The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

Storage of electromagnetic field energy in matter

The partitioning, uniqueness and form of field energy stored in matter, and its properties as a state function, is established. Consequently, the first and second laws apply to the nonfield and field parts of the internal energy as separate entities. This provides a bridge between thermodynamics and the classical theory of electromagnetism. Presentation of the temperature as the sum of nonfield and field contributions is used to establish field dependent barriers to temperature decrease toward the absolute zero, and the existence of field induced temperature jumps. These temperature jumps appear at the instant the field is switched on, or turned off. The partitioning of field and nonfield energies is illustrated for a specific case of an ideal gas, and the heat absorbed by the field is derived in terms of difference in adiabatic magnetization. Finally, the current, restrictive, form of electromagnetic field energy density is redefined with respect to the effect of field energy stored outside the system boundaries. Received 6 June 2000 / Received in final form 26 March 2002 Published online 24 September 2002 RID="a" ID="a"e-mail: zimmels@tx.technion.ac.il     

Complexity of two-dimensional patterns

To describe quantitatively the complexity of two-dimensional patterns we introduce a complexity measure based on a mean information gain. Two types of patterns are studied: geometric ornaments and patterns arising in random sequential adsorption of discs on a plane (RSA). For the geometric ornaments analytical expressions for entropy and complexity measures are presented, while for the RSA patterns these are calculated numerically. We compare the information-gain complexity measure with some alternative measures and show advantages of the former one, as applied to two-dimensional structures. Namely, this does not require knowledge of the “maximal” entropy of the pattern, and at the same time sensitively accounts for the inherent correlations in the system. Received 12 November 1999     

Fractional approach,quantum statistics,and non-crystalline solids at very low temperatures

We consider nonlocal effects, obtained by incorporating fractional derivatives in the kinetic energy of a conventional Hamiltonian, to analyze physical properties of non-crystalline solids at very low temperature. By using thermal Green functions, we deduce some experimentally observable quantities such as the particle momentum distribution function, the particle energy distribution function, and the specific heat. The agreement between the results obtained for the specific heat and the experimental data suggests that the approach presented here may be useful as a phenomenological model to investigate thermal properties of non-crystalline solids at low temperature.     

Nonadditive entropy: The concept and its use

The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to construct the exact differential dS = Q/T , where Q is the heat transfer and the absolute temperature T its integrating factor. A few years later, in the period 1872-1877, it was shown by Boltzmann that this quantity can be expressed in terms of the probabilities associated with the microscopic configurations of the system. We refer to this fundamental connection as the Boltzmann-Gibbs (BG) entropy, namely (in its discrete form) , where k is the Boltzmann constant, and {p _i} the probabilities corresponding to the W microscopic configurations (hence ∑^W _i=1 p _i = 1 . This entropic form, further discussed by Gibbs, von Neumann and Shannon, and constituting the basis of the celebrated BG statistical mechanics, is additive. Indeed, if we consider a system composed by any two probabilistically independent subsystems A and B (i.e., , we verify that . If a system is constituted by N equal elements which are either independent or quasi-independent (i.e., not too strongly correlated, in some specific nonlocal sense), this additivity guarantees S_BG to be extensive in the thermodynamical sense, i.e., that in the N ≫ 1 limit. If, on the contrary, the correlations between the N elements are strong enough, then the extensivity of S_BG is lost, being therefore incompatible with classical thermodynamics. In such a case, the many and precious relations described in textbooks of thermodynamics become invalid. Along a line which will be shown to overcome this difficulty, and which consistently enables the generalization of BG statistical mechanics, it was proposed in 1988 the entropy . In the context of cybernetics and information theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form S_q is, for any q 1 , nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies . This form will turn out to be extensive for an important class of nonlocal correlations, if q is set equal to a special value different from unity, noted q_ent (where ent stands for entropy . In other words, for such systems, we verify that , thus legitimating the use of the classical thermodynamical relations. Standard systems, for which S_BG is extensive, obviously correspond to q _ent = 1 . Quite complex systems exist in the sense that, for them, no value of q exists such that S_q is extensive. Such systems are out of the present scope: they might need forms of entropy different from S_q, or perhaps --more plainly-- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with S_q, the q -generalizations of the Central Limit Theorem and of its extended Lévy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of q -exponentials, q -Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations --in high-energy physics and elsewhere-- are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms versus distinct regimes of a single physical mechanism. This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.     

On some natural and model 2D bimodal random cellular structures

The topological and metric properties of a few natural 2D random cellular structures, namely an armadillo shell structure and young soap froths, which are formed from two classes of cells, large and small, have been characterized. The topological properties of a model generated from a Kagome tiling, which mimics such random binary structures, have also been exactly calculated. The distribution of the number of cell sides is bimodal for the structures investigated. In contrast to the classical Aboav-Weaire law for homogeneous 2D random cellular structures, nm(n), the mean total number of edges of neighbouring cells of cells with n sides does not vary linearly with n. Only the nm(i, n) (i=1,2) determined separately for every class of cells are linear in n for all investigated structures. Topological properties and correlations between metric and topological properties are finally compared with the predictions of various literature models. Received: 24 December 1997 / Revised: 7 April 1998 / Accepted: 20 April 1998     

Statistical mechanics of money: how saving propensity affects its distribution

We consider a simple model of a closed economic system where the total money is conserved and the number of economic agents is fixed. Analogous to statistical systems in equilibrium, money and the average money per economic agent are equivalent to energy and temperature, respectively. We investigate the effect of the saving propensity of the agents on the stationary or equilibrium probability distribution of money. When the agents do not save, the equilibrium money distribution becomes the usual Gibb's distribution, characteristic of non-interacting agents. However with saving, even for individual self-interest, the dynamics becomes cooperative and the resulting asymmetric Gaussian-like stationary distribution acquires global ordering properties. Intriguing singularities are observed in the stationary money distribution in the market, as functions of the marginal saving propensity of the agents. Received 2 May 2000     

Revealing of the simplest mechanisms of a structure formation

In this paper, results of investigations of the simplest mechanisms of a structure formation are presented. In frameworks of the suggested model the main attention was focused on such characteristics as wiring of the system, clusters formation, dynamics of the wiring. The idea to take into account an influence of the environment factor is employed in the proposed model. Investigations of systems with such principle of a structure formation reveal that the system's dynamics has typical features of self-organized criticality phenomenon. In the avalanche-like processes, which occur in the wiring dynamics, a power law was found with the index close to 1.4. It is independent on the environment factor (which in a sense can be considered as system parameter). The system wiring is approximated pretty well by the Gaussian distribution. The size of the system does not play any role in the dynamics of the model. Received 10 March 1999 and Received in final form 24 May 1999     

Diagonalization of the elliptic Ruijsenaars model. Correspondence with the Belavin model

Studied is the elliptic Ruijsenaars model, which is a difference analogue of the Calogero-Sutherland-Moser model. Using a novel relationship between the elliptic Ruijsenaars operator and the transfer matrix of the Belavin model, we diagonalize the Ruijsenaars operator by the algebraic Bethe ansatz method. Received: 29 January 1998 / Accepted: 17 April 1998     

Combinatorial basis and non-asymptotic form of the Tsallis entropy function

Using a q-analog of Boltzmann's combinatorial basis of entropy, the non-asymptotic non-degenerate and degenerate combinatorial forms of the Tsallis entropy function are derived. The new measures – supersets of the Tsallis entropy and the non-asymptotic variant of the Shannon entropy – are functions of the probability and degeneracy of each state, the Tsallis parameter q and the number of entities N. The analysis extends the Tsallis entropy concept to systems of small numbers of entities, with implications for the permissible range of q and the role of degeneracy.     

q-deformed Fermions

This is a study of q-Fermions resulting from q-deformed algebra of harmonic oscillators arising from two distinct algebras. Employing the first algebra, the Fock states are constructed for the generalized Fermions obeying Pauli exclusion principle. The distribution function and other thermodynamic properties such as the internal energy and entropy are derived. Another generalization of fermions from a different q-deformed algebra is investigated which deals with q-fermions not obeying the exclusion principle. Fock states are constructed for this system. The basic numbers appropriate for this system are determined as a direct consequence of the algebra. We also establish the Jackson Derivative, which is required for the q-calculus needed to describe these generalized Fermions.     

Adsorption of a Gaussian random copolymer chain at an interface

We consider the adsorption of an isolated, Gaussian, random, and quenched copolymer chain at an interface. We first propose a simple analytical method to obtain the adsorption/depletion transition, by averaging over the disorder the partition function instead of the free energy. The adsorption thresholds obtained by previous authors at a solid/liquid and at a liquid/liquid interface for multicopolymer chains can be rederived using this method. We also compare the adsorption thresholds obtained for bimodal and for Gaussian disorder; they only agree for small disorder. We focus on the specific case of an ideally flat asymmetric liquid/liquid interface, and consider the situation where the chain is composed of monomers of two different chemical species A and B. The replica method is developed for this case. We show that the Hartree approximation, coupled to a replica symmetry assumption, leads to the same adsorption thresholds as obtained from our general method. In order to describe the properties of the adsorbed (or depleted) chain, we develop a new approximation for long chains, within the framework of the replica theory. In most cases, the behavior of a random copolymer chain can be mapped onto that of a homopolymer chain at an asymmetric attractive interface. The values of the effective adsorption energy are different for a random and a periodic copolymer chain. Finally, we consider the case of uncorrelated annealed disorder. The behavior of an annealed chain can be mapped onto that of a homopolymer chain at an asymmetric non attractive interface; hence, an annealed chain cannot adsorb at an asymmetric interface. Received 21 January 1999     

Performance analysis of a thermosize micro/nano heat engine

In a recent paper [A. Sisman, I. Muller, Phys. Lett. A 320 (2004) 360] the thermodynamic properties of ideal gases confined in a narrow box were examined theoretically. The so-called “thermosize effects” similar to thermoelectric effects, such as Seebeck-like thermosize effect, Peltier-like thermosize effect and Thomson-like thermosize effect, were analyzed. Like the thermoelectric generator, based on the thermosize effects we have established a model of micro/nano scaled ideal gas heat engine cycle which includes two isothermal and two isobaric processes. The expressions of power output and efficiency of this cycle in the two cases of reversible and irreversible heat exchange are derived and the optimal performance characteristics of the heat engine is discussed by some numerical example. The results obtained here will provide theoretical guidance for the design of micro/nano scaled device.     

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy . More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices q in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation q_stat+q_sens=2 is numerically satisfied, q_stat and q_sens being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.     

Scale-invariant cellular automata and self-similar Petri nets

Two novel computing models based on an infinite tessellation of space-time are introduced. They consist of recursively coupled primitive building blocks. The first model is a scale-invariant generalization of cellular automata, whereas the second one utilizes self-similar Petri nets. Both models are capable of hypercomputations and can, for instance, “solve” the halting problem for Turing machines. These two models are closely related, as they exhibit a step-by-step equivalence for finite computations. On the other hand, they differ greatly for computations that involve an infinite number of building blocks: the first one shows indeterministic behavior, whereas the second one halts. Both models are capable of challenging our understanding of computability, causality, and space-time.     

Statistical physics and practical training of soft-committee machines

Equilibrium states of large layered neural networks with differentiable activation function and a single, linear output unit are investigated using the replica formalism. The quenched free energy of a student network with a very large number of hidden units learning a rule of perfectly matching complexity is calculated analytically. The system undergoes a first order phase transition from unspecialized to specialized student configurations at a critical size of the training set. Computer simulations of learning by stochastic gradient descent from a fixed training set demonstrate that the equilibrium results describe quantitatively the plateau states which occur in practical training procedures at sufficiently small but finite learning rates. Received 16 December 1998     

Quantum Brownian motion and the second law of thermodynamics

We consider a single harmonic oscillator coupled to a bath at zero temperature. As is well-known, the oscillator then has a higher average energy than that given by its ground state. Here we show analytically that for a damping model with arbitrarily discrete distribution of bath modes and damping models with continuous distributions of bath modes with cut-off frequencies, this excess energy is less than the work needed to couple the system to the bath, therefore, the quantum second law is not violated. On the other hand, the second law may be violated for bath modes without cut-off frequencies, which are, however, physically unrealistic models. An erratum to this article is available at .     

Nonextensive interpretation of radiative recombination in electron cooling

An interest for the low-energy range of the nonextensive distribution function arises from the study of radiative recombination in electron cooling devices in particle accelerators, whose experimentally measured reaction rates are much above the theoretical prediction. The use of generalized distributions, that differ from the Maxwellian in the low energy part (due to subdiffusion between electron and ion bunches), may account for the observed rate enhancement. In this work, we consider the isotropic distribution function and we propose a possible experiment for verifying the existence of a cut-off in the generalized momentum distribution, by measuring the spectrum of the X-rays emitted from radiative recombination reactions.