Breakdown of Onsager Symmetry in Neoclassical Transport Theory

Neoclassical transport theory is developed in a Lagrangian rather than the usual Eulerian formulation. We show that an underlying asymmetry exists in the neoclassical pinch and bootstrap effects and demonstrate the physical basis of the Onsagesymmetry relationship in the pinch-bootstrap duality. It is suggested that low frequency turbulence can destroy the bootstrap current at levels too low to effect the Ware Pinch. Neoclassical transport theory has long predicted the existence of a "bootstrap current" driven by the radial density gradient and a pinch effect driven by the toroidal electrical field. Both effects have substantial practical implications for tokamak reactors. Neither, however, has been confirmed by experiment. Inward plasma flow has been clearly, if indirectly, observed although not (3) quantitatively explained by the neoclassical pinch. The bootstrap current (6) has not been seen even in experiments where it should have been easily detected. (4) The pinch effect is often explained as a purely kinematic, collisionless inward flow of trapped particles. The underlying mechanism of the bootstrap current is somewhat obscure, although it clearly is a collision driven phenomenon involving circulating particles. These two phenomena display an Onsager symmetry relation and have equal coefficients. Why effects involving different classes of particles should be Onsager conjugate has never been satisfactorily explained. This letter considers the questions of Onsager symmetry and the non-observance of the bootstrap current in light of a recently developed.Lagrangian formulation of neoclassical theory.(8) This formulation has the virtue of being a direct expression of the elementary kinematics and collision processes, allowing the various flows to be identified at the microscopic level. We show the net kinematic contributions to both pinch and bootstrap current to be small and symmetric. However, the individual kinematic processes involved in the pinch are large and there is an underlying asymmetry of