“The book presents an overwhelmingly rich display of deep ideas in the theory and applications of invariant manifolds for dynamical systems, which appear in physical and chemical kinetics, as well as in biology.

An example of a result and exposition is provided by a stunningly beautiful treatment of the systems with inheritance in Chapter 14, combining mathematical rigor with a wide display of concrete applications.

The book should become an all time classics.”

**Mikhail
Shubin,**

*Matthews
Distinguished University Professor*

*in
Northeastern University, Boston, USA.*

* *

“This book is a treasure box filled with a variety of ideas on the reduced description of kinetics. The authors do not only summarize their pioneering work --- at the heart of which is the "invariant manifold method" --- but also put it nicely into perspective and add new material (e.g., "invariant grids", "films") to compose a well-balanced picture of physical and chemical kinetics. The elegant idea of "films" consisting of macroscopically definable nonequilibrium states offers new opportunities for a deeper understanding of irreversibility. 25 years of experience in the field are condensed into deep physical insight, powerful mathematical techniques, and a wealth of illustrating examples and useful references.”

**Hans
Christian Öttinger, **

*Professor of
Polymer Physics, ETH Zürich*

“Statistical mechanics is the branch of theoretical physics which endeavours to explain the macroscopic behaviour of natural systems in terms of the fundamental interactions between their microscopic constituents. It comes in two flavors: equilibrium and non-equilibrium. At equilibrium, only time-asymptotic states are relevant to the macroscopic description, and consequently the time-evolution

of the system plays no role. As a result, many elegant and general

results can be established, which place equilibrium statistical mechanics on a very solid foundational and operational basis. The same is definitely not true for non-equilibrium statistical mechanics, especially for systems far-off equilibrium. For such systems, the dynamics does play a vital role and consequently the task of establishing general principles appears much harder and way less accomplished than for the equilibrium case.

One of the few powerful and general ideas which help unravelling the complexities of non-equilibrium systems is the notion of separation between 'slow' and 'fast' degrees of freedom. Loosely speaking, the former describe the large-scale, macroscopic dynamics, whereas the latter associate with microscopic fluctuations around the slow modes.

Of course, the 'slow/fast' dichotomy is a rather fuzzy notion, which necessitates quantitative specification in order to acquire genuine scientific status.

Such quantitative underpinning is provided by the theory of invariant manifolds, that is subsets of the system phase-space which act as attractors of the long-time dynamics of the system. Providing criteria to identify, construct and classify the invariant manifolds of complex systems, thereby offering a quantitative tool for the investigation of non-equilibrium statistical systems, is a task of utmost importance in modern non-equilibrium statistical mechanics.

This task makes the central core of the present monograph.

The authors are highly reputed experts in the field and provide a very remarkable, and much needed, self-contained introduction to this fascinating subject, along with a systematic treatment of advanced topics which the authors themselves have largely contributed to develop.

The resulting monograph is a unique and very precious contribution to modern non-equilibrium statistical mechanics. It will make a very useful and enjoyable entry in the library of graduate students and professional in the very many disciplines, such as applied math, physics, chemistry and biology, which share non-equilibrium statistical mechanics as a common conceptual background.”

**Sauro Succi,
**

*Professor,
Research Director*

*Istituto
Applicazioni Calcolo, CNR-IAC*

*Roma, Italy*