We illustrate the relation between the scattering phase appearing in the Friedel sum rule and the phase of the transmission amplitude for quantum scatterers connected to two one-dimensional leads. Transmission zero points cause abrupt phase changes +/-π of the phase of the transmission amplitude. In contrast, the Friedel phase is a continuous function of energy. We investigate these scattering phases for simple scattering problems and illustrate the behavior of these models by following the path of the transmission amplitude in the complex plane as a function of energy. We verify the Friedel sum rule for these models by direct calculation of the scattering phases and by direct calculation of the density of states

A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integrals for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit.

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.

The Lyapunov spectrum corresponding to a periodic orbit for a two dimensional many particle system with hard core interactions is discussed. Noting that the matrix to describe the tangent space dynamics has the block cyclic structure, the calculation of the Lyapunov spectrum is attributed to the eigenvalue problem of 16 × 16 reduced matrices regardless of the number of particles. We show that there is the thermodynamic limit of the Lyapunov spectrum in this periodic orbit. The Lyapunov spectrum has a step structure, which is explained by using symmetries of the reduced matrices.