Transport coefficients and Lyapunov exponents

After introducing the viscosity of a fluid macroscopically and microscopically as well as the Lyapunov exponents of the fluid, the SLLOD equations of motion with a Gaussian thermostat and Lees-Edwards boundary conditions for the motion of particles in a sheared fluid in a nonequilibrium stationary state are discussed. An explicit expression, due to Posch and Hoover, for the viscosity is then derived in terms of the sum of all Lyapunov exponents, illustrating the direct connection between irreversible entropy production (due to viscous heating) and phase space contraction. A symmetry of the Lyapunov spectrum allows this expression to be reduced to a simple relation between the viscosity and the two maximal Lyapunov exponents of the fluid in the stationary state. A numerical check of this relation for fluids consisting of 108 and 864 particles is presented. Finally, similar relations for other transport coefficients and the connection with other work are discussed.