Compatibility of Carnot efficiency with finite power in an underdamped Brownian Carnot cycle in small temperature-difference regime

Abstract

We study the possibility of achieving the Carnot efficiency in a finite-power underdamped Brownian Carnot cycle. Recently, it was reported that the Carnot efficiency is achievable in a general class of finite-power Carnot cycles in the vanishing limit of the relaxation times. Thus, it may be interesting to clarify how the efficiency and power depend on the relaxation times by using a specific model. By evaluating the heat-leakage effect intrinsic in the underdamped dynamics with the instantaneous adiabatic processes, we demonstrate that the compatibility of the Carnot efficiency and finite power is achieved in the vanishing limit of the relaxation times in the small temperature-difference regime. Furthermore, we show that this result is consistent with a trade-off relation between power and efficiency by explicitly deriving the relation of our cycle in terms of the relaxation times.

Achieving Carnot efficiency in a finite-power Brownian Carnot cycle with arbitrary temperature difference

“…where λ h (t ) and λ c (t ) are the time evolution of λ(t ) in the isothermal processes with temperatures T h and T c , respectively, and w j ( j = 1, 2, 3, 4) are positive parameters. This protocol is inspired by the optimal protocol in the overdamped Brownian Carnot cycle with the instantaneous adiabatic processes [16,27] and used in our previous study [18]. From Eqs.…”

“…However, the power P defined as the output work per unit time vanishes in the quasistatic limit. Since the compatibility of the Carnot efficiency and finite power may not be forbidden by the second law of thermodynamics, many efforts have been made to achieve it [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. This possibility, however, may be reconsidered in terms of the recently formulated trade-off relation between the efficiency and power in heat engines [21][22][23][24][25],…”

“…In Ref. [18], we studied the Brownian Carnot cycle to demonstrate the feasibility of the above scenario. We used an underdamped Brownian Carnot cycle, where the instantaneous change of the potential and temperature of the heat bath is used as an adiabatic process [17,[26][27][28].…”

Success versus failure: Efficient heat devices in thermodynamics

“…In between these thresholds and the corresponding maximum Carnot values, a fairly good agreement with experimental results for both HE and RE is found. Lately, the role of stability in optimization processes has been addressed, first, with a compromise between fluctuations and operation regimes in small systems for cyclic and steady-state processes [15,16,20] and, later on, addressing a possible role of stability of operation regimes in an optimization mechanism, which could favor evolution and adaptation [21], with possible applications under more realistic energetic demands [22,23]. Recently, it has also been discussed that in a variety of natural phenomena the time evolution of nonequilibrium states follows entropy demands [24].…”

Geometric Bounds on the Power of Adiabatic Thermal Machines

“…The question how this trade-off can be formulated quantitatively for meso-and micro-scale thermal machines, and whether it can be overcome in special situations, has attracted significant interest over the last decade [2][3][4][5][6][7][8][9][10][11][12]. As a result, a variety of trade-off relations that bound the power of different types of machines in terms of a dimensionless figure of merit were discovered, first in linear-response [13][14][15][16][17] and then far from equilibrium [18][19][20][21][22][23][24][25][26][27].…”