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arXiv 1503.01434cond-mat.stat-mech
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小热机有限时间热力学的动力学模型

A kinetic model for the finite-time thermodynamics of small heat engines

Luca Cerino, Andrea Puglisi, Angelo Vulpiani

摘要:

我们研究了一个分子发动机,该发动机由封闭在一个巨大的活塞和恒温器之间的$N \sim 10^2$分子气体构成。作用在活塞上的力和恒温器的温度以有限的周期$τ$周期性地变化。在绝热极限$τ\to \infty$中,即使对于有限尺寸的$N$,平均功和热也能重现热力学值,恢复卡诺效率结果。该系统表现出停滞时间$τ^*$,其中净功为零:对于$τ<τ^*$,它消耗功而不是产生功,充当制冷机或散热器。在$τ>τ^*$下,最大功率下的效率接近Curzorn-Ahlborn极限。功和热的涨落近似地表现出高斯行为。基于动力学理论,我们建立了一个三变量朗之万模型,其中活塞的位置和速度与气体的内能线性耦合在一起。该模型再现了系统的许多特征,如功的符号反转,最大功率下的效率和波动的近似形状。模型的进一步简化允许分析计算平均功,解释其对$τ$的非平凡依赖性。

英文摘要:

We study a molecular engine constituted by a gas of $N \sim 10^2$ molecules enclosed between a massive piston and a thermostat. The force acting on the piston and the temperature of the thermostat are cyclically changed with a finite period $τ$. In the adiabatic limit $τ\to \infty$, even for finite size $N$, the average work and heats reproduce the thermodynamic values, recovering the Carnot result for the efficiency. The system exhibits a stall time $τ^*$ where net work is zero: for $τ<τ^*$ it consumes work instead of producing it, acting as a refrigerator or as a heat sink. At $τ>τ^*$ the efficiency at maximum power is close to the Curzorn-Ahlborn limit. The fluctuations of work and heat display approximatively a Gaussian behavior. Based upon kinetic theory, we develop a three-variables Langevin model where the piston's position and velocity are linearly coupled together with the internal energy of the gas. The model reproduces many of the system's features, such as the inversion of the work's sign, the efficiency at maximum power and the approximate shape of fluctuations. A further simplification in the model allows to compute analytically the average work, explaining its non-trivial dependence on $τ$.

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