By Biryukov O.N.
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Additional info for A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk
The Poisson structure used to describe reversible dynamics is thus an abstract generalization of the concepts of classical mechanics. The energy E is the most natural generator of time translations for reversible systems, and the antisymmetry and Jacobi identity are the abstract properties used in generalizing the usual Poisson bracket of classical mechanics to capture the essence of reversibility. This Poisson bracket allows us to formulate Hamiltonian dynamical systems. Classical mechanics not only provides an important example of a symplectic structure but also is the prototype of all finite-dimensional symplectic structures.
Thermodyn. , Physica A 285 (2000) 448; Jou & Casas-Vbzquez, J. Non-NewtonianFluid Mech. , J. Non-Equilib. Thermodyn. , J. Non-Equilib. Thermodyn. , J. Non-Equilib. Thermodyn. 28 (2003) 51; Pasquali & Scriven, J. NonNewtonian Fluid Mech. 120 (2004) 101. 1 and Appendix B. 20 INTRODUCTION I 0 I I 4 2% - Fig. 4 Variables for two subsystems exchanging heat and volume. 4). Use x = (q, v, E l , E z ) , where v is the velocity of the wall. Assume that the subsystems contain the same number of particles ( N I = N2 = N ) , and that their equilibrium thermodynamics is given by the entropy function S ( E ,V, N).
I Steps Toward Frameworks of Thermodynamics at and Beyond Equilibrium Equilibrium Beyond Equilibrium “work” + “heat” dE = dW dQ “reversible” + “irreversible” dd xt - &eversible . &reversible dE = -pdV - = L . E dt 6x + dE = -pdV + /IdN + dQ + /IdN + TdS + thermodynamics can be built, expresses a geometric structure in the space of all thermodynamic variables. In particular, it naturally leads to Legendre transformations between thermodynamic potentials, depending on which variables one chooses as the independent ones.
A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk by Biryukov O.N.