov process with time-independent transition rates that are determined by time-averaged quantities associated with the original system. These time-averaged quantities are ${\bar{{ \mathcal J }}}_{{ij}}$, defined in equation (21), and

$$\begin{eqnarray}&&{\theta }_{{ij}}\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }\displaystyle \frac{{\bar{{ \mathcal J }}}_{{ij}}}{{{ \mathcal J }}_{{ij}}(t)}\mathrm{ln}\displaystyle \frac{{\pi }_{i}(t){w}_{{ij}}(t)}{{\pi }_{j}(t){w}_{{ji}}(t)}{\rm{d}}{t}.\end{eqnarray} \tag{ 31 }$$

Both quantities are anti-symmetric, i.e., ${\bar{{ \mathcal J }}}_{{ij}}=-{\bar{{ \mathcal J }}}_{{ji}}$ and ${\theta }_{{ij}}=-{\theta }_{{ji}}$. Moreover, from the definition in equation (31), ${\bar{{ \mathcal J }}}_{{ij}}$ and θ_ij have the same sign. We assume without loss of generality that ${\bar{{ \mathcal J }}}_{{ij}}$ and θ_ij are non-negative .

From equation (20), ${\sigma }^{* }$ can be written as ${\sigma }^{* }={\sum }_{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\theta }_{{ij}}$. The transition rates associated with this auxiliary process are denoted by r_ij and the stationary distribution associated with this process is denoted by _pi. The stationary probability currents of this auxiliary process are the time-averaged currents ${\bar{{ \mathcal J }}}_{{ij}}$, hence, we have the constraint

$$\begin{eqnarray}&&{p}_{i}{r}_{{ij}}-{p}_{j}{r}_{{ji}}={\bar{{ \mathcal J }}}_{{ij}}.\end{eqnarray} \tag{ 32 }$$

Furthermore, if we impose

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{{ij}}}{{r}_{{ji}}}={{\rm{e}}}^{{\theta }_{{ij}}},\end{eqnarray} \tag{ 33 }$$

then the rate of entropy production of the auxiliary process is ${\sigma }^{* }$, i.e., ${\sigma }^{* }={\sum }_{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}\mathrm{ln}({r}_{{ij}}/{r}_{{ji}})$. From the conditions in equations (32) and (33), we obtain

$$\begin{eqnarray}&&{r}_{{ij}}=\displaystyle \frac{{\bar{{ \mathcal J }}}_{{ij}}{{\rm{e}}}^{{\theta }_{{ij}}}}{{{\rm{e}}}^{{\theta }_{{ij}}}{p}_{i}-{p}_{j}}.\end{eqnarray} \tag{ 34 }$$

The reversed rate r_ji is then given by

$$\begin{eqnarray}&&{r}_{{ji}}=\displaystyle \frac{{\bar{{ \mathcal J }}}_{{ij}}}{{{\rm{e}}}^{{\theta }_{{ij}}}{p}_{i}-{p}_{j}}.\end{eqnarray} \tag{ 35 }$$

Equation (34) defines a class of stationary states that have entropy production ${\sigma }^{* }$. Since transition rates are non-negative, the stationary probability must satisfy the constraint ${{\rm{e}}}^{{\theta }_{{ij}}}{p}_{i}-{p}_{j}\geqslant 0$. One possible stationary probability that fulfills this constraint for any model is the uniform distribution ${p}_{i}=1/{\rm{\Omega }}$ for i = 1, 2, ..., Ω, since θ_ij ≥ 0.

We can now provide the following physical interpretation for ${\sigma }^{* }$. This rate quantifies the thermodynamic cost to maintain a nonequilibrium stationary state that is determined by the transition rates in equation (34). There are different stationary probabilities that fulfill equation (34), hence, this nonequilibrium stationary state is not unique but rather a class of nonequilibrium stationary states. The network topology of this class of nonequilibrium stationary states is the same as the network topology of the periodically driven system, furthermore, the stationary currents are the same as the time-averaged currents of the periodically driven system. As an example, consider a colloidal particle driven by an external periodic protocol, such as the model represented in figure 1(b). For such molecular pump we can think of a colloidal particle driven by a fixed force that reaches a nonequilibrium stationary state. The force that drives this particle and the specific transition rates that determine its dynamics are obtained from time-averaged quantitative associated with the original molecular pump. The rate ${\sigma }^{* }$ quantifies the energetic cost of driving the colloidal particle with such fixed force.

In our proof we use the theory for 2.5 large deviations for periodically driven systems developed in [48]. At the level 2.5 the joint distribution of all empirical densities defined in equation (3) and all empirical currents defined in equation (5) is considered. In our notation ρ(t) represents a vector with the empirical densities that has dimension Ω and J(t) is a vector with the empirical currents that has dimension M, where M is the number of unordered pairs of states with non-zero transition rates. The advantage of considering this level of large deviations is that the rate function can be calculated exactly as

$$\begin{eqnarray}&&{I}_{2.5}^{{\rm{cur}}}[{(J(t))}_{t\in [0,\tau ]},{(\rho (t))}_{t\in [0,\tau ]}]=\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\psi ({J}_{{ij}}(t),{G}_{{ij}}(t),{a}_{{ij}}(t)),\end{eqnarray} \tag{ 36 }$$

where

$$\begin{eqnarray}&&{G}_{{ij}}(t)\equiv {\rho }_{i}(t){w}_{{ij}}(t)-{\rho }_{j}(t){w}_{{ji}}(t),\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&{a}_{{ij}}(t)\equiv 2\sqrt{{\rho }_{i}(t){\rho }_{j}(t){w}_{{ij}}(t){w}_{{ji}}(t)},\end{eqnarray} \tag{ 38 }$$

and

$$\begin{eqnarray}&&\psi (J,G,a)=\sqrt{{G}^{2}+{a}^{2}}-\sqrt{{J}^{2}+{a}^{2}}+J[{\sinh }^{-1}(J/a)-{\sinh }^{-1}(G/a)].\end{eqnarray} \tag{ 39 }$$

Note that the quantities G and a depend on the empirical density ρ. The empirical density and current in equation (36) fulfill the constraint

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{\rho }_{i}(t)+\displaystyle \sum _{j\ne i}{J}_{{ij}}(t)=0,\end{eqnarray} \tag{ 40 }$$

for all states i. To simplify the notation we write ${I}_{2.5}^{{\rm{cur}}}[J(t),\rho (t)]$ instead of the lhs of equation (36).

The name level 2.5 large deviations can also refer to the rate function associated with the joint probability of the empirical density and the empirical flow defined in equation (4). The rate function with the empirical current can be obtained from the rate function with the empirical flow [48].

An important technique in large deviation theory is the so called contraction [16–19], for which the rate function associated with a coarse-graining of the number of variables can be obtained from the original rate function. Hence, the rate function for an arbitrary current J_α can be obtained from a contraction of ${I}_{2.5}^{{\rm{cur}}}[J(t),\rho (t)]$, which leads to the expression

$$\begin{eqnarray}&&{I}_{\alpha }(x)=\mathop{\inf }\limits_{J(t),\rho (t)}{I}_{2.5}^{{\rm{cur}}}[J(t),\rho (t)],\end{eqnarray} \tag{ 41 }$$

where J(t) and ρ(t) are such that they fulfill equation (40) and the relation

$$\begin{eqnarray}&&\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}(t){J}_{{ij}}(t)=x.\end{eqnarray} \tag{ 42 }$$

In particular, this relation leads to the inequality

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant {I}_{2.5}^{{\rm{cur}}}[\tilde{J}(t),\tilde{\rho }(t)]=\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\psi ({\tilde{J}}_{{ij}}(t),{\tilde{G}}_{{ij}}(t),{\tilde{a}}_{{ij}}(t)),\end{eqnarray} \tag{ 43 }$$

where $\tilde{G}$ and $\tilde{a}$ are functions of $\tilde{\rho }$ as in (37) and (38). This inequality is valid for any pair of vectors that fulfill the constraints

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{\tilde{\rho }}_{i}(t)+\displaystyle \sum _{j\ne i}{\tilde{J}}_{{ij}}(t)=0,\end{eqnarray} \tag{ 44 }$$

for all states i, and

$$\begin{eqnarray}&&\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}{\tilde{J}}_{{ij}}(t){\alpha }_{{ij}}(t)=x.\end{eqnarray} \tag{ 45 }$$

The inequality [15]

$$\begin{eqnarray}&&\psi ({J}_{{ij}},{G}_{{ij}},{a}_{{ij}})\leqslant \displaystyle \frac{1}{4}\displaystyle \frac{{[{J}_{{ij}}-{G}_{{ij}}]}^{2}}{{G}_{{ij}}}\mathrm{ln}\displaystyle \frac{{\rho }_{i}{w}_{{ij}}}{{\rho }_{j}{w}_{{ji}}}\end{eqnarray} \tag{ 46 }$$

together with equation (43), leads to

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\displaystyle \frac{1}{4}\displaystyle \frac{{[{\tilde{J}}_{{ij}}(t)-{\tilde{G}}_{{ij}}(t)]}^{2}}{{\tilde{G}}_{{ij}}(t)}\mathrm{ln}\displaystyle \frac{{\tilde{\rho }}_{i}(t){w}_{{ij}}(t)}{{\tilde{\rho }}_{j}(t){w}_{{ji}}(t)}.\end{eqnarray} \tag{ 47 }$$

We are now left with the problem of finding a judicious choice of $(\tilde{J}(t),\tilde{\rho }(t))$ that fulfills the constraints in equation (44) and in equation (45). One such choice is

$$\begin{eqnarray}&&{\tilde{\rho }}_{i}(t)={\pi }_{i}(t),\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&{\tilde{J}}_{{ij}}(t)={{ \mathcal J }}_{{ij}}(t)+\displaystyle \frac{(x-{{ \mathcal J }}_{\alpha }){K}_{{ij}}}{{\displaystyle \sum }_{i^{\prime} \lt j^{\prime} }{K}_{i^{\prime} j^{\prime} }{\bar{\alpha }}_{i^{\prime} j^{\prime} }},\end{eqnarray} \tag{ 49 }$$

where

$$\begin{eqnarray}&&{\bar{\alpha }}_{{ij}}\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\alpha }_{{ij}}(t){\rm{d}}{t}.\end{eqnarray} \tag{ 50 }$$

The time-independent parameters K_ij are anti-symmetric, i.e., K_ij = −K_ji, and satisfy

$$\begin{eqnarray}&&\displaystyle \sum _{j\ne i}{K}_{{ij}}=0,\end{eqnarray} \tag{ 51 }$$

for all states i. Using this choice in equation (47), we obtain

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\sigma }_{K}^{* }}{4{{ \mathcal J }}_{K}^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2},\end{eqnarray} \tag{ 52 }$$

where

$$\begin{eqnarray}&&{{ \mathcal J }}_{K}\equiv \displaystyle \sum _{i\lt j}{K}_{{ij}}{\bar{\alpha }}_{{ij}},\end{eqnarray} \tag{ 53 }$$

and

$$\begin{eqnarray}&&{\sigma }_{K}^{* }\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }\displaystyle \sum _{i\lt j}\displaystyle \frac{{({K}_{{ij}})}^{2}}{{{ \mathcal J }}_{{ij}}(t)}\mathrm{ln}\displaystyle \frac{{\pi }_{i}(t){w}_{{ij}}(t)}{{\pi }_{j}(t){w}_{{ji}}(t)}{\rm{d}}{t}.\end{eqnarray} \tag{ 54 }$$

The global bound in equation (52), together with equation (25), leads to

$$\begin{eqnarray}&&{D}_{\alpha }\geqslant \displaystyle \frac{{{ \mathcal J }}_{K}^{2}}{{\sigma }_{K}^{* }}.\end{eqnarray} \tag{ 55 }$$

4.2.1. Generic choice for K

Due to the constraint in equation (51), K_ij can be seen as the current of some auxiliary Markov process with time-independent transition rates in the stationary state. A natural choice of K_ij is to consider the time-integrated probability current, as defined in equation (21), i.e.,

$$\begin{eqnarray}&&{K}_{{ij}}\equiv {\bar{{ \mathcal J }}}_{{ij}}.\end{eqnarray} \tag{ 56 }$$

For this choice

$$\begin{eqnarray}&&{{ \mathcal J }}_{K}=\displaystyle \sum _{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\bar{\alpha }}_{{ij}},\end{eqnarray} \tag{ 57 }$$

and ${\sigma }_{K}^{* }={\sigma }^{* }$, where ${\sigma }^{* }$ is defined in equation (20). For currents with time-independent increments α_ij(t) = α_ij, we obtain ${\sum }_{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\bar{\alpha }}_{{ij}}={{ \mathcal J }}_{\alpha }$, where ${{ \mathcal J }}_{\alpha }$ is given by equation (9), and the bound in equation (52) becomes the bound in equation (19). For currents with time-dependent increments, which include the rate of extracted work and the rate of heat flow in a heat engine driven by periodic temperature variation, the rate ${{ \mathcal J }}_{K}$ in equation (57) is, in general, different from the average current ${{ \mathcal J }}_{\alpha }$.

4.2.2. Other possible choices for K

We now derive a local quadratic bound on I_α(x) that leads to the first inequality in equation (27). For a and G fixed, a Taylor expansion of the function ψ(J, G, a) for J around the value G, leads to

$$\begin{eqnarray}&&\psi (J,G,a)=\displaystyle \frac{{(J-G)}^{2}}{2\sqrt{{G}^{2}+{a}^{2}}}+{\rm{o}}(| J-G{| }^{2}).\end{eqnarray} \tag{ 58 }$$

Applying this Taylor expansion to equation (43) with $\tilde{\rho }$ and $\tilde{J}$ given by (48) and (49), respectively, we obtain the local bound

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\tilde{\sigma }}_{K}}{4{{ \mathcal J }}_{K}^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2}+{\rm{o}}(| x-{{ \mathcal J }}_{\alpha }{| }^{2}),\end{eqnarray} \tag{ 59 }$$

where ${{ \mathcal J }}_{K}$ is defined in equation (53) and

$$\begin{eqnarray}&&{\tilde{\sigma }}_{K}\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }{\rm{d}}{t}\displaystyle \sum _{i\lt j}\displaystyle \frac{2{({K}_{{ij}})}^{2}}{{\pi }_{i}(t){w}_{{ij}}(t)+{\pi }_{j}(t){w}_{{ji}}(t)}.\end{eqnarray} \tag{ 60 }$$

The local bound in equation (59) together with equation (25) leads to

$$\begin{eqnarray}&&{D}_{\alpha }\geqslant \displaystyle \frac{{{ \mathcal J }}_{K}^{2}}{{\tilde{\sigma }}_{K}}.\end{eqnarray} \tag{ 61 }$$

A generic model-independent choice for K is the one given in equation (56), i.e., ${K}_{{ij}}={\bar{{ \mathcal J }}}_{{ij}}$. If, in addition, the increments are time-independent, the bound in equation (61) becomes the trade-off relation between speed and precision in equation (30). We recall that from equation (29), ${\sigma }_{K}^{* }\geqslant {\tilde{\sigma }}_{K}$, thus, the bound in equation (61) is stronger than the bound in equation (55).

Here, we stress that the bounds for time-periodic transition rates derived above imply new bounds for the case of time-independent transition rates that lead to a nonequilibrium stationary state. For time-independent transition rates, and for currents with time-independent increments, the terms in equation (52) become

$$\begin{eqnarray}&&{{ \mathcal J }}_{K}\equiv \displaystyle \sum _{i\lt j}{K}_{{ij}}{\alpha }_{{ij}},\end{eqnarray} \tag{ 62 }$$

and

$$\begin{eqnarray}&&{\sigma }_{K}^{* }\equiv \displaystyle \sum _{i\lt j}\displaystyle \frac{{({K}_{{ij}})}^{2}}{{{ \mathcal J }}_{{ij}}}\mathrm{ln}\displaystyle \frac{{\pi }_{i}{w}_{{ij}}}{{\pi }_{j}{w}_{{ji}}}.\end{eqnarray} \tag{ 63 }$$

Hence, from equation (52) we have the bound

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\sigma }_{K}^{* }}{4{{ \mathcal J }}_{K}^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2}.\end{eqnarray} \tag{ 64 }$$

For ${K}_{{ij}}={{ \mathcal J }}_{{ij}}$, equation (64) becomes the known parabolic bound for stationary states from [14, 15]. Furthermore, for time-independent transition rates equation (61) becomes

$$\begin{eqnarray}&&{D}_{\alpha }\geqslant \displaystyle \frac{{{ \mathcal J }}_{K}^{2}}{{\tilde{\sigma }}_{K}},\end{eqnarray} \tag{ 65 }$$

where

$$\begin{eqnarray}&&{\tilde{\sigma }}_{K}\equiv \displaystyle \sum _{i\lt j}\displaystyle \frac{2{({K}_{{ij}})}^{2}}{{\pi }_{i}{w}_{{ij}}+{\pi }_{j}{w}_{{ji}}}.\end{eqnarray} \tag{ 66 }$$

This bound is tighter then the bound on the diffusion coefficient that follows from equation (64). For the case ${K}_{{ij}}={{ \mathcal J }}_{{ij}}$, equation (65) becomes an even stronger bound than the thermodynamic uncertainty relation, as discussed in section 3.

We can obtain a bound different from the global bound in equation (52) by considering a choice for ${\tilde{J}}_{{ij}}(t)$ that is different from the one in equation (49). We write the stationary distribution of a master equation with frozen transition rates w_ij(t) as μ_i(t). This quantity is known as accompanying density [54]. Due to the periodicity of w_ij(t) we have μ_i(t) = μ_i(t + τ). We consider the bound in equation (47) with ${\tilde{\rho }}_{i}(t)={\pi }_{i}(t)$ and

$$\begin{eqnarray}&&{\tilde{J}}_{{ij}}\equiv {{ \mathcal J }}_{{ij}}(t)+{c}_{1}(t){M}_{{ij}}(t)+{c}_{2}(t){K}_{{ij}},\end{eqnarray} \tag{ 67 }$$

where c₁(t) and c₂(t) are time-periodic functions, K_i,j is anti-symmetric and fulfill the relation in equation (51), and

$$\begin{eqnarray}&&{M}_{{ij}}(t)\equiv {\mu }_{i}(t){w}_{{ij}}(t)-{\mu }_{j}(t){w}_{{ji}}(t).\end{eqnarray} \tag{ 68 }$$

Since ${\sum }_{j\ne i}{M}_{{ij}}(t)=0$, which comes from the definition of the accompanying density μ_i(t), this choice fulfills the constraint in equation (44). Setting ${K}_{{ij}}={\bar{{ \mathcal J }}}_{{ij}}$, c₁(t) = c₁, and c₂(t) = c₂, the constraint in equation (45) applied to the choice in equation (67), leads to

$$\begin{eqnarray}&&{c}_{1}=(x-{{ \mathcal J }}_{\alpha })q{{ \mathcal J }}_{\mu }^{-1},\end{eqnarray} \tag{ 69 }$$

$$\begin{eqnarray}&&{c}_{2}=(x-{{ \mathcal J }}_{\alpha })(1-q){\left(\displaystyle \sum _{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\bar{\alpha }}_{{ij}}\right)}^{-1}.\end{eqnarray} \tag{ 70 }$$

where q is an arbitrary real number and

$$\begin{eqnarray}&&{{ \mathcal J }}_{\mu }\equiv \displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}(t){M}_{{ij}}(t){\rm{d}}{t}.\end{eqnarray} \tag{ 71 }$$

The bound in equation (47) then becomes

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{(x-{{ \mathcal J }}_{\alpha })}^{2}}{4\tau }{\int }_{0}^{\tau }\displaystyle \sum _{i\lt j}\displaystyle \frac{{\left({M}_{{ij}}(t)q{{ \mathcal J }}_{\mu }^{-1}+(1-q){\left({\displaystyle \sum }_{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\bar{\alpha }}_{{ij}}\right)}^{-1}{\bar{{ \mathcal J }}}_{{ij}}\right)}^{2}}{{{ \mathcal J }}_{{ij}}(t)}\mathrm{log}\displaystyle \frac{{\pi }_{i}(t){w}_{{ij}}(t)}{{\pi }_{j}(t){w}_{{ji}}(t)}{\rm{d}}{t}.\end{eqnarray} \tag{ 72 }$$

Minimization over the single parameter q gives the tightest bound on the large deviation function. For q = 0 we obtain the bound in equation (52) with ${K}_{{ij}}={\bar{{ \mathcal J }}}_{{ij}}$. However, for q = 1 we obtain a bound that cannot be obtained from equation (52), which reads

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\sigma }_{\mu }^{* }}{4{{ \mathcal J }}_{\mu }^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2},\end{eqnarray} \tag{ 73 }$$

where

$$\begin{eqnarray}&&{\sigma }_{\mu }^{* }=\displaystyle \frac{1}{\tau }{\int }_{0}^{\tau }\displaystyle \sum _{i\lt j}\displaystyle \frac{{({M}_{{ij}}(t))}^{2}}{{{ \mathcal J }}_{{ij}}(t)}\mathrm{log}\displaystyle \frac{{\pi }_{i}(t){w}_{{ij}}(t)}{{\pi }_{j}(t){w}_{{ji}}(t)}{\rm{d}}{t}.\end{eqnarray} \tag{ 74 }$$

The master equation for the model with a stochastic protocol reads

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{P}_{i}^{n}=\displaystyle \sum _{j\ne i}({P}_{j}^{n}{w}_{{ji}}^{n}-{P}_{i}^{n}{w}_{{ij}}^{n})+\gamma ({P}_{i}^{n-1}-{P}_{i}^{n}),\end{eqnarray} \tag{ A.1 }$$

where n − 1 = N − 1 for n = 0 and ${P}_{i}^{n}$ is the time-dependent distribution. The stationary distribution of state (i, n) is denoted by ${\pi }_{i}^{n}$. The stationary distribution of the state n of the protocol is given by ${\pi }^{n}\equiv {\sum }_{i}{\pi }_{i}^{n}=1/N$, which comes from the solution of the master equation (A.1) for the stationary distribution. The conditional probability for the system to be in state i given that the protocol is in state n is written as $\pi (i| n)={\pi }_{i}^{n}/{\pi }^{n}\,=\,N{\pi }_{i}^{n}$. Consider a time-periodic Markov process with rates w_ij(t) and period τ. If the transition rates fulfill the relation ${w}_{{ij}}^{n}={w}_{{ij}}(t=n\tau /N)$ and γ = N/τ, then, in the limit $N\to \infty ,\pi (i| n)\to {\pi }_{i}(t)$ [40], where $n=[{tN}/\tau ]$ and $[\cdot ]$ denotes the integer part. Therefore, if we consider the average elementary current ${{ \mathcal J }}_{{ij}}^{n}\equiv {\pi }_{i}^{n}{w}_{{ij}}^{n}-{\pi }_{j}^{n}{w}_{{ji}}^{n}$ in the limit of $N\to \infty$, we obtain

$$\begin{eqnarray}&&N{{ \mathcal J }}_{{ij}}^{n}\to {{ \mathcal J }}_{{ij}}(t),\end{eqnarray} \tag{ A.2 }$$

where $n=[{tN}/\tau ]$. This relation is important for the connection between the cases of a deterministic and stochastic protocols.

A stochastic trajectory is denoted by ${({b}_{t})}_{0\leqslant t\leqslant {t}_{f}}$, where t_f is the final time. Note that a state of the Markov process here is specified by the variable that determines the state of the system i and the variable that determines the state of the protocol n. The stochastic trajectory has a fluctuating number of jumps N_f, the time interval between two jumps is denoted Δ t_k, with k = 0, 1, ..., N_f, and the state of the Markov process during the time interval Δ t_k is denoted b_k.

The empirical density of state (i, n), which is the fraction of time spent in this state, is defined as

$$\begin{eqnarray}&&{\rho }_{i}^{n}=\displaystyle \frac{1}{{t}_{f}}\displaystyle \sum _{k=0}^{{N}_{f}}{\rm{\Delta }}{t}_{k}{\delta }_{{b}_{k},(i,n)},\end{eqnarray} \tag{ A.3 }$$

${\delta }_{{b}_{k},(i,n)}$ is the Kronecker delta between the state of the trajectory b_k and the state (i, n). The notation here in the appendix is different from the notation in the main text for the case of a deterministic protocol. If we compare equation (A.3) with equation (3), we see that here the upper index in ${\rho }_{i}^{n}$ refers to the state of the stochastic protocol and is equivalent to t in ${\rho }_{i}^{(m)}(t)$, for which the upper index m refers to the time interval of the stochastic trajectory. For a more compact notation we do not keep the dependence of the fluctuating quantities on the time interval t_f.

The empirical current from state (i, n) to state (j, n) reads

$$\begin{eqnarray}&&{J}_{{ij}}^{n}=\displaystyle \frac{1}{{t}_{f}}\displaystyle \sum _{k=1}^{{N}_{f}}({\delta }_{{b}_{k-1},(i,n)}{\delta }_{{b}_{k},(j,n)}-{\delta }_{{b}_{k-1},(j,n)}{\delta }_{{b}_{k},(i,n)}).\end{eqnarray} \tag{ A.4 }$$

For the case of a stochastic protocol, we also consider the empirical flow (or unidirectional current) from state (i, n) to state (i, n + 1), where n + 1 = 0 for n = N − 1, which is defined as

$$\begin{eqnarray}&&{C}_{i}^{n}=\displaystyle \frac{1}{{t}_{f}}\displaystyle \sum _{k=1}^{{N}_{f}}{\delta }_{{b}_{k-1},(i,n)}{\delta }_{{b}_{k},(i,n+1)}.\end{eqnarray} \tag{ A.5 }$$

The average of this empirical flow in the stationary state is ${{ \mathcal C }}_{i}^{n}\equiv \langle {C}_{i}^{n}\rangle =\gamma {\pi }_{i}^{n}$.

A generic fluctuating current is written as

$$\begin{eqnarray}&&{J}_{\alpha }\equiv \displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}^{n}{J}_{{ij}}^{n},\end{eqnarray} \tag{ A.6 }$$

where ${\alpha }_{{ij}}^{n}=-{\alpha }_{{ji}}^{n}$ are the increments. If we compare this expression with equation (7), which is the expression for a deterministic protocol, we see that an integral over a period divided by the period τ for a deterministic protocol becomes a sum over n divided by the total number of states of the protocol N for a stochastic protocol. Note that the factor 1/N does not appear in front of the sum in the rhs of equation (A.6) due to equation (A.2). The average current in the stationary state reads

$$\begin{eqnarray}&&{{ \mathcal J }}_{\alpha }\equiv \langle {J}_{\alpha }\rangle =\displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}^{n}{{ \mathcal J }}_{{ij}}^{n}.\end{eqnarray} \tag{ A.7 }$$

The rate function associated with J_α is defined as

$$\begin{eqnarray}&&{\rm{Prob}}({J}_{\alpha }\approx x)\sim \exp [-{t}_{f}{I}_{\alpha }(x)],\end{eqnarray} \tag{ A.8 }$$

where ∼ means asymptotic equality in the limit ${t}_{f}\to \infty$. The scaled cumulant generating function for a stochastic protocol is defined as

$$\begin{eqnarray}&&{\lambda }_{\alpha }(z)\equiv \mathop{\mathrm{lim}}\limits_{{t}_{f}\to \infty }\displaystyle \frac{1}{{t}_{f}}\mathrm{ln}\langle \exp ({t}_{f}{J}_{\alpha }z)\rangle .\end{eqnarray} \tag{ A.9 }$$

These two quantities are related by a Legendre–Fenchel transform, as in equation (14).

Similar to equation (21) and equation (50) for a deterministic protocol, we define

$$\begin{eqnarray}&&{\bar{{ \mathcal J }}}_{{ij}}\equiv \displaystyle \sum _{n=0}^{N-1}{{ \mathcal J }}_{{ij}}^{n}\end{eqnarray} \tag{ A.10 }$$

and

$$\begin{eqnarray}&&{\bar{\alpha }}_{{ij}}\equiv \displaystyle \frac{1}{N}\displaystyle \sum _{n=0}^{N-1}{\alpha }_{{ij}}^{n},\end{eqnarray} \tag{ A.11 }$$

respectively. Furthermore, we define

$$\begin{eqnarray}&&{{ \mathcal J }}_{K}\equiv \displaystyle \sum _{i\lt j}{K}_{{ij}}{\bar{\alpha }}_{{ij}},\end{eqnarray} \tag{ A.12 }$$

which is equivalent to (53),

$$\begin{eqnarray}&&{\sigma }_{K}^{* }\equiv \displaystyle \frac{1}{{N}^{2}}\displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}\displaystyle \frac{{({K}_{{ij}})}^{2}}{{{ \mathcal J }}_{{ij}}^{n}}\mathrm{ln}\displaystyle \frac{{\pi }_{i}^{n}{w}_{{ij}}^{n}}{{\pi }_{j}^{n}{w}_{{ji}}^{n}},\end{eqnarray} \tag{ A.13 }$$

which is equivalent to equation (54), and

$$\begin{eqnarray}&&{\tilde{\sigma }}_{K}\equiv \displaystyle \frac{1}{{N}^{2}}\displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}\displaystyle \frac{2{({K}_{{ij}})}^{2}}{{\pi }_{i}^{n}{w}_{{ij}}^{n}+{\pi }_{j}^{n}{w}_{{ji}}^{n}},\end{eqnarray} \tag{ A.14 }$$

which is equivalent to equation (60). The parameter K_ij in these equations is anti-symmetric, i.e., K_ij = −K_ji, and thus fulfill ${\sum }_{j\ne i}\,{K}_{{ij}}=0$ for all i.

We now consider the joint distribution of the vector of empirical densities ρ, the vector of empirical currents J, and the vector of the empirical flow C. The level 2.5 rate function [46] for this Markov process reads

$$\begin{eqnarray}&&{I}_{2.5}[J,C,\rho ]=\displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}\psi ({J}_{{ij}}^{n},{G}_{{ij}}^{n},{a}_{{ij}}^{n})+\displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i}\left({C}_{i}^{n}\mathrm{ln}\displaystyle \frac{{C}_{i}^{n}}{{\rho }_{i}^{n}\gamma }+\gamma {\rho }_{i}^{n}-{C}_{i}^{n}\right),\end{eqnarray} \tag{ A.15 }$$

where

$$\begin{eqnarray}&&{G}_{{ij}}^{n}\equiv {\rho }_{i}^{n}{w}_{{ij}}^{n}-{\rho }_{j}^{n}{w}_{{ji}}^{n},\end{eqnarray} \tag{ A.16 }$$

and

$$\begin{eqnarray}&&{a}_{{ij}}^{n}\equiv 2\sqrt{{\rho }_{i}^{n}{\rho }_{j}^{n}{w}_{{ij}}^{n}{w}_{{ji}}^{n}}.\end{eqnarray} \tag{ A.17 }$$

The quantities in this rate function fulfill the constraint

$$\begin{eqnarray}&&({C}_{i}^{n}-{C}_{i}^{n-1})+\displaystyle \sum _{j\ne i}{J}_{{ij}}^{n}=0,\end{eqnarray} \tag{ A.18 }$$

for all i and n.

Applying a contraction to obtain I_α(x) from I_2.5[J, C, ρ], as in equation (41) for a deterministic protocol, and setting ${\rho }_{i}^{n}={\pi }_{i}^{n}$ and ${C}_{i}^{n}=\gamma {\pi }_{i}^{n}$, we obtain

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}\psi ({\tilde{J}}_{{ij}}^{n},{{ \mathcal J }}_{{ij}}^{n},2\sqrt{{\pi }_{i}^{n}{\pi }_{j}^{n}{w}_{{ij}}^{n}{w}_{{ji}}^{n}}),\end{eqnarray} \tag{ A.19 }$$

where ${\tilde{J}}_{{ij}}^{n}$ fulfill the constraints

$$\begin{eqnarray}&&\displaystyle \sum _{n=0}^{N-1}\displaystyle \sum _{i\lt j}{\tilde{J}}_{{ij}}^{n}{\alpha }_{{ij}}^{n}=x\end{eqnarray} \tag{ A.20 }$$

and

$$\begin{eqnarray}&&\gamma ({\pi }_{i}^{n}-{\pi }_{i}^{n-1})+\displaystyle \sum _{j\ne i}{\tilde{J}}_{{ij}}^{n}=0,\end{eqnarray} \tag{ A.21 }$$

for all i and n.

The global bound on large deviations is obtained by setting

$$\begin{eqnarray}&&{\tilde{J}}_{{ij}}^{n}={{ \mathcal J }}_{{ij}}^{n}+\displaystyle \frac{(x-{{ \mathcal J }}_{\alpha }){K}_{{ij}}}{{\displaystyle \sum }_{i\lt j}{K}_{{ij}}{\bar{\alpha }}_{{ij}}}.\end{eqnarray} \tag{ A.22 }$$

and by using the inequality in equation (46). With these operations, equation (A.19) becomes

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\sigma }_{K}^{* }}{4{({\bar{{ \mathcal J }}}_{K})}^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2},\end{eqnarray} \tag{ A.23 }$$

which is the global bound for a stochastic protocol.

The choice in equation (A.22) and the Taylor expansion in equation (58), together with equation (A.19) lead to the local bound

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{\tilde{\sigma }}_{K}}{4{{ \mathcal J }}_{K}^{2}}{(x-{{ \mathcal J }}_{\alpha })}^{2}+{\rm{o}}(| x-{{ \mathcal J }}_{\alpha }{| }^{2}).\end{eqnarray} \tag{ A.24 }$$

Using the relation (25) for the diffusion coefficient we obtain the bound

$$\begin{eqnarray}&&{D}_{\alpha }\geqslant \displaystyle \frac{{{ \mathcal J }}_{K}^{2}}{{\tilde{\sigma }}_{K}}.\end{eqnarray} \tag{ A.25 }$$

The choice ${K}_{{ij}}={\bar{{ \mathcal J }}}_{{ij}}$ for a stochastic protocol leads to bounds similar to the bounds discussed in section 4.2.1 for a deterministic protocol.

A bound similar to the bound in equation (72) for a stochastic protocol can be obtained by setting ${\tilde{\rho }}_{i}^{n}={\pi }_{i}^{n}$ and

$$\begin{eqnarray}&&{\tilde{J}}_{{ij}}^{n}\equiv {{ \mathcal J }}_{{ij}}^{n}+{c}_{1}{M}_{{ij}}^{n}+{c}_{2}{\bar{{ \mathcal J }}}_{{ij}},\end{eqnarray} \tag{ A.26 }$$

where

$$\begin{eqnarray}&&{M}_{{ij}}^{n}\equiv {\mu }_{i}^{n}{w}_{{ij}}^{n}-{\mu }_{j}^{n}{w}_{{ji}}^{n},\end{eqnarray} \tag{ A.27 }$$

and ${\mu }_{i}^{n}$ is the solution of the stationary master equation ${\sum }_{j\ne i}({\mu }_{i}^{n}{w}_{{ij}}^{n}-{\mu }_{j}^{n}{w}_{{ji}}^{n})=0$. Defining

$$\begin{eqnarray}&&{{ \mathcal J }}_{\mu }\equiv \displaystyle \frac{1}{N}\displaystyle \sum _{n}\displaystyle \sum _{i\lt j}{\alpha }_{{ij}}^{n}{M}_{{ij}}^{n},\end{eqnarray} \tag{ A.28 }$$

and setting

$$\begin{eqnarray}&&{c}_{1}=(x-{{ \mathcal J }}_{\alpha })q{{ \mathcal J }}_{\mu }^{-1}\end{eqnarray} \tag{ A.29 }$$

$$\begin{eqnarray}&&{c}_{2}=(x-{{ \mathcal J }}_{\alpha })(1-q){\left(\displaystyle \sum _{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\bar{\alpha }}_{{ij}}\right)}^{-1},\end{eqnarray} \tag{ A.30 }$$

leads to the fulfillment of the constraint in equation (A.20). With this choice for ${\tilde{\rho }}_{i}^{n}$ and ${\tilde{J}}_{{ij}}^{n}$, the bound in equation (A.19) becomes

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{(x-{{ \mathcal J }}_{\alpha })}^{2}}{4}\displaystyle \frac{1}{{N}^{2}}\displaystyle \sum _{n}\displaystyle \sum _{i\lt j}\displaystyle \frac{{\left({M}_{{ij}}^{n}q{{ \mathcal J }}_{\mu }^{-1}+(1-q){\left({\displaystyle \sum }_{i\lt j}{\bar{{ \mathcal J }}}_{{ij}}{\bar{\alpha }}_{{ij}}\right)}^{-1}{\bar{{ \mathcal J }}}_{{ij}}\right)}^{2}}{{{ \mathcal J }}_{{ij}}^{n}}\mathrm{log}\displaystyle \frac{{\pi }_{i}^{n}{w}_{{ij}}^{n}}{{\pi }_{j}^{n}{w}_{{ji}}^{n}}.\end{eqnarray} \tag{ A.31 }$$

In particular, for q = 1 we obtain

$$\begin{eqnarray}&&{I}_{\alpha }(x)\leqslant \displaystyle \frac{{(x-{{ \mathcal J }}_{\alpha })}^{2}{\sigma }_{\mu }^{* }}{4{{ \mathcal J }}_{\mu }^{2}}\end{eqnarray} \tag{ A.32 }$$

where

$$\begin{eqnarray}&&{\sigma }_{\mu }^{* }=\displaystyle \frac{1}{{N}^{2}}\displaystyle \sum _{n}\displaystyle \sum _{i\lt j}\displaystyle \frac{{({M}_{{ij}}^{n})}^{2}}{{{ \mathcal J }}_{{ij}}^{n}}\mathrm{log}\displaystyle \frac{{\pi }_{i}^{n}{w}_{{ij}}^{n}}{{\pi }_{j}^{n}{w}_{{ji}}^{n}}.\end{eqnarray} \tag{ A.33 }$$