Abstract
The hallmark of active matter is the autonomous directed motion of its microscopic constituents driven by consumption of energy resources. This 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 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.
Highlights
Active materials are those in which each component extracts energy from the environment to produce directed motion [1,2,3]
Building the thermodynamics of active matter is a major challenge of modern nonequilibrium statistical mechanics
The irreversibility of active dynamics has recently attracted much attention, since it provides an unambiguous measure of the distance from equilibrium: It is quantified by the explicit entropy production rate (EPR), which compares forward and time-reversed realizations of the dynamics [30,41,42,56,57,58,59,60,62,68]
Summary
Active materials are those in which each component extracts energy from the environment to produce directed motion [1,2,3]. After identifying the relevant forces and currents for a given theory, the field dynamics are formulated by postulating linear relations between them These theories were originally designed to capture the effect of external drives, such as temperature gradients or electric fields, yet extensions to systems with internal activity, such as active gels, are shown to successfully reproduce the behavior of living materials [70,71,72]. We show that the heat rate can be generically decomposed into a homogeneous background contribution, independent of active fields, and a contribution given in terms of the statistics of the active fields This decomposition allows one to quantify how the structure and dynamics of active fields affect where heat is dissipated, opening the door to estimating and comparing the energetic cost associated with different emerging orders.
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