The analytical and computational studies of various isolated classical Hamiltonian systems including long-range interactions suggest that the N→∞ and t→∞ limits do not commute for entire classes of initial conditions. This is, for instance, the case for inertial planar rotators whenever...

Statistical mechanics can only be ultimately justified in terms of microscopic dynamics (classical, quantum, relativistic, or any other). It is known that Boltzmann–Gibbs statistics is based on the hypothesis of exponential sensitivity to the initial conditions, mixing and ergodicity in Gibbs...

The optimization of the recently generalized entropy of the form Sq ≡ 1 − ∝dx[p(x)]q∼/(q−1) with the constraints ∝dxp(x) = 1 and 〈x2〉q ≡ ∝dxx2[p(x)]q = 1 yields the Student's t-distribution for q > 1, and the r-distribution for q < 1.

We study a discrete N-component spin ferromagnet with Hamiltonian βH= −NK∑〈i,j〉(Si·Sj)−N2L∑〈i,j〉(Si·Sj)2 in a semi-infinite cubic lattice. The coupling constants at the surface, Ks and Ls, are allowed to be different from the bulk ones, KB and LB. Using a simple real-space...

We discuss the non-Boltzmannian nature of quasi-stationary states in the Hamiltonian mean field (HMF) model, a paradigmatic model for long-range interacting classical many-body systems. We present a theorem excluding the Boltzmann–Gibbs exponential weight in Gibbs Γ-space of microscopic...

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann–Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular...

We discuss and illustrate a new stochastic algorithm (generalized simulated annealing) for computationally finding the global minimum of a given (not necessarily convex) energy/cost function defined in a continuous D-dimensional space. This algorithm recovers, as particular cases, the so-called...

We study, through molecular dynamics, a conservative two-dimensional Lennard-Jones-like gas (with attractive potential ∝r−α). We consider the effect of the range index α of interactions, number of particles, total energy and particle density. We detect negative specific heat when the...

We numerically study a one-dimensional system of N classical localized planar rotators coupled through interactions which decay with distance as 1/rα (α≥0). The approach is a first principle one (i.e., based on Newton’s law), and yields the probability distribution of momenta. For α large...

It seems reasonable to consider concavity (with regard to all probability distributions) and stability (under arbitrarily small deformations of any given probability distribution) as necessary for an entropic form to be a physical one in the thermostatistical sense. Most known entropic forms,...