%'e propose a new definition of natural invariant measure for trajectory segments of finite duration for a many-particle system. On this basis we give an expression for the probability of fluctuations in the shear stress of a fluid in a nonequilibrium steady state far from equilibrium. In particular we obtain a formula for the ratio that, for a finite time, the shear stress reverses sign, violating the second law of thermodynamics. Computer simulations support this formula.

In recent years the interaction between dynamical systems theory and non-equilibrium statistical mechanics has been enormous. The discovery of fluctuation theorems as a fundamental structure common to almost all non-equilibrium systems, and the connections with the free energy calculation methods of Jarzynski and Crooks, have excited both theorists and experimentalists. This graduate-level book charts the development and theoretical analysis of molecular dynamics as applied to equilibrium and non-equilibrium systems. Designed for both researchers in the field and graduate students of physics, it connects molecular dynamics simulation with the mathematical theory to understand non-equilibrium steady states. It also provides a link between the atomic, nano, and macro worlds. The book ends with an introduction to the use of non-equilibrium statistical mechanics to justify a thermodynamic treatment of non-equilibrium steady states, and gives a direction to further avenues of exploration.

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Boundary effects in the stepwise structure of the Lyapunov spectra and corresponding wavelike structure of the Lyapunov vectors are discussed numerically in quasi-one-dimensional systems of many hard disks. Four different types of boundary conditions are constructed by combinations of periodic boundary conditions and hard-wall boundary conditions, and each leads to different stepwise structures of the Lyapunov spectra. We show that for some Lyapunov exponents in the step region, the spatial y component of the corresponding Lyapunov vector deltaq(yj), divided by the y component of momentum p(yj), exhibits a wavelike structure as a function of position q(xj) and time t. For the other Lyapunov exponents in the step region, the y component of the corresponding Lyapunov vector deltaq(yj) exhibits a time-independent wavelike structure as a function of q(xj). These two types of wavelike structure are used to categorize the type and sequence of steps in the Lyapunov spectra for each different type of boundary condition.

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