Search

We give a review of the Fermi–Pasta–Ulam (FPU) problem from the perspective of its possible impact on the foundations of physics, concerning the relations between classical and quantum mechanics. In the first part we point out that the problem should be looked upon in a wide sense (whether the equilibrium Gibbs measure is attained) rather than in the original restricted sense (whether energy equipartition is attained). The second part is devoted to some very recent results of ours for an FPU-like model of an ionic crystal, which has such a realistic character as to reproduce in an impressively good way the experimental infrared spectra. Since the existence of sharp spectral lines is usually considered to be a characteristic quantum phenomenon, even unconceivable in a classical frame, this fact seems to support a thesis suggested by the original FPU result. Namely, that the relations between classical and quantum mechanics are much subtler than usually believed, and should perhaps be reconsidered under some new light.

We give a review of the Fermi–Pasta–Ulam (FPU) problem from the perspective of its possible impact on the foundations of physics, concerning the relations between classical and quantum mechanics. In the first part we point out that the problem should be looked upon in a wide sense (whether the equilibrium Gibbs measure is attained) rather than in the original restricted sense (whether energy equipartition is attained). The second part is devoted to some very recent results of ours for an FPU-like model of an ionic crystal, which has such a realistic character as to reproduce in an impressively good way the experimental infrared spectra. Since the existence of sharp spectral lines is usually considered to be a characteristic quantum phenomenon, even unconceivable in a classical frame, this fact seems to support a thesis suggested by the original FPU result. Namely, that the relations between classical and quantum mechanics are much subtler than usually believed, and should perhaps be reconsidered under some new light.

It was recently shown that the experimental infrared spectra of ionic crystals at room temperature are very well reproduced by classical realistic models, and here new results are reported on the temperature dependence of the spectra, for the LiF crystal. The principal aim of the present work is however to highlight the deep analogy existing between the problem of spectra in ionic crystal models on the one hand, and that of energy equipartition in the Fermi–Pasta–Ulam model, on the other. Indeed at low temperatures the computations of the spectra show that the dynamics of the considered system is not completely chaotic, so that the use of the Boltzmann–Gibbs statistics is put in question, as in the Fermi–Pasta–Ulam case. Here, however, at variance with the equipartition problem, a first positive indication is given on the modifications that should be introduced in a classical statistical treatment: the new results at low temperatures show that it is indeed the Clausius identification of temperature that has to be modified. In fact, at very low temperatures the theoretical spectra fail to reproduce the experimental ones, if the temperature is taken as proportional to mean kinetic energy, but agreement is recovered through the only expedient of introducing a suitable temperature rescaling. Analogous results are also found in connection with thermal expansion.

A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old Fermi–Pasta–Ulam problem, which gave origin to such discussions, and with the optical spectral lines, the existence of which was recently proven to be possible in classical models, just in virtue of such a persistence.

The acquisition of an invasive phenotype is a prerequisite for metastasization, yet it is not clear whether or to which extent the invasive phenotype is linked to other features characteristic of metastatic cells. We selected an invasive subpopulation from the triple negative breast cancer cell line MDA-MB-231, performing repeated cycles of preparative assays of invasion through Matrigel covered membranes. The invasive sub-population of MDA-MB-231 cells exhibits stronger migratory capacity as compared to parental cells confirming the highly invasive potential of the selected cell line. Prolonged cultivation of these cells did not abolish the invasive phenotype. ArrayCGH, DNA index quantification and karyotype analyses confirmed a common genetic origin of the parental and invasive subpopulations and revealed discrete structural differences of the invasive subpopulation including increased ploidy and the absence of a characteristic amplification of chromosome 5p14.1-15.33. Gene expression analyses showed a drastically altered expression profile including features of apocrine breast cancers and of invasion related matrix-metalloproteases and cytokines. The invasive cells showed accelerated proliferation, increased apoptosis, and an altered pattern of chemo-sensitivity with lower IC50 values for drugs affecting the mitotic apparatus. However, the invasive cell population is significantly less tumorigenic in orthotopic mouse xenografts suggesting that the acquisition of the invasive capacity and the achievement of metastatic growth potential are distinct events.