Part 1: Interesting
recent approach for analyzing the irregularity of parameter-dependent
transport coefficients in terms of a multifractal formalism stimulated
by time-series analysis, see A.Faccini,
S.Wimberger and A.Tomadin, Physica A 376,
266--274 (2007) and further references therein
Section 1.1: Some
literature about transient chaos should have been cited such as, e.g., Z.Kaufmann, H.Lustfeld, A.Nemeth,
P.Szepfalusy, Phys. Rev. Lett. 78, 4031--4034 (1997) and further
Chapter 3: Progress in
mathematical analysis of Ruelle's susceptibility formula, see preprint
by V.Baladii, arXiv:math.DS/0612852
(2007) as well as B.Cessac
and J.-A. Sepulchre, Physica D 225, 13--28 (2007).
Seems that these articles as well as Ruelle's papers cited in these
works provide a rigorous mathematical approach towards the regularity
irregularity of observables such as current and diffusion coefficient
of one-dimensional maps. The model analyzed in Chapter 3 seems to
provide an example where the situation is not that simple.
Section 6.2: There is
further interesting work on intermittent maps by Z.Kaufmann, H.Lustfeld, J.Bene, Phys. Rev.
53, 1416--1421 (1996).
Section 8.3: Recent
results on single-molecule spectroscopy allow to experimentally resolve
the paths of single molecules diffusing through mesoporous structures,
see J.Kirstein et al., Nature
Materials 6, 303--310 (2007) as well as a popular
science online presentation (in German). This looks like an
extremely promising experimental technique for learning about the
microscopic origin of diffusion in periodic structures.
Chapter 9: There is some
new work on the bouncing ball billiard by A. de Wijn, H.Kantz, Phys. Rev. E 75,
046214 (2007). By mistake the authors did not refer to previous
work where this model was introduced originally, see the book for
such references. One of these references was later on referred to in an
erratum in PRE.
Part 2: two new
papers by G.Gallavotti presenting his view about thermostats, phase
space contraction and entropy production: Chaos 16, 043114 (2006) and J. Stat. Mech., P10011/1--9 (2006)
Section 10.3: The velocity
probability distribution functions Eqs.(10.19) and (10.20) for a
subsystem interacting with a finite dimensional thermal reservoir
should actually be called Schlueter distributions, as was pointed out
Shutler, S.V. Springham and J.C. Martinez, Phys. Scr. 76, 466--469
(2007); see also
Schlueter's original work on this problem in V.A. Schlueter, Z. Naturf. A 3, 350--360
Chapters 13 and 18: There
is an interesting line of research on relations between
Kolmogorov-Sinai and `physical' (thermodynamic) entropies starting from
the work by M.Dzugutov, E.Aurell and
A.Vulpiani, Phys.Rev.Lett. 81, 1762--1765 (1998). This research
seems to be mainly motivated by computer simulations on more realistic
interacting many-partivcle systems suggesting the existence of
universal scaling laws between both types of entropies. Detailed
connections to the analytical works by Gaspard, Dorfman, Evans, Hoover
et al. discussed in Part 2 of the book seem to pose an open question.
See also V.Latora, M.Baranger, Phys.
Rev. Lett. 82, 520--523 (1999); A.Samanta et al., Phys. Rev. Lett. 92,
145901/1--4 (2004); M.Falcioni et al., Phys. Rev. E 71, 016118/1--8
(2005) and further literature cited therein.
Experimental results on cell migration of zebra fish keratocytes show
the existence of bimodal velocity distributions. The data was explained
theoretically by introducing a velocity-dependent friction coefficient
that keeps the cell velocity constant on average thus providing a first
example where the concept of active Brownian particles has indeed
successfully been applied in order to understand cell migration. See
work by Simon Flyvbjerg
MPIPKS Dresden, to be published, presented on the recent DPG Spring
meeting, see Verhandl. DPG (VI) 42 (2007)
Excellent review about fluctuation relations from the stochastic point
of view by R.J.Harris and
(2007), to be published in J.Stat.Mech.
Section 17.3: New
analytical results on Fourier's law in anharmonic chains by K.Aoki, J.Lukkarinen and H.Spohn, J. Stat.
Phys. 124, 1105--1129 (2006).
Section 17.4.2: As
already mentioned in the book, there is a close link between the study
of diffusion in polygonal billiards and diffusion in piecewise
isometries such as a non-chaotically kicked oscillator model. In J.H.Lowenstein, G.Poggiaspalla, F.Vivaldi,
Dynamical Systems 20, 413--451 (2005) the authors find a highly
nontrivial parameter dependence of the exponents of the time dependence
of the mean square displacement on parameter variation, similar to what
has been observed numerically in polygonal billiards. However, their
model enables exact analytical solutions on a rigorous mathematical
In fact, Lowenstein et al. also find subdiffusion in their Hamiltonian
dynamical system contradicting statements, based on theoretical
arguments, that subdiffusion cannot exist in Hamiltonian systems, see
p.360ff of the book. Another counterexample to this statement is
provided by A.Loskutov and A.
Ryabov, J.Stat. Phys. 108, 995--1014 (2002), who report velocity
subdiffusion (``velocity cooling'') for a
Hamiltonian particle billiard with vibrating walls.