Statistical Features of High-Dimensional Hamiltonian Systems

Abstract

In this short review we propose a critical assessment of the role of chaos for the thermalization of Hamiltonian systems with high dimensionality. We discuss this problem for both classical and quantum systems. A comparison is made between the two situations: some examples from recent and past literature are presented which support the point of view that chaos is not necessary for thermalization. Finally, we suggest that a close analogy holds between the role played by Kinchin's theorem for high-dimensional classical systems and the role played by Von Neumann's theorem for many-body quantum systems.

Thermalization without chaos in harmonic systems

“…As a final remark, let us notice that a generalization of the present results to quantum mechanics may provide further insights on the thermalization mechanism in quantum isolated systems [37], a theoretical issue which is the object of renewed interest thanks to the possibilities, offered by new technologies, to manipulate nanostructured materials suitable for quantum computing tasks. The point made by our findings on the irrelevance of chaos for the thermalization of harmonic systems are in fact quite similar to the Von Neumann quantum ergodic theorem approach [38,39]. According to the latter thermalization is in fact related to an appropriate choice of observables, rather than to the spectral properties of the Hamiltonian.…”

Generalised Gibbs Ensemble for spherically constrained harmonic models

“…Most of previous similar studies focused on quantum models, and several review articles summarise methods and results [1][2][3][4][5][6][7][8][9]. Less attention has been paid to classical out of equilibrium macroscopic integrable systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and we contribute here to their better understanding.…”

Generalised Gibbs Ensemble for spherically constrained harmonic models

“…Most of previous similar studies focused on quantum models, and several review articles summarise methods and results [1][2][3][4][5][6][7][8][9]. Less attention has been paid to classical out of equilibrium macroscopic integrable systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and we contribute here to their better understanding.…”