By Marco Pettini
This publication covers a brand new rationalization of the beginning of Hamiltonian chaos and its quantitative characterization. the writer specializes in major parts: Riemannian formula of Hamiltonian dynamics, delivering an unique perspective concerning the courting among geodesic instability and curvature homes of the mechanical manifolds; and a topological idea of thermodynamic part transitions, concerning topology alterations of microscopic configuration house with the new release of singularities of thermodynamic observables. The booklet includes quite a few illustrations all through and it'll curiosity either mathematicians and physicists.
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Extra resources for Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Interdisciplinary Applied Mathematics, 33)
Example text
At the dawning of the computer era, as admirably realized by E. Fermi, J. von Neumann, and S. Ulam, a new insight into the foundations of statistical mechanics became possible through the “long-time” numerical solution of the differential equations of motion of a collection of up to a few hundreds of interacting particles, nothing with respect to the Avogadro number, but surprisingly enough to discover an unsuspected richness of the dynamics and, later on, enough to give birth to so-called molecular dynamics, a numerical ab initio computational method to estimate macroscopic properties (such as 17 18 Chapter 2 Background in Physics viscosity, specific heat, magnetic susceptibility, and elastic constants) of real materials.
35). Taking the logarithm of both sides gives log ΩN (E) = [ log ZN (β ) + β E ] + log 2π ∂ 2 log ZN (β ) ∂β 2 − 12 + ··· . 36) The first term in the right-hand side of this equation is O(N ), whereas the second term in square brackets is O(log N ), so that at large N it can be ignored in comparison with the first. Hence log ΩN (E) ≈ log ZN (β ) + β E . 37) In the thermodynamic limit N → ∞, this approximate relation becomes exact in the form 1 E 1 log ΩN (E) = lim log ZN (β ) + β N →∞ N N →∞ N N lim .
Y2N ) = Xi . 1 Statistical Mechanics ∂(x1 , . . , Xi , . . , x2N ) = ∂(y1 , . . , y2N ) 2N k=1 2N ∂Xi αki ∂yk 2N = k=1 2N = r=1 21 r=1 ∂Xi ∂xr ∂xr ∂yk 2N αki = r=1 ∂Xi ∂xr 2N k=1 ∂xr αki ∂yk ∂Xi ∂(x1 , . . , xi−1 , xi , xi+1 , . . , x2N ) , ∂xr ∂(y1 , . . 12) but if r = i, then the Jacobian matrix has two rows that are equal and thus its determinant vanishes. In conclusion, ∂ det(J) = ∂t 2N 2N i=1 r=1 2N = i=1 ∂Xi ∂(x1 , . . , xr , . . , x2N ) δri ∂xr ∂(y1 , . . 14) which proves that the Liouville measure µ(At ) = At dx1 · · · dx2N in Γ -space is invariant under the natural Hamiltonian motion.