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Absence of Thermodynamic Uncertainty Relations with Asymmetric Dynamic Protocols


 Many versions of Thermodynamic Uncertainty Relations (TUR) have recently been discovered, which impose lower bounds on relative fluctuations of integrated currents in irreversible dissipative processes, and suggest that there may be fundamental limitations on the precision of small scale machines and heat engines. In this work we rigorously demonstrate that TUR can be evaded by using dynamic protocols that are asymmetric under time-reversal. We illustrate our results using a model heat engine using two-level systems, and also discuss heuristically the fundamental connections between TUR and time-reversal symmetry.


I. INTRODUCTION
In recent years, several classes of inequalities have been discovered, which indicate tradeoffs between various aspects of non-equilibrium processes. For example, many versions of thermodynamic uncertainty relations (TUR)  [23] impose lower bounds of integrated current fluctuations in terms of entropy production, which signify a trade-off between dissipation and precision for small-scale machines. Another example is the so-called speed-limit inequalities [24][25][26][27][28][29][30][31], which indicate a trade-off between efficiency and power output for general Markov processes. Most recently, Dechant and Sasa [32] derived an upper bound for non-equilibrium response function in terms of fluctuations and relative entropy between the perturbed and reference states, which generated immediate interests [21,33,34]. Given the large number of works published in recent years, it is highly desirable and urgent to understand whether these inequalities are consequences of more fundamental aspects of non-equilibrium physics, such as Fluctuation Theorems and Markovian property, or rather depend on specific system details.
The current situation of TUR is particularly vibrant and complex [5]. The first version of TUR, which was proposed [1] and proved [2] for continuous time Markov jump processes with local detailed balance, has the form: δQ 2 / Q 2 ≤ 2/ Σ , where Q is certain integrated current, and Σ the average entropy production. It was very quickly generalized to other types of irreversible processes, including finite-time processes [11,14,15,22] discrete time processes [17,18], and most recently quantum processes [13,21]. Its relation to the linear response theory has also been clarified [19]. More recently, Timpanaro et. al. [6] and Hasegawa et. al. [7] proved tighter TUR bounds in the form of δQ 2 / Q 2 ≤ f ( Σ ) for general processes with symmetric dynamic protocols. For small Σ , f ( Σ ) ∼ 2/ Σ , so that the original TUR is restored. For large Σ , f ( Σ ) vanishes exponentially. (Note that in the limit of large entropy production Σ → ∞, both the original TUR bound in Refs. [1,2] and the generalized bounds in Refs. [6,7] vanish. This is completely expected, since large Σ means either large system size or long time. Either way, the law of large number come to play, so relative fluctuation of entropy production always reduces to zero. In another word, various TUR bounds discovered for symmetric protocols are effective only for small size systems in short time.) These generalized TUR has their origin in Fluctuation Theorem [35][36][37], and hence are universal and independent of model details. The deep connection between fluctuation of entropy production and Fluctuation Theorem was previously studied by Merhav and Kafri [38].
While almost all previous works concern processes with dynamic protocols that are symmetric under time-reversal, employment of asymmetric dynamic protocol provides another dimension for the issue of TUR. Results about asymmetric dynamic protocols are scarce, but did indicate a very different scenario. Barrato and Seifert [39] showed that Brownian clock driven periodically can achieve arbitrary high precision with arbitrary low dissipation.
Chun et. al. [40] showed that TUR may not work for system coupled to a magnetic field, which explicitly breaks the time-reversal symmetry. More recently, Cangemi et. al. [41,42] discovered violation of TUR in quantum energy converters with asymmetric protocols. In light of these results, it is highly desirable to know whether in principle TUR-bound can always be evaded via clever design of asymmetric dynamic protocols. This question, which is independent of model details, is of fundamental importance for study of small scale machines [43][44][45][46][47][48][49].
In this letter, we shall rigorously establish the absence of TUR-bound for processes with asymmetric dynamic protocols. We shall minimize the relative current fluctuations with the constraints of (1) fixed entropy production and (2) satisfaction of Detailed Fluctuation Theorem, which is valid for processes with asymmetric protocols, and show that the lower bound for the current fluctuations is strictly zero. Hence all TUR-bounds can be evaded via clever designs of asymmetric dynamic protocols. To illustrate our results, we also discuss a model of heat engine whose output fluctuation can be tuned arbitrarily small. Finally we present a heuristic discussion in terms of Detailed Fluctuation Theorem why TUR-bounds exist in processes with symmetric dynamic protocols but not in those with asymmetric dynamic protocols. Our results provide important insights for design of high precision microscopic machines and heat engines.

II. FLUCTUATION THEOREM
Consider an irreversible stochastical process with path probability distribution P U [γ], where U = {λ(t), t ∈ (0, T )} denotes the dynamic protocol, and λ(t) is the time-dependent external parameter, and (0, T ) is the time-range of the process. Associated with this process is the backward process, with a reversed protocolŪ = {λ(t) = λ * (−t), t ∈ (−T, 0)} and a path probability distribution PŪ [γ]. Note that λ * is related to λ via reversal of odd parameters, such as magnetic field. Assuming that there is separation of time-scale, and that all slow variables included in the model, the entropy production Σ[γ] of the forward process along the path γ satisfies [52,53] , i.e., entropy production changes sign under the reversal of both the path and the dynamic protocol. Hereγ is the time-reversal of the path forward γ, which is obtained from γ via reversal of both time and odd variables. Equation (2.1) has been established for classical systems on very general ground [52,53]. It is also known to hold for certain quantum systems [54]. We shall be interested in asymmetric protocols i.e., U =Ū , which means that the forward and backward processes are macroscopically different.

Now consider a certain integrated current
Generalizing a theorem due to van der Broeck and Cleuren [55], we can readily obtain a generalized version of Detailed Fluctuation Theorem (DFT): All statistical properties of Σ and Q for the forward and backward processes can be obtained from p(σ, q) andp(σ, q). If the dynamic protocol is symmetric,Ū = U andp = p, and Eq. (2.4) reduces to: which was the starting point of studies in Ref. [6,7]. Our aim is to minimize the variances δQ 2 and δΣ 2 under the constraint of fixed averages Σ and Q , for all distributions that satisfying Eq. (2.4).

III. MINIMIZATION OF FLUCTUATIONS
In this section we will show that DFT (2.4) implies no TUR bound for fluctuations.
Any continuous probability distribution can be approximated, to an arbitrary precision, by a discrete distribution. Hence we only need to study discrete distributions. We shall use the term N-point distribution to denote a probability distribution in the form we can always find a pair of 2-point distributions p 2 (σ, q),p 2 (σ, q) such that, comparing with p N (σ, q), p 2 (σ, q) has same averages Σ , Q and smaller variances δΣ 2 and δQ 2 .
The detailed proof of this lemma is shown in SI.
Let us further try to find the pair p 2 (σ, q),p 2 (σ, q) that produces the minimal fluctuations δΣ 2 and δQ 2 . We shall first treat the simple case Q = Σ, so that we only need to minimize δΣ 2 with fixed Σ . Let us write: (3.1c) Using Eqs. (3.1) we find averages and fluctuations of Σ for the forward and backward pro-cesses: If σ 1 = 0 (σ 2 = 0), p = 1 (p = 0), the entropy production is always zero, which corresponds to a reversible process. This is not what we aim to study in this work, hence we shall assume that neither of σ 1 , σ 2 vanish. From Eq. (3.1c) we see to make p or 1 − p both positive, σ 1 , σ 2 must have different signs. Let us assume σ 1 < 0 < σ 2 .
Let us now consider the general case where Q is an integrated current distinct from entropy production. The joint distributions of Σ and Q have the following forms: p 2 (σ, q) = p e −σ 1 δ(σ + σ 1 )δ(q + q 1 ) where p is still given by Eq. (3.1c). Whilst the average entropy productions are still given by Eqs. (3.2), the average and variance of Q can be calculated using Eqs. (3.4). Recall that we fix σ 2 > 0 and let σ 1 → −∞. In this limit, we have for the forward process: For all reasonable physical models, we expect that entropy production scales quadratically (or at least polynomially) with the current. Hence q 1 diverges with σ 1 , whereas q 2 remains finite. From Eqs. (3.5) we find that δQ 2 converges to zero, whereas Q remains fixed.
The relative fluctuation of Q converges to zero: For the backward process we have hence both δQ 2 and Q diverge with q 1 . The relative fluctuation of Q converges to the same limit as in Eq. (3.3f): Naturally one may wonder whether it is possible to make current fluctuations small for both the forward and backward process. This is impossible. In Refs. [8,9] the following inequality was derived from DFT (2.4):  non-vanishing TUR bound for current fluctuations. This does not contradict our theory.
Instead our theory means that the TUR bound discovered in Ref. [56] can be evaded if more sophisticated protocols are used.

IV. A MODEL HEAT ENGINE
We shall now discuss a concrete irreversible process with finite Σ and vanishingly small current fluctuations. Consider a two-level system with a Hamiltonian where the energy of the excited state ∆ that can be tuned externally. We shall construct an irreversible cycle that starts from a Gibbs stateρ(∆) = Z −1 e −βĤ(∆) , illustrated by panel (a) of Fig. 1 (top left), which can be obtained by connecting the system to a heat bath with temperature T = 1/β. As illustrated by the black arrows in Fig. 1, the cycle consists of the following four steps (a) 1. Disconnection of the system from the heat bath.
3. Equilibration process: We reconnect the system to the bath, and the system reequilibrates to a new Gibbs stateρ = Z −1 e −βĤ(0) =Î/2. This step is irreversible process with an entropy increase.

4.
Isothermal process: With the system connected to the bath, the gap is tuned isotatically back to ∆ so that the density matrix returns to the initial stateρ(∆) = Z −1 e −βĤ(∆) .
The only integrated current in this problem is the entropy production. Since the process is cyclic, the entropy production is related to the work done by the external force via Σ = βW .
Let us consider the adiabatic process. Along the path γ 1 , the system remains in the excited state, and the external force does work −∆. Along the path γ 2 , the work is identically zero.
During the isothermal process, two path coincide, and the work done equals to the change of system free energy, which is T log 2/(1 + e −β∆ ). Hence the total works along two paths are respectively: (4.2b) Hence the path probabilities and the entropy production of γ 1 , γ 2 are respectively This is a special case of Eqs. (3.1), with only one independent parameter p (or equivalently β∆).
The backward process is obtained by reversing the dynamic protocols, shown as all red arrows in Fig. 1. In the backward process, the system starts from the equilibrium statê ρ = Z −1 e −βĤ(∆) (a), and goes through consecutively isothermal, disconnection, and adiabatic, and connection steps. Note the reversed operations of disconnection/connection with the bath is connection/disconnection with the bath. For example, in the forward process, Figure 2: A cartoon of pdf p(σ, q) for processes with symmetric dynamic protocol. The high peak in the right is centered at (q 0 , σ 0 ), whereas the mirror peak in the left, which is dictated by the Detailed Fluctuation Theorem, is located at (−q 0 , −σ 0 ). The mirror peak can be diminished by using asymmetric dynamic protocols, thereby evading TUR bounds.
the system gets disconnected from the bath with ∆ > 0, whereas in the backward process, disconnection happens when ∆ = 0. This shows explicitly that the protocol is indeed asym-  We easily verify that Eqs. (4.3) satisfy the entropy production formula, Eq. (2.1).

V. HEURISTIC DISCUSSION
We conclude our work with a heuristic discussion about TUR bounds in general irreversible processes. We aim, via tuning of dynamic protocols, to reduce the fluctuations of Σ and Q, with their averages fixed and finite. The distribution p(σ, q) then should have significant values only near a single point (σ 0 , q 0 ). Everywhere else p(σ, q) must be negligibly small. However, if the dynamic protocol is symmetric under time-reversal, Eq. (2.5) must be respected, according to which there is a peak at the mirror point (−σ 0 , −q 0 ), as illustrated in Fig. 2. Equation (2.5) also dictates that the probabilities of the peak and of the mirror peak are respectively 1/(1 + e −σ 0 ) and 1/(1 + e σ 0 ). Using these probabilities one easily obtain Σ = σ 0 tanh(σ 0 /2) and δQ 2 / Q 2 = 2 csch 2 (σ 0 /2). The latter is in fact exactly the sharp TUR bound obtained recently by Timpanaro et. al. [6]. For processes with asymmetric dynamic protocols, however, the relevant Detailed Fluctuation Theorem is Eq. (2.4) rather than Eq. (2.5). But Eq. (2.4) does not imposes any constraint on p(σ, q).
Instead, it determinesp(σ, q) in terms of p(σ, q). Hence by using asymmetric protocols, the mirror peak can be arbitrarily diminished, and the TUR bound can be evaded, with average entropy production finite. Note that this heuristic discussion is applicable to models with continuous distributions of dynamic paths as well.