The hallmark of active matter is the autonomous directed motion of its microscopic constituents driven by consumption of energy resources. This motion leads to the emergence of large-scale dynamics and structures without any equilibrium equivalent. Though active field theories offer a useful hydrodynamic description, it is unclear how to properly quantify the energetic cost of the dynamics from such a coarse-grained description. We provide a thermodynamically consistent framework to identify the energy exchanges between active systems and their surrounding thermostat at the hydrodynamic level. Based on linear irreversible thermodynamics, we determine how active fields couple with the underlying reservoirs at the basis of nonequilibrium driving. This approach leads to evaluating the rate of heat dissipated in the thermostat, as a measure of the cost to sustain the system away from equilibrium, which is related to the irreversibility of the active field dynamics. We demonstrate the applicability of our approach in two popular active field theories: (i) the dynamics of a conserved density field reproducing active phase separation and (ii) the coupled dynamics of density and polarization describing motile deformable droplets. Combining numerical and analytical approaches, we provide spatial maps of dissipated heat, compare them with the irreversibility measure of the active field dynamics, and explore how the overall dissipated heat varies with the emerging order.

中文翻译：

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活动场理论的热力学：耦合到储层的能量成本

活性物质的标志是其微观成分由能源消耗驱动的自主定向运动。这种运动导致没有任何平衡当量的大规模动力学和结构的出现。尽管活性场理论提供了有用的流体动力学描述，但尚不清楚如何从这种粗粒度描述中正确量化动力学的能量成本。我们提供了一个热力学一致的框架，以在流体动力学水平上识别有源系统与其周围恒温器之间的能量交换。基于线性不可逆热力学，我们在非平衡驱动的基础上确定活动场如何与下伏储层耦合。这种方法导致评估恒温器中的散热率，作为维持系统远离平衡的成本的度量，这与活性场动力学的不可逆性有关。我们证明了我们的方法在两种流行的活性场理论中的适用性：（i）再现活性相分离的守恒密度场的动力学和（ii）描述运动可变形液滴的密度和极化的耦合动力学。结合数值和分析方法，我们提供了散热的空间图，将它们与活动场动力学的不可逆性度量进行比较，并探索整体散热如何随出现的顺序而变化。(i) 再现活性相分离的守恒密度场的动力学和 (ii) 描述运动可变形液滴的密度和极化的耦合动力学。结合数值和分析方法，我们提供了散热的空间图，将它们与活动场动力学的不可逆性度量进行比较，并探索整体散热如何随出现的顺序而变化。(i) 再现活性相分离的守恒密度场的动力学和 (ii) 描述运动可变形液滴的密度和极化的耦合动力学。结合数值和分析方法，我们提供了散热的空间图，将它们与活动场动力学的不可逆性度量进行比较，并探索整体散热如何随出现的顺序而变化。