PaperlayerView exact revision on arXiv

Source-linked AI summary

Cost and Precision of Brownian Clocks

Andre C Barato, Udo Seifert

arXiv:1610.07960v2cond-mat.stat-mechphysics.bio-ph

TL;DR

The paper asks how thermodynamic cost limits the precision of Brownian clocks. It compares fixed-force clocks with periodically driven clocks and uses a stochastic-protocol bipartite Markov representation to calculate dispersion. Fixed-force clocks have a precision cost, whereas periodic driving can achieve arbitrarily high precision with arbitrarily low dissipation, within the model’s scope.

  • Problem

    The paper investigates the minimal energy budget required for a Brownian clock to achieve a specified precision.

  • Method

    It maps a periodically driven clock to a bipartite Markov process with stochastic protocol changes, enabling analytical calculation of current, dispersion, and dissipation.

  • Results

    A fixed-force clock obeys a nonzero precision–cost bound, while a periodically driven clock can approach arbitrarily small uncertainty with vanishingly small dissipation.

  • Takeaways & Limitations

    Periodic external driving is fundamentally different from fixed thermodynamic forcing because high clock precision need not require a minimum energy budget.

  • Takeaways & Limitations

    The paper’s explicit model uses a simple ring geometry, although its authors state that the conclusion remains sufficient as an in-principle result for periodically driven systems.

Abstract

from arXiv · show

Brownian clocks are biomolecular networks that can count time. A paradigmatic example are proteins that go through a cycle thus regulating some oscillatory behaviour in a living system. Typically, such a cycle requires free energy often provided by ATP hydrolysis. We investigate the relation between the precision of such a clock and its thermodynamic costs. For clocks driven by a constant thermodynamic force, a given precision requires a minimal cost that diverges as the uncertainty of the clock vanishes. In marked contrast, we show that a clock driven by a periodic variation of an external protocol can achieve arbitrary precision at arbitrarily low cost. This result constitutes a fundamental difference between processes driven by a fixed thermodynamic force and those driven periodically. As a main technical tool, we map a periodically driven system with a deterministic protocol to one subject to an external protocol that changes in stochastic time intervals, which simplifies calculations significantly. In the non-equilibrium steady state of the resulting bipartite Markov process, the uncertainty of the clock can be deduced from the calculable dispersion of a corresponding current.

I. INTRODUCTION

The paper asks how much energy a Brownian clock needs to achieve a specified precision. It contrasts fixed-force clocks with periodically driven clocks and develops a stochastic-protocol framework for analyzing the latter.

  • The central question is the minimal energy budget required for a Brownian clock to achieve a given precision.
  • Fixed thermodynamic-force clocks and periodically driven clocks form two distinct classes with different precision–dissipation relations.
  • A periodic external protocol can produce high precision with an arbitrarily small energy budget, unlike a clock driven by a fixed thermodynamic force.
  • Because deterministic periodic steady states make dispersion calculations difficult, the paper studies protocols that change at random time intervals.
  • Representing the clock and stochastic protocol as a bipartite Markov process enables analytical calculation of clock dispersion and identification of high-precision, low-dissipation parameters.

II. BROWNIAN CLOCK DRIVEN BY A FIXED THERMODYNAMIC FORCE

For a Brownian clock driven by a constant thermodynamic force, precision, clock structure, and energetic cost obey a universal thermodynamic uncertainty relation. Achieving small uncertainty therefore requires a nonzero energetic price.

  • A biased random walk on a ring counts time through net clockwise revolutions, with backward crossings assigned negative clock-time increments.
  • Uniform transition rates minimize Cε2 at fixed affinity, yielding the uncertainty relation Cε2 ≥ 2.
  • For precision ε = 10^-2 over one hour, both slow and fast designs require at least 10^4 elementary steps, with N_min = 167 and 3, respectively.
  • The slow and fast one-hour designs have minimum affinities approximately 333 and 5.55, yet both have an overall cost bounded by 20000.
  • Because ATP hydrolysis releases approximately 20kBT under physiological conditions, the bound implies consumption of 1/(10ε2) ATP molecules for uncertainty ε.

A. Model Definition

The periodic-clock model uses a ring whose energies and barriers rotate during a protocol period. Replacing deterministic protocol changes with stochastic jumps produces an analytically tractable bipartite description.

  • The protocol period τ is divided into N intervals, and the energies and barriers shift one step clockwise between intervals.
  • The clock has N states with time-dependent forward and backward rates determined by rotating site energies and energy barriers.
  • The clock position i and energy-label variable α transform differently: rates rotate forward for i, while α undergoes an effective backward transition.
  • The model replaces deterministic protocol changes with stochastic changes occurring at rate γ = N/τ, while retaining the same clock-counting variable X.
  • For a continuous deterministic periodic protocol, a bipartite process can be constructed with the same probability distribution as the periodic steady state.

B. Optimal Time-Scales and Energy Barriers

The stochastic-protocol clock can outperform the fixed-force uncertainty–cost bound by choosing suitable time scales and energy barriers. In an appropriate limit, its precision–dissipation product becomes very small.

  • Analytical expressions for current, entropy production, and diffusion yield Cε2 as a function of the transition rates.
  • For N = 3, the minimum product is Cε2 ≃ 1.33651, below the fixed-force limit 2, and the minimum decreases as N increases through N = 6.
  • The optimum uses χ1 = χ2 = … = χN−1 = χ ≫ γ and (χN)^−1 → 0, corresponding to a strongly separated transition-rate scale.
  • Entropy production equals the work rate from protocol variation, with each protocol jump contributing the system’s energy change after the jump.

C. Dissipation-less Clock I: Simple Profile

A simple energy profile shows that periodically driven clocks can make the cost–precision product arbitrarily small, unlike clocks driven by a fixed thermodynamic force.

  • C. Dissipation-less Clock I: Simple Profile: The section introduces the simple profile before comparing it with an optimized energy profile that minimizes Cϵ2.The displayed Fig. 4 profile is the optimized profile, not the simple profile analyzed here.
  • C. Dissipation-less Clock I: Simple Profile: The product Cϵ2 is computed from the clock's dispersion, dissipation, and current through Cϵ2 = 2Dσ/J2.This expression is evaluated for the simple profile using the model's current and diffusion formulas.
  • C. Dissipation-less Clock I: Simple Profile: For N = 64 and E = 5.7, the simple profile achieves Cϵ2 ≃0.11, and the product can approach zero when e^E ≫ N ≫ E.The resulting dissipation-less clock contrasts with the fixed-force bound Cϵ2 ≥2.
  • C. Dissipation-less Clock I: Simple Profile: The profile's large barrier makes the particle follow the moving barrier nearly unidirectionally, with current set by the barrier velocity.The condition e^E ≫ N keeps the particle away from the highest-energy state, while N ≫ E keeps dissipation small.

D. Dissipation-less Clock II: Optimal Profile

An optimized energy profile suppresses dissipation and makes the cost–precision product vanish with increasing clock size; deterministic driving reduces diffusion further.

  • D. Dissipation-less Clock II: Optimal Profile: The optimal profile becomes flatter in the middle as N grows, with J → γ/N and D → γ/(2N^2).These limits correspond to a unidirectional random walk performed by the moving barrier.
  • D. Dissipation-less Clock II: Optimal Profile: For an optimal N = 64 profile, Cϵ2 ≃0.0047 and scales as N−2, while uncertainty ϵ = 10−2 costs approximately 47kBT.The reported cost is much lower than the 20000kBT fixed-force cost for the same precision.
  • D. Dissipation-less Clock II: Optimal Profile: Other profiles also yield dissipation-less precision, including Eα = −α/N^φ with 0 < φ < 1, for which Cϵ2 ∼ N−φ.The optimized profile uses the same large-barrier mechanism while suppressing dissipation as N−2.
  • D. Dissipation-less Clock II: Optimal Profile: For a deterministic protocol, simulations up to N = 8 find the same J and σ but smaller D, so Cϵ2 also vanishes for large N.The smaller diffusion results from removing randomness in protocol waiting times.

E. Numerical Case Study

For N = 3, an externally driven clock reaches Cϵ2 ≃1.33651, outperforming the optimal fixed-affinity tradeoff over the region shown in the contour plot.

  • E. Numerical Case Study: For large B and x, the externally driven N = 3 clock reaches its minimum Cϵ2 ≃1.33651.The case study uses E1 = 0, E2 = −1.21938, E3 = −1.43550, χ1 = χ2 = 1, χ3 = 10−B, and γ = 10−x.
  • E. Numerical Case Study: The external-protocol result is below the fixed-affinity optimum (A/3) coth(A/6) in the contour-plot region identified by the caption.That fixed-affinity expression is the optimal product for N = 3 under a constant thermodynamic force.
  • E. Numerical Case Study: For the optimal fixed-affinity clock, Cϵ2 = (A/3) coth(A/6) increases with affinity and reaches 2 as A → 0.Increasing A enlarges the region where the externally driven clock has a smaller product.

IV. DISCUSSION AND CONCLUSION

The paper shows that externally driven Brownian clocks can achieve small uncertainty with vanishingly small dissipation, unlike clocks driven by fixed thermodynamic forces. Its stochastic-protocol framework makes the relevant dispersion analytically calculable, while the thermodynamic cost depends on whether protocol generation is included.

  • IV. DISCUSSION AND CONCLUSION: External periodic protocols can achieve high precision with vanishingly small dissipation, unlike fixed-force clocks constrained by a thermodynamic uncertainty relation.The result is established for a specific model and demonstrates that Cε^2 can approach zero under an external protocol.
  • IV. DISCUSSION AND CONCLUSION: The fixed-force thermodynamic uncertainty relation applies beyond ring geometries to multicyclic networks, supporting the broader scope of the cost bound.The paper notes that the relation is valid for any multicyclic network of states, not only the model geometry considered.
  • IV. DISCUSSION AND CONCLUSION: Stochastic-time protocol changes convert the clock and protocol into a bipartite Markov process, enabling standard steady-state methods to calculate diffusion-related quantities.This framework was crucial to obtaining the main result and avoids the unavailable analogous calculation for standard deterministic periodic steady states.
  • IV. DISCUSSION AND CONCLUSION: Counting the entropy production required to generate the external stochastic protocol restores the thermodynamic uncertainty relation for the full bipartite process.The distinction therefore depends on treating the protocol as truly external; chemically generated protocols incur an additional thermodynamic cost.
  • IV. DISCUSSION AND CONCLUSION: Experimental confirmation of both the fixed-force cost bound and the low-dissipation high-precision limit remains an open challenge.Suggested experimental platforms include single molecules, colloidal particles, and small electronic systems.

Appendix A: External protocols that change at stochastic times

The appendix considers periodic protocols whose changes occur at stochastic times.

  • Appendix A: External protocols that change at stochastic times: The appendix considers a theoretical framework for systems driven by protocols that change at stochastic times.

1. Two state model

The two-state model compares a continuously driven periodic steady state with a stochastic protocol whose discrete energy changes approximate it as L increases. The stochastic construction reproduces the periodic probabilities and work behavior in the large-L limit.

  • Model: The stochastic protocol represents a two-state system whose lower state has energy 0 and upper state has time-dependent energy En.The protocol advances from En to En+1 at rate γ, with En sampled over L subdivisions of the period.
  • Two-state comparison: As L increases, the stochastic protocol’s conditional probability P(u|n) converges to the periodic steady-state probability RPS(t).The stochastic protocol uses γ = L/τ, with P(u|n) compared against RPS(t).
  • Equivalence: For any periodic steady state, a bipartite stochastic process can be constructed whose stationary probability converges to the periodic distribution as L →∞.The external protocol and system together form a process with 2 × L states in the two-state example.
  • Work: For large L, the work rate ˙w of the stochastic protocol approaches the periodic-steady-state rate ˙wPS.At smaller L, the rates differ quantitatively but both increase with ω.

2. General theory

The general theory embeds a system and a randomly changing external protocol in a bipartite Markov process. This representation preserves periodic steady-state probabilities in the continuum limit and makes clock-current dispersion analytically calculable.

  • Thermodynamics: The bipartite process permits entropy production to be decomposed into contributions from system affinities and external protocol variation.For the Sec. III model, only the work associated with external protocol variation contributes.
  • Current fluctuations: Random protocol changes simplify analysis because the clock’s diffusion coefficient can be calculated from an elementary probability current.The current increments for a jump from a to b and decrements for the reverse jump, enabling calculation of its current and diffusion coefficient.
  • Clock observable: The clock’s elapsed time is represented by a fluctuating ring current that counts forward revolutions and assigns backward crossings a negative increment.This current avoids over-counting sequences involving backward motion across the ring boundary.
  • Construction: The stationary distribution is obtained by discretizing the period into L intervals and assigning each interval time-independent transition rates.The resulting matrices Mn approximate the periodic generator at times nτ/L.
  • Periodic-state equivalence: A bipartite process with γ = L/τ becomes equivalent to the periodic steady state as L →∞.The full process has N × L states, while the protocol cycles irreversibly through its L states.
Loading 1610.07960v2…