Landau Damping and Entropy

It is shown that the well-known LANDAu-damping entropy paradox can be resolved by considering a macroscopic entropy as defined by CLAUSIUS instead of the microscopic entropy defined by BOLTZMANN. Although both entropy definitions are identical for equilibrium configurations, the same in general is not true for non-equilibrium configurations. To extend the entropy definition of CLAUSIUS to non-equilibrium configurations requires a generalization of the concept of temperature to nonequilibrium configurations. A natural generalization of the concept of temperature to non-equilibrium configurations is to put the temperature proportional to the mean square velocity fluctuation. In this way a macroscopic entropy can be defined which is then proportional to the logarithm of the mean square velocity fluctuation. In the case of L A N D A U damping it can be shown that two states having the same statistical permutability and thus the same microscopic entropy may have different mean square velocity fluctuations. One should therefore consider, besides the microscopic disorder defined by BOLTZMANN, which is proportional to the logarithm of the permutability, a macroscopic disorder which is proportional to the logarithm of the mean square velocity fluctuation. In the case of LANDAU damping the microscopic entropy does not change with time; the macroscopic entropy, however, increases steadily with time.