By Marco Pettini
This publication covers a brand new clarification of the beginning of Hamiltonian chaos and its quantitative characterization. the writer makes a speciality of major parts: Riemannian formula of Hamiltonian dynamics, delivering an unique point of view concerning the courting among geodesic instability and curvature houses of the mechanical manifolds; and a topological thought of thermodynamic section transitions, bearing on topology alterations of microscopic configuration house with the new release of singularities of thermodynamic observables. The ebook comprises quite a few illustrations all through and it'll curiosity either mathematicians and physicists.
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Extra info for Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Interdisciplinary Applied Mathematics, 33)
Kammerling Onnes in 1908, brought about an advancement of the problem of classifying those transitional phenomena that are not ordinary changes of state. 19 K. In particular, at this temperature discontinuities were observed of the thermal dilatation coeﬃcient, of the dielectric constant, and of the constant-volume speciﬁc heat. The two forms of liquid helium were denoted by He I and He II. 19 K, He I was stable, whereas He II was stable below this temperature. This was called the λ-transition because of the shape of the graph of speciﬁc heat as a function of T .
Instead of sketching here this standard presentation of the bases of ensemble theory of statistical mechanics, we propose an equivalent conceptual construction, which we could deﬁne as a “top-down” approach, based on an old and almost forgotten work by L. ” In other words, the second law of thermodynamics is the founding physical principle of ensemble statistical mechanics. 7 Let us now recall Szilard’s work, which surprised many, including Einstein and von Laue, in which the author showed that the second law of thermodynamics provides information not only about the mean values of macroscopic observables but also about their ﬂuctuation properties.
49) where F (N, V, T ) is the Helmholtz free energy computed in the canonical ensemble. 50) where v = V/N is the speciﬁc volume (inverse density), and where we have used P (v) = −(1/N )(∂F/∂v) for the pressure of the system. It is an experimental fact that (∂P/∂v) ≤ 0 holds always true. 50) we have σN /N → ∞ as N → ∞. 48), we ﬁnally obtain the Helmholtz free energy, from which all the other thermodynamic functions can be derived. 6 Phase Transitions Phase transitions involve abrupt major changes of the physical properties of macroscopic objects when a thermodynamic parameter is even slightly varied across a critical value.