Structure Of Statistical Mechanics Research Articles

Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos

We consider both the dynamics within and towards the supercycle attractors along the period-doubling route to chaos to analyze the development of a statistical-mechanical structure. In this structure the partition function consists of the sum of the attractor position distances known as supercycle diameters and the associated thermodynamic potential measures the rate of approach of trajectories to the attractor. The configurational weights for finite 2N, and infinite , periods can be expressed as power laws or deformed exponentials. For a finite period the structure is undeveloped in the sense that there is no true configurational degeneracy, but in the limit this is realized together with the analog property of a Legendre transform linking entropies of two ensembles. We also study the partition functions for all N and the action of the central limit theorem via a binomial approximation.

Laws of Zipf and Benford, intermittency, and critical fluctuations

We describe precise equivalences between theoretical descriptions of: (i) size-rank and first-digit laws for numerical data sets, (ii) intermittency at the transition to chaos in nonlinear maps, and (iii) cluster fluctuations at criticality. The equivalences stem from a common statistical-mechanical structure that departs from the usual via a one-parameter deformation of the exponential and logarithmic functions. The generalized structure arises when configurational phase space is incompletely visited such that the accessible fraction has fractal properties. Thermodynamically, the common focal expression is an (incomplete) Legendre transform between two entropy (or Massieu) potentials. The theory is in quantitative agreement with real size-rank data and it naturally includes the bends or tails observed for small and large rank.

Statistical mechanics of gravitating systems ... and some curious history of Chandra’s rare misses!

Chandra’s academic life had several phases each culminating in a monograph describing that subject. I shall deal with aspects of his work in the two earliest phases. I shall describe the overall structure of statistical mechanics of gravitating systems, the relevance of isothermal sphere in the mean-field approximation and issues related to collisional relaxation and dynamical friction in self-gravitating system of particles. There are several curious features in the history of these topics which I comment upon.

Renewal stochastic processes with correlated events: Phase transitions along time evolution

We consider renewal stochastic processes generated by nonindependent events from the perspective that their basic distribution and associated generating functions obey the statistical-mechanical structure of systems with interacting degrees of freedom. Based on this fact we look briefly into the less-known case of processes that display phase transitions along time. When the density distribution ψ{n}(t) for the occurrence of the nth event at time t is considered to be a partition function, of a "microcanonical" type for n "degrees of freedom" at fixed "energy" t, one obtains a set of four partition functions of which that for the generating function variable z and Laplace transform variable ε, conjugate to n and t, respectively, plays a central role. These partition functions relate to each other in the customary way and in accordance to the precepts of large deviations theory, while the entropy, or Massieu potential, derived from ψ{n}(t) satisfies an Euler relation. We illustrate this scheme first for an ordinary renewal process of events generated by a simple exponential waiting-time distribution ψ(t). Then we examine a process modeled after the so-called Hamiltonian mean-field model that is representative of agents that perform a repeated task with an associated outcome, such as an opinion poll. When a sequence of (many) events takes place in a sufficiently short time the process exhibits clustering of the outcome, but for larger times the process resembles that of independent events. The two regimes are separated by a sharp transition, technically of the second order. Finally we point out the existence of a similar scheme for random-walk processes.

Q-deformed statistical-mechanical structure in the dynamics of the Feigenbaum attractor

We show that the two complementary parts of the dynamics associated to the Feigenbaum attractor, inside and towards the attractor, form together a q-deformed statistical-mechanical structure. A time-dependent partition function produced by summing distances between neighboring positions of the attractor leads to a q-entropy that measures the ratio of ensemble trajectories still away at a given time from the attractor (and the repellor). The values of the q-indexes are given by the attractor's universal constants, while the thermodynamic framework is closely related to that first developed for multifractals.

Dynamics towards the Feigenbaum attractor

As an essential component in the demonstration of an atypical, q-deformed, statistical mechanical structure in the dynamics of the Feigenbaum attractor we expose, at a previously unknown level of detail, the features of the dynamics of trajectories that either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of the rank structure of the fractal boundaries between attractor and repellor positions for the family of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The power law with log-periodic modulation associated to the rate of approach of trajectories towards the attractor (and to the repellor) is explained in terms of the progression of gap formation. vi) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated. We discuss the function of these properties in the atypical thermodynamic framework existing at the period-doubling transition to chaos.

Labyrinthine pathways towards supercycle attractors in unimodal maps

Abstract As an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.