A generalization of the thermodynamic uncertainty relation to periodically driven systems
The thermodynamic uncertainty relation expresses a universal
trade-off between precision and entropy production, which applies in
its original formulation to current observables in steady-state
systems. We generalize this relation to periodically time-dependent
systems and, relatedly, to a larger class of inherently
time-dependent current observables. In the context of heat engines
or molecular machines, our generalization applies not only to the
work performed by constant driving forces, but also to the work
performed while changing energy levels. The entropic term
entering the generalized uncertainty relation is the sum of local
rates of entropy production, which are modified by a factor that
refers to an effective time-independent probability
distribution. The conventional form of the thermodynamic uncertainty
relation is recovered for a time-independently driven steady state
and, additionally, in the limit of fast driving. We illustrate our
results for a simple model of a heat engine with two energy levels.
Keywords: current fluctuations, heat engines, entropy production
Dated: July 19, 2019
One of the main objectives of stochastic thermodynamics is to relate thermodynamic properties of a small system to the statistical fluctuations of its currents, for example the mechanical work, dissipated heat or delivered chemical output [1, 2]. A recent development in this spirit is the thermodynamic uncertainty relation (TUR)
which relates the relative fluctuations of a current, characterized by its diffusion coefficient and average , to the total rate of entropy production [3, 4]. In the following, we set Boltzmann’s constant to unity.
The TUR (1) applies universally to current observables of steady-state systems that can be modeled in continuous time using time-independent Markovian dynamics, either on a discrete network or in continuous space [5, 6, 7]. While this covers already a large class of systems and observables, recent efforts to push the limits of applicability of the TUR even further have been fruitful, leading to variants for, e.g., finite time [8, 9] and first-passage time fluctuations [10, 11]. However, there are various settings of stochastic systems for which a direct application of the TUR fails, calling for modifications that generalize Eq. (1). Such settings include the discrete-time case [12, 13, 14], ballistic transport and coherent dynamics , and systems in linear response with asymmetric Onsager matrices .
In this paper, we focus on time-periodically driven systems as a further prominent setting for which the conventional form (1) of the TUR does not hold . Roughly speaking, the driving protocol itself serves here as an exact external clock that can enable the currents of the system proper to reach a precision that surpasses the limit set by its rate of entropy production. Hence, the TUR can be restored by adding the thermodynamic cost for the external driving to the entropy production of the system proper . Furthermore, systems driven by time-symmetric protocols show similarities to the discrete-time case, allowing for a generalization of the TUR in which the exponential of the entropy production per period enters . Recent work on large deviation theory for arbitrary periodic driving has led to bounds on the large deviation function for current fluctuations [19, 20, 21], which generalize similar bounds that imply the TUR for time-independent driving .
Applied to molecular motors and steady-state heat engines, the TUR yields a fundamental bound on the efficiency, which depends only on the fluctuations of measurable currents [23, 24]. However, paradigmatic models and experimental realizations of stochastic heat engines often use externally controlled, time periodic protocols [25, 26], to which the TUR in its original formulation does not apply . The generalizations of the TUR following from the large deviation bounds in Ref.  apply mainly to current observables that count jumps in Markovian networks with time-independent increments, which covers for example the cycle current generated by a stochastic pump [28, 29]. Instead, the current observables most relevant for heat engines are of an entirely different type. In particular, the work performed on the system is given by the change of the energy of the state that is currently occupied by the system. The generalized thermodynamic uncertainty relation (GTUR) we derive here applies to a broad class of current observables in periodically driven systems, which includes the currents relevant for heat engines. In this generalization, the entropy production in Eq. (1) is replaced by an effective entropy production, which can be larger than and which depends on a comparison between the currents in the periodic stationary state and in a time-independent state of reference. We illustrate the GTUR and an implied generalized bound on the efficiency for a simple two-level heat engine that is alternatingly coupled to two different heat baths.
We consider a Markovian dynamics on a network of states with transition rates from state to that are time-dependent and periodic with period . These rates must be thermodynamically consistent and thus have to obey the local detailed balance condition
where is a possibly time-dependent inverse temperature, the energy difference between internal states and , and a driving affinity caused, e.g., by an external non-conservative force or a chemical reaction supplied by chemostats. These transition rates define a master equation
where the dot denotes a time-derivative and where the periodic matrix has the entries
The entries of vector in (3) give the probability that state is occupied at time . Furthermore, is the time-dependent exit rate and the Kronecker delta. This periodically driven system converges for long times into a periodic stationary state , which is the unique periodic and normalized solution of (3).
A stochastic trajectory of length is characterized by an occupation variable , which is one if state is occupied at time and zero, otherwise. The variable counts the directed total number of jumps from to observed up to time . In contrast to steady-state systems, a current can also depend on the occupation and not only on jumps . For an example, consider work that is performed while driving the energy levels without an external non-conservative force, analogously to the definition of work used in the Jarzynski relation . The associated time-averaged power can be expressed through the occupation variable as
where we use the sign convention such that is positive when work is delivered on average by the system. In the following, such currents that only depend on the occupation are called “occupation currents”, whereas currents that only depend on jumps are called “jump currents”. An example for a jump current is the entropy production 
A general current consisting of two parts, an occupation current and a jump current, reads
where is the instantaneous change of a time-periodic state variable and is the increment associated with a transition from to at time . Averages of currents sampled over one period of the periodic stationary state can be expressed as
where denotes the average over all trajectories in the periodic stationary state and
denotes the periodic stationary probability current. We have used that and . The average of the power (5) delivered while the system is in state is obtained for and reads
The average of the fluctuating entropy production in (6) is obtained for , yielding the average rate of entropy production
Fluctuations of currents in the ensemble of trajectories with are quantified via the scaled cumulant generating function
which is in the following referred to as the “generating function”. Its long-time limit is . Denoting derivatives for with , the average current follows as and the diffusion coefficient associated with that current is given by
The calculation of these quantities using time-ordered exponentials is sketched in A.
Our main result generalizes the TUR (1) to systems driven into a periodic stationary state and is called in the following the generalized thermodynamic uncertainty relation (GTUR). It is valid for all currents defined in (7). The GTUR reads
with the effective rate of entropy production
is the instantaneous periodic stationary entropy production rate associated with the link . The term is an effective current
caused by an time-independent effective density . It will in general not satisfy a conservation law. The effective density is a set of free variation parameters that have to fulfill the condition . For time-independent transition rates, the effective densities can be chosen as the stationary state . Then, the effective currents are the stationary ones and . Hence, (14) assumes the conventional form of the TUR.
We emphasize that the bound (14) has a broader applicability than two earlier generalizations of the TUR. First, it is not restricted to time-symmetric driving as the one in Ref. . Second, our generalization applies not only to currents with time-independent increments, which Ref.  focuses on. Consequently, as we will show below, our bound on precision is not trivial for two-level systems, where all currents with time-independent increments must vanish.
Two different choices for have an immediate physical interpretation. The first choice is defined through
as the stationary solution of the master equation with time-averaged transition rates. If the driving frequency is large compared to the entries of , the periodic stationary state converges to this effective density , see B.
The second choice for the variation parameters is a simple time average over the periodic stationary state
i.e., the average fraction of the total time spent in a state during one period.
These two choices become equivalent in the limiting case of large driving frequencies or for linear response around a genuine non-equilibrium steady state. In leading order, as discussed in B, the periodic stationary state is then time-independent and solves Eq. (18). Consequently, the currents and become the same, which leads to and thus restores the original form (1) of the TUR. However, in those limiting cases where both currents vanish in zeroth order, in particular in linear response around an equilibrium state, and differ in leading order and remains different from .
Illustration: Two level heat engine.
We consider a heat engine that is coupled alternatingly to two different heat baths. It has two states with one energy periodically driven, such that
Here, is an amplitude and an offset with respect to the energy of the first state. In the first half of the period, , the temperature is fixed at a cold inverse temperature and in the second half, , it is fixed at a hot inverse temperature . We choose the individual rates symmetrically according to the local detailed balance condition in (2) as
where determines the basic time scale for particle jumps. A schematic representation of the engine is shown in fig. 1.
For the analysis shown in Fig. 2, we vary the rate amplitude and keep all other parameters fixed. The periodic stationary distribution yielding , , and and the diffusion constants for the respective currents are calculated numerically using the methods outlined in A. The left-hand side (l.h.s) of the GTUR (14) for the two choices in (18) and (19) as well as the l.h.s. of the corresponding steady state TUR (1) are shown for the power (Fig. 2a) and for the entropy production (Fig. 2b) as currents of interest.
For small , i.e., in the fast driving limit , the periodic stationary state approaches a time-independent state, as shown in Fig. 2c. Then, the two choices for the GTUR and the TUR become identical for small , as explained in B. Differences between the two choices for can be seen for larger . In this regime, the choice (18) becomes better than the choice (19) for both currents. Furthermore, the TUR for power is strongly violated for large . Here, the GTUR does hold and becomes sharper again. For the entropy production, the GTUR is less sharp for large where again the TUR does not hold.
Bound on efficiency of heat engines.
The trade-off relation between power, efficiency and constancy, derived in  as a consequence of the TUR, applies to steady-state heat engines, but in general not to periodically driven systems . The GTUR derived here generalizes this trade-off relation and bounds the efficiency of periodically driven heat engines as we show in the following. The formally similar trade-off described in Ref.  applies to periodically driven engines, but does not make reference to power fluctuations.
The efficiency of a heat engine is given by
where is the total output power of the heat engine defined in (10) and is the heat current flowing into the system from the hot reservoir. This efficiency is always bounded by the Carnot efficiency . Following the analogous calculations from Ref. , the efficiency of a periodically driven heat engine is bounded due to the GTUR (14) by the stronger relation
where is the diffusion coefficient (13) of the fluctuating output power.
As an example, we consider the heat engine from fig. 1 and vary the rate amplitude . The effective entropy production is calculated from the choice (18) for the effective density. The quantities entering the bound (23) are shown in fig. 3a and the efficiency of the heat engine and the bounds and are shown in fig. 3b. For small rate amplitudes , where the GTUR assumes the form of the TUR, the new bound based on the GTUR becomes identical to the bound for steady-state heat engines, called here . In the regime of large rate amplitudes, , both the efficiency and the power increase to finite limiting values. Since at the same time fluctuations decrease, the bound becomes rather strong with the actual efficiency being only about below this bound, whereas the bound no longer holds.
We now derive our main result shown in (14). For this purpose, we bound the generating function by introducing an auxiliary dynamics with path weight . A similar formalism has been introduced in Ref.  for continuous degrees of freedoms. The weight of paths from the periodic stationary state is denoted by , so that
where the summation indicates a path integral over all trajectories , and where the occupation variable and the jump variable refer implicitly to these trajectories. We split up as , where is the path weight conditioned in the system being in state at time , which in turn is associated with the probability . Likewise, for the path weight of the auxiliary dynamics, we split , with an a priori arbitrary initial distribution .
The generating function in (12) for a current can be written in terms of and as
where denotes the average over all trajectories in the auxiliary dynamics. This generating function can be bounded by using Jensen’s inequality as
Next, the path weight is chosen such that it produces in analogy to (24) a density , which is the periodic stationary solution for the auxiliary rates , i.e.,
In the following, we denote the currents associated with (27) as
and the corresponding traffic, or activity, as
The second term in (26) can be written as a Kullback-Leibler divergence between the initial distribution of the original periodic stationary distribution and the initial distribution of the auxiliary dynamics
We evaluate the third term in (26) by calculating the fraction of the two path weights for the same trajectory
where are the exit rates of the auxiliary dynamics. Inserting (32) into (26) leads to terms containing averages with the path weight for and , for which we can use (27). We express the rates in terms of the current and the associated traffic as
After optimizing the third term in (26) with respect to the traffic, the optimal rates read
The densities and currents of the auxiliary dynamics must fulfill the conditions
for all and , which guarantee that a matching set of auxiliary transition rates can be found.
Now, we choose a suitable ansatz for the densities and currents of the auxiliary dynamics. One can easily verify that the ansatz
with an arbitrary small optimization parameter fulfills the conditions (38), if . Using this ansatz, one can expand (35) up to order for small to obtain a local bound on the generating function after an optimization with respect to the parameter . Additionally, we restrict ourselves to observation times that are multiples of the period. Then, (35) reads up to
Using the inequality , one obtains the bound given in (14) with an additional term arising from the Kullback-Leibler divergence. Taking the long-time limit , one obtains exactly the GTUR in Eq. (14) with the effective entropy production (15).
As an aside, we note that in the case of time-independent rates the ansatz (39) becomes , , where the upper index “s” denotes the stationary distribution and currents. Using a quadratic bound [4, 32] on in (36), and performing an optimization with respect to leads to the quadratic bound on , which implies the finite-time TUR [8, 9]. In the long-time limit , this lower bound on the generating function becomes equivalent to the upper bound on the large deviation function [22, 4]. In Ref. , such a quadratic bound on has led to a global quadratic bound on the large deviation function for jump currents with time-independent increments in a periodically driven system. The corresponding local bound, though formally similar, is different from the GTUR derived here.
Unlike most variants of the TUR, the present generalization (14) is not a consequence of a simple, usually quadratic, global bound on the large deviation function or generating function. Technically, choosing small and consequently small in Eq. (40) is necessary to ensure that the specific ansatz (39) for the density is positive. From a more general perspective, we note that the fluctuations of occupation currents are always limited to a finite range that is set by those realizations of that maximize or minimize in Eq. (7), which rules out the existence of any global quadratic upper bound on the large deviation function.
Finally, the local quadratic bound in (40) is valid for small enough and also at finite time. This leads to Eq. (41) as a generalization of the finite-time uncertainty relation [8, 9] to periodically driven systems. Here, the Kullback-Leibler divergence (31), which leads to the second term of , does not vanish. This term quantifies the difference between the initial distribution of the periodic stationary state and an effective time-independent distribution. For large driving frequencies the periodic stationary state converges to an effective density , as we show in B, and hence the Kullback-Leibler divergence vanishes. Moreover, the Kullback-Leibler divergence can be brought to vanish by choosing .
We have generalized the TUR to time-periodically driven systems and to a larger class of current observables. This class includes the currents most relevant for periodically driven heat engines, in particular the work associated with changing the energy level of a state occupied by the system.
Our generalization restores the ordinary form of the TUR for the special case of large driving frequencies. Hence, for large driving frequencies precision has a universal minimal cost. This is somewhat remarkable, because although the system can be described by a time-independent distribution a one-to-one mapping of a periodically driven system to a steady-state system fails at the description of currents. Thus, we extend the applicability of the TUR and the ensuing trade-off relations to heat engines driven solely by fast alterations of energy levels and temperature.
For moderate or low driving frequencies one has to compare the periodic stationary currents with the associated effective currents in a time-independent state of reference. One can then predict whether a larger entropy production or smaller currents than the effective ones are needed for a higher precision. Furthermore, due to the generalization of the TUR, one can bound the efficiency of heat engines and hence predict whether an engine is able to work close to Carnot efficiency or not. Finally, we note that by setting , the bound on efficiency for heat engines can be adapted to isothermal engines transforming work to heat or work to work.
A formulation of the GTUR for overdamped Brownian motion is straightforward, either by performing the continuum limit on a finely discretized state space or by redoing the derivation using the path weights pertaining to Langevin dynamics.
Work funded in part by the ERC under the EU Horizon 2020 Programme via ERC grant agreement 740269.
Appendix A Calculation of cumulants
The solution of the time-dependent master equation (3) for an initial distribution is formally given by the time-ordered exponential
defining the evolution operator . There exists a unique initial condition that corresponds to the periodic stationary state . This initial condition can be determined using the periodicity of , which leads to the eigenvalue equation
Hence, this initial distribution is the eigenvector of with eigenvalue one.
with the tilted evolution operator and the tilted generator with entries . In the long-time limit, the generating function follows as
with being the maximal eigenvalue of . For the illustration of the GTUR and the TUR for the two-level heat engine, we have calculated in a small region around , yielding through numerical differentiation.
Appendix B Limiting cases
In the limit of fast driving, where for all transition rates at all times, the time-ordered exponential in (42) can be expanded as
where stands generically for the scaling of all transition rates. The eigenvalue equation (43) is then given in leading order by (18). Then, the periodic stationary state is in leading order time-independent and given by the first choice of the effective density , i.e., Due to its time-independence, the leading order of is also captured by the second choice for (19). Consequently, the periodic stationary currents and the effective currents, while still being time-dependent, become equal in the leading zeroth order, i.e., and thus in Eq. (14).
Another special case where the original form of the TUR is restored is the one where the transition rates become time-independent and correspond to a genuine non-equilibrium steady state, which leads to non-zero stationary currents . Remarkably, this result goes beyond the classical statement of the TUR if the increments and are still periodically time-dependent. However, in the limiting case of time-independent transition rates that correspond to a non-driven system, the currents and both vanish and differ in leading order, such that does not approach .
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