Abstract
The fourth lecture was devoted to the derivation of the theory of active gels that we have made with Frank Jülicher, Jacques Prost and other collaborators. This theory describes the behavior of active matter when momentum is conserved. It is a hydrodynamic theory, based on Lars Onsager's theory of weakly out-of-equilibrium systems, and is therefore in principle applicable in the vicinity of thermodynamic equilibrium. It generalizes the hydrodynamics of nematic liquid crystals by introducing active effects. We have proposed this theory to describe the cytoskeleton of eukaryotic cells, but it can be applied to any active system that possesses the same symmetries (and is fluid).
Active effects are taken into account in the theory by introducing a new flux-force couple associated with energy consumption. The thermodynamic force is the elementary energy dissipated (produced with the hydrolysis of an ATP molecule for the cytoskeleton) and the flux is the number of ATP molecules hydrolyzed. The essential result of the theory is that, without any additional hypothesis, a local active constraint appears which depends on the orientation of the constituents. This stress can be contractile if it contracts the active material in the direction of the constituents' orientation, and extensive if it expands it. The spontaneous flows that are a general feature of active systems are due to gradients in this active stress. Any orientation gradient creates an active stress gradient, which, according to the local force equilibrium equation, must be balanced by a viscous stress gradient. An orientation gradient therefore creates flow in an activesystem .
