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By Kox A.J.

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Example text

Furthermore, it is required that the Poisson matrix L ( x ) be antisymmetric and the friction matrix M ( x ) symrnetric*l and positive semidefinite. 5) and the symmetry properties are very important for the formulation of proper L- and M-matrices when modeling concrete beyond-equilibrium situations. At this point, we have not really incorporated the powerful geometric structure behind Hamiltonian dynamics, as discussed in connection with the possibility of a reversible-irreversible separation.

For this task, various geometric structureshave been introduced in classical mechanics, namely symplectic, Poisson and Dirac structures. These structures provide suitable linear operators that translate energy gradients into vector fields &eversible. The mathematically interested reader can find an extensive discussion of these geometric structures in Appendix B. To understand the framework of beyond-equilibrium thermodynamics, however, it is sufficient to accept that there exists a well-established geometric structure behind terms that are “under mechanistic control” and that it allows us to express reversible time evolution in terms of energy gradients (as will be illustrated by simple examples in the following subsection).

4 Guide Through the Book To read this book on beyond-equilibrium thermodynamics, a firm background in equilibrium thermodynamics and statistical mechanics (for Part I1 only) is required. For the reader’s convenience, a brief summary of the structure of equilibrium thermodynamics, including some comments on statistical mechanics, is provided in Appendix A. This appendix should be considered as a reminder rather than as a tutorial. Chapter 1 plays a very special role, and it should be read at least twice, at the beginning and at the end.

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