Nonequilibrium transients and steady states from Hamiltonian dynamics (4h)

1) Title: Transport and the escape rate formalism (1h)

Abstract: In this lecture, it is shown that the transport properties (diffusion,
viscosity,…) can be formulated in terms of first passage problems and they
can be associated with nonequilibrium transients characterized by an escape
rate of trajectories in phase space. This formulation is suitable to relate the
transport coefficients to the characteristic quantities of phase-space dynamics
such as the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time,
or the fractal dimensions. The formalism applies to Hamiltonian systems
satisfying Liouville’s theorem.

2) Title: Hydrodynamic modes and nonequilibrium steady states (1h)

Abstract: If the dynamical system sustaining the transport properties is
spatially periodic, a Fourier transform can be performed in order to reduce the
nonequilibrium time evolution to a dynamics with quasi-periodic boundary
conditions. The linear operator ruling the time evolution of probability
depends on the wave number introduced by the Fourier transform and admits
Pollicott-Ruelle resonances giving the decay rates of the corresponding
hydrodynamic modes of relaxation toward equilibrium. These modes are
given by singular Schwartz-type distributions with fractal cumulative
functions. The nonequilibrium steady states can be derived from these modes.

3) Title: Ab initio derivation of entropy production (1h)

Abstract: On the basis of the construction of the hydrodynamic modes and the
corresponding steady states, it is possible to carry out the ab initio calculation
of the thermodynamic entropy production starting from the Hamiltonian
dynamics statisfying Liouville’s theorem. The singular character of the
distributions representing the hydrodynamic modes and the steady states turns
out to play an important role in the non-vanishing of entropy production.

4) Title: Time asymmetry in nonequilibrium statistical mechanics (1h)

Abstract: The distributions representing the hydrodynamic modes and the
nonequilibrium steady states are shown to break the time-reversal symmetry at
the level of the statistical description. This symmetry breaking finds its origin
in the selection of initial conditions in Newtonian mechanics. For
nonequilibrium steady states, the time asymmetry manifests itself in the
temporal disorder of the fluctuating paths of the system. The thermodynamic
entropy production can be expressed in terms of the characteristic quantities of
this temporal disorder in the paths and the time-reversed paths. The relation to
the fluctuation theorem is discussed.