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The power of a critical heat engine
Michele Campisi, Rosario Fazio
TL;DR
The paper addresses whether Carnot efficiency can be approached at finite power, a question framed by the conventional power-efficiency trade-off. It uses finite-size scaling for interacting quantum Otto engines near second-order phase transitions and finds that critical exponents can enable this asymptotic approach without sacrificing power per resource, subject to practical and exact-point limitations.
Problem
The central question is whether a heat engine can approach Carnot efficiency while delivering finite power, despite the power-efficiency trade-off.
Method
The paper studies interacting N-body quantum Otto engines using finite-size scaling near a second-order phase transition.
Results
α − zν > 0 gives more-than-linear performance-rate scaling, while critical heat-capacity divergence enhances work output and can support approaching Carnot efficiency without sacrificing power per resource.
Takeaways & Limitations
Critical scaling provides a route to approaching the Carnot point asymptotically while retaining fixed power per constituent, with the approach rate determined by critical indices.
Takeaways & Limitations
Exactly at the Carnot point, quantum Otto engines deliver zero work and zero power; the result therefore concerns an asymptotic approach, while implementation also requires global coupling and increasingly accurate control.
Abstract
from arXiv · showhide
Since its inception about two centuries ago thermodynamics has sparkled continuous interest and fundamental questions. According to the second law no heat engine can have an efficiency larger than Carnot's efficiency. The latter can be achieved by the Carnot engine, which however ideally operates in infinite time, hence delivers null power. A currently open question is whether the Carnot efficiency can be achieved at finite power. Most of the previous works addressed this question within the Onsager matrix formalism of linear response theory. Here we pursue a different route based on finite-size-scaling theory. We focus on quantum Otto engines and show that when the working substance is at the verge of a second order phase transition diverging energy fluctuations can enable approaching the Carnot point without sacrificing power. The rate of such approach is dictated by the critical indices, thus showing the universal character of our analysis.
INTRODUCTION
The paper asks whether finite power can coexist asymptotically with Carnot efficiency and proposes using interactions and critical scaling in quantum Otto engines to improve this trade-off. Near a second-order phase transition, critical exponents can make the performance rate grow faster than linearly with system size.
- INTRODUCTION: The open problem is whether a heat engine can deliver finite power at Carnot efficiency, despite the usual power-efficiency trade-off.The paper measures this trade-off through the deviation Δ η = η_C − η from Carnot efficiency.
- INTRODUCTION: An array of N identical engines in parallel increases power linearly while leaving efficiency, and therefore Δη, unchanged.This gives performance rate scaling as Π̇ ∼ N but requires a proportional increase in resources.
- INTRODUCTION: A genuine improvement requires Π̇ ∼ N^(1+a), with a > 0, so Carnot efficiency can be approached while power per resource remains fixed.Under this scaling, Δη ∼ N^−a tends to zero while P ∼ N.
- INTRODUCTION: The proposed route is to introduce interactions among N constituents and study a quantum Otto engine near a second-order phase transition.The paper reports universal anomalous scaling of the performance rate in this regime.
- INTRODUCTION: The condition α − zν > 0 yields more-than-linear performance-rate scaling, combining heat-capacity and relaxation-time critical contributions.The stronger condition α − zν > 1 supports asymptotic approach to Carnot efficiency without sacrificing power per resource.
- INTRODUCTION: The performance scaling is linked to the working substance’s heat capacity, which can diverge at criticality and thereby enhance the work output.The paper identifies this heat-capacity dependence as a previously unnoticed mechanism underlying the result.
N-body quantum Otto engine
The quantum Otto engine is a four-stroke cycle whose efficiency remains below Carnot's bound. Near the Carnot point, performance is governed by the output-work slope, which can become superlinear when the working substance has anomalous heat-capacity scaling.
- Cycle structure: The quantum Otto engine uses thermal isolation for Hamiltonian changes and thermalisation with the baths at fixed Hamiltonian parameters.Its four strokes alternate between parameter changes and bath thermalisation.
- Efficiency: For β1 < β2, engine operation requires λ2/λ1 ≥ β1/β2, while efficiency satisfies η ≤ ηC.The efficiency is smaller than the Carnot efficiency in the stated operating regime.
- Performance: Near the Carnot point, Π is determined by the slope ∂Wout/∂∆η at ∆η = 0, so superlinear performance requires that slope to grow faster than N.For N independent devices in parallel, the slope and performance scale only linearly with N.
- Thermal mechanism: Larger heat capacity increases the heat exchanged during thermalisation and therefore increases the output work near the Carnot point.The relevant specific heat is cK(θ) = −(1/N)θ^2∂UK/∂θ.
- Critical scaling: At a second order phase transition, finite-size scaling predicts c̄K ∼ N^α/(dν), yielding Π ∼ N^1+α/(dν).The specific-heat peak height scales anomalously while its width scales as δ ∼ N^−1/(dν).
- Performance rate: The performance rate additionally depends on cycle time, which is dominated by thermalisation and scales as Trelax ∼ N^z/d.Thus both heat-capacity and relaxation-time critical exponents contribute to the rate.
Critical engine design
The proposed design tunes a scalable working substance to its critical point and fixes output work according to the relaxation-time scaling. With suitable critical exponents, efficiency approaches Carnot asymptotically at fixed power per constituent, while the linear-response region narrows.
- Critical engine design: The working substance is chosen with a second order transition, and λ rescales its critical temperature to match bath 1.The Hamiltonian is implemented as HWS(t) = λ(t)K with λ(t) ∈ [λ1, λ2].
- Fixed-power construction: Choosing Wout = N^1+z/d w makes power per constituent fixed because the cycle time scales as T ∼ N^z/d.Here w is held fixed while λ2 determines the corresponding efficiency.
- Carnot approach: When α − νz > 0, ∆η ∼ N^−(α−νz)/(dν), so the efficiency approaches ηC as N increases.The approach follows from the competition between output-work scaling and the performance scaling set by the heat-capacity slope.
- Illustration: For Dy2Ti2O7-inspired indices α ≃ 0.38 and zν ≃ −0.7, the exponent characterising the approach reaches approximately −0.67.The plotted Carnot point is ηC = 1/2 at fixed rescaled work w = 0.1.
- Finite-size regime: The linear region around ∆η = 0 shrinks with increasing N because the specific-heat peak narrows, yet the fixed-rescaled-work intercept remains inside that region.The construction uses universal critical scaling rather than details of a particular microscopic model.
DISCUSSION
The work develops a finite-size-scaling route toward critical powerful Carnot engines, while clarifying that exact Carnot efficiency still yields zero power and implementation faces technological challenges.
- DISCUSSION: The work confirms that phase transitions can support asymptotic approach to Carnot efficiency at finite power through universality and finite-size scaling.It also accounts for criticality's effect on operation time and therefore power.
- DISCUSSION: Exactly at the Carnot point, all quantum Otto engines deliver null work and null power; the claim concerns asymptotic approach while retaining power per constituent.The approach can be made arbitrarily close to Carnot efficiency without sacrificing power per constituent.
- DISCUSSION: The analysis uses a condition ∆η ≪1 that differs from linear response, so the two bath temperatures need not be close.This distinguishes the treatment from prior analyses based on β2 −β1 ≪β2 or η ≪1.
- DISCUSSION: Realising critical powerful Carnot engines is constrained by technological challenges, including global control of λ(t) and increasingly precise control of λ1 and λ2.The required accuracy of λ1 and λ2 must increase as the Carnot point is approached asymptotically.
- DISCUSSION: Increasing specific heat is identified as a general route to improving working-substance performance, achievable through more constituents or other Hamiltonian parameters.The result in Eq. (9) follows from the linear approximation for ∆η ≪1 and is not restricted to critical phenomena.
Authors Contribution
M. C. conceived the idea, while M. C. and R. F. conducted the work, analysed the results, and drew the conclusions.
- Authors Contribution: M. C. conceived the idea and, with R. F., carried out the work, analysed the results, and drew the conclusions.