100 research outputs found
On the derivation of power-law distributions within classical statistical mechanics far from the thermodynamic limit
We show that within classical statistical mechanics without taking the
thermodynamic limit, the most general Boltzmann factor for the canonical
ensemble is a q-exponential function. The only assumption here is that
microcanonical distributions have to be separable from of the total system
energy, which is the prerequisite for any sensible measurement. We derive that
all separable distributions are parametrized by a mathematical separation
constant Q which can be related to the non-extensivity q-parameter in Tsallis
distributions. We further demonstrate that nature fixes the separation constant
Q to 1 for large dimensionality of Gibbs Gamma-phase space. Our results will be
relevant for systems with a low-dimensional Gamma-space, for example
nanosystems, comprised of a small number of particles or for systems with a
dimensionally collapsed phase space, which might be the case for a large class
of complex systems.Comment: 8 pages, contribution to Next Sigma Phi, Crete 200
Generalized (c,d)-entropy and aging random walks
Complex systems are often inherently non-ergodic and non-Markovian for which
Shannon entropy loses its applicability. In particular accelerating,
path-dependent, and aging random walks offer an intuitive picture for these
non-ergodic and non-Markovian systems. It was shown that the entropy of
non-ergodic systems can still be derived from three of the Shannon-Khinchin
axioms, and by violating the fourth -- the so-called composition axiom. The
corresponding entropy is of the form and depends on two system-specific scaling exponents, and . This
entropy contains many recently proposed entropy functionals as special cases,
including Shannon and Tsallis entropy. It was shown that this entropy is
relevant for a special class of non-Markovian random walks. In this work we
generalize these walks to a much wider class of stochastic systems that can be
characterized as `aging' systems. These are systems whose transition rates
between states are path- and time-dependent. We show that for particular aging
walks is again the correct extensive entropy. Before the central part
of the paper we review the concept of -entropy in a self-contained way.Comment: 8 pages, 5 eps figures. arXiv admin note: substantial text overlap
with arXiv:1104.207
Socio-economical dynamics as a solvable spin system on co-evolving networks
We consider social systems in which agents are not only characterized by
their states but also have the freedom to choose their interaction partners to
maximize their utility. We map such systems onto an Ising model in which spins
are dynamically coupled by links in a dynamical network. In this model there
are two dynamical quantities which arrange towards a minimum energy state in
the canonical framework: the spins, s_i, and the adjacency matrix elements,
c_{ij}. The model is exactly solvable because microcanonical partition
functions reduce to products of binomial factors as a direct consequence of the
c_{ij} minimizing energy. We solve the system for finite sizes and for the two
possible thermodynamic limits and discuss the phase diagrams.Comment: 5 pages 3 fig
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