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Register a new userUniversal Bounds on Fluctuations in Continuous Thermal Machines
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We study bounds on ratios of fluctuations in steady-state time-reversal heat engines controlled by multi affinities. In the linear response regime, we prove that the relative fluctuations (precision) of the output current (power) is always lower-bounded by the relative fluctuations of the input current (heat current absorbed from the hot bath). As a consequence, the ratio between the fluctuations of the output and input currents are bounded both from above and below, where the lower (upper) bound is determined by the square of the averaged efficiency (square of the Carnot efficiency) of the engine. The saturation of the lower bound is achieved in the tight-coupling limit when the determinant of the Onsager response matrix vanishes. Our analysis can be applied to different operational regimes, including engines, refrigerators, and heat pumps. We illustrate our findings in two types of continuous engines: two-terminal coherent thermoelectric junctions and three-terminal quantum absorption refrigerators. Numerical simulations in the far-from-equilibrium regime suggest that these bounds apply more broadly, beyond linear response.
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We study universal aspects of fluctuations in an ensemble of noninteracting continuous quantum thermal machines in the steady state limit. Considering an individual machine, such as a refrigerator, in which relative fluctuations (and high order cumulants) of the cooling heat current to the absorbed heat current, $eta^{(n)}$, are upper-bounded, $eta^{(n)}leq eta_C^n$ with $ngeq 2$ and $eta_C$ the Carnot efficiency, we prove that an {it ensemble} of $N$ distinct machines similarly satisfies this upper bound on the relative fluctuations of the ensemble, $eta_N^{(n)}leq eta_C^n$. For an ensemble of distinct quantum {it refrigerators} with components operating in the tight coupling limit we further prove the existence of a {it lower bound} on $eta_N^{(n)}$ in specific cases, exemplified on three-level quantum absorption refrigerators and resonant-energy thermoelectric junctions. Beyond special cases, the existence of a lower bound on $eta_N^{(2)}$ for an ensemble of quantum refrigerators is demonstrated by numerical simulations.
We study the statistics of the efficiency in a class of isothermal cyclic machines with realistic coupling between the internal degrees of freedom. We derive, under fairly general assumptions, the probability distribution function for the efficiency. We find that the macroscopic efficiency is always equal to the most likely efficiency, and it lies in an interval whose boundaries are universal as they only depend on the input and output thermodynamic forces, and not on the details of the machine. The machine achieves the upper boundary of such an interval only in the limit of tight coupling. Furthermore, we find that the tight coupling limit is a necessary, yet not sufficient, condition for the engine to perform close to the reversible efficiency. The reversible efficiency is the least likely regardless of the coupling strength, in agreement with previous studies. By using a large deviation formalism we derive a fluctuation relation for the efficiency which holds for any number of internal degrees of freedom in the system.
When engineering microscopic machines, increasing efficiency can often come at a price of reduced reliability due to the impact of stochastic fluctuations. Here we develop a general method for performing multi-objective optimisation of efficiency and work fluctuations in thermal machines operating close to equilibrium in either the classical or quantum regime. Our method utilises techniques from thermodynamic geometry, whereby we match optimal solutions to protocols parameterised by their thermodynamic length. We characterise the optimal protocols for continuous-variable Gaussian machines, which form a crucial class in the study of thermodynamics for microscopic systems.
We consider a molecular machine described as a Brownian particle diffusing in a tilted periodic potential. We evaluate the absorbed and released power of the machine as a function of the applied molecular and chemical forces, by using the fact that the times for completing a cycle in the forward and the backward direction have the same distribution, and that the ratio of the corresponding splitting probabilities can be simply expressed as a function of the applied force. We explicitly evaluate the efficiency at maximum power for a simple sawtooth potential. We also obtain the efficiency at maximum power for a broad class of 2-D models of a Brownian machine and find that loosely coupled machines operate with a smaller efficiency at maximum power than their strongly coupled counterparts.
Subdiffusive transport in tilted washboard potentials is studied within the fractional Fokker-Planck equation approach, using the associated continuous time random walk (CTRW) framework. The scaled subvelocity is shown to obey a universal law, assuming the form of a stationary Levy-stable distribution. The latter is defined by the index of subdiffusion alpha and the mean subvelocity only, but interestingly depends neither on the bias strength nor on the specific form of the potential. These scaled, universal subvelocity fluctuations emerge due to the weak ergodicity breaking and are vanishing in the limit of normal diffusion. The results of the analytical heuristic theory are corroborated by Monte Carlo simulations of the underlying CTRW.
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