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.

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.