Boltzmann-Gibbs Distribution Research Articles

Detection of Subsurface Bubbles with Discrete Electromagnetic Geotomography

Abstract In electromagnetic geotomography, the inversion is very ill-posed due to the limita- tion of an angular range of projections, and consequently the reconstructed image is different than the true image. Nevertheless, the reconstruction can be much im- proved if we are interested only in a piece of information on the interior volume of the examined object, e.g. detection of some anomalies (bubbles). We assume that the image can be represented by binary values, hence: Discrete Electromagnetic Geotomography (DEG). In this paper, we discuss two discrete image reconstruction algorithms. The first one uses the D.C. programming to find the integer solution to the penalized least-squares problem. The penalty term is based on the Gibbs distribution. In the second one, the mean field annealing is applied to maximize the Gibbs-Boltzmann distribution associated with the penalized least-squares function. The numerical results are presented for noise-free and noisy data.

Thermodynamic control of germanium quantum dot growth on silicon

Strained epitaxial growth of Ge on Si(0 0 1) produces self-assembled, nanometer scale islands, or quantum dots. We study this growth by atomistic simulation, computing the energy of island structures to determine when and how islanding occurs. The distribution of island sizes on a surface is determined by the relation of island energy to size. Applying the calculated energy per atom to the Boltzmann–Gibbs distribution, we predict size distributions as functions of coverage and temperature. The peak populations around 86,000 atoms (35 nm wide) compare favorably with experiment.

The p-sphere and the geometric substratum of power-law probability distributions

Links between power law probability distributions and marginal distributions of uniform laws on p-spheres in R n show that a mathematical derivation of the Boltzmann–Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters p and n to the value of the non-extensivity parameter q that characterizes these power laws in the context of non-extensive statistics.

Anharmonic Oscillator Driven by Additive Ornstein–Uhlenbeck Noise

We present an analytical study of a nonlinear oscillator subject to an additive Ornstein-Uhlenbeck noise. Known results are mainly perturbative and are restricted to the large dissipation limit (obtained by neglecting the inertial term) or to a quasi-white noise (i.e., a noise with vanishingly small correlation time). Here, in contrast, we study the small dissipation case (we retain the inertial term) and consider a noise with finite correlation time. Our analysis is non perturbative and based on a recursive adiabatic elimination scheme: a reduced effective Langevin dynamics for the slow action variable is obtained after averaging out the fast angular variable. In the conservative case, we show that the physical observables grow algebraically with time and calculate the associated anomalous scaling exponents and generalized diffusion constants. In the case of small dissipation, we derive an analytic expression of the stationary Probability Distribution Function (P.D.F.) which differs from the canonical Boltzmann-Gibbs distribution. Our results are in excellent agreement with numerical simulations.

An analytic treatment of the Gibbs–Pareto behavior in wealth distribution

We develop a general framework, based on Boltzmann transport theory, to analyze the distribution of wealth in societies. Within this framework we derive the distribution function of wealth by using a two-party trading model for the poor people, while for the rich people a new model is proposed, where interaction with wealthy entities (huge reservoir) is relevant. At equilibrium, the interaction with wealthy entities gives a power-law (Pareto-like) behavior in the wealth distribution, while the two-party interaction gives a Boltzmann–Gibbs distribution.

Deterministic generation of the Boltzmann–Gibbs distribution and the free energy calculation from the Tsallis distribution

A method for constructing the Boltzmann–Gibbs distributions is presented by combining the re-weighting technique and the deterministic dynamical equation that produces the Tsallis distributions. The ability to produce the Boltzmann–Gibbs distributions is shown by calculating the distribution, the energy with higher moments, and the Helmholtz free energy. From numerical simulations of a small system that represents a fundamental chemical interaction model, correct answers were obtained within a wide range of temperatures. An application to a peptide system is also shown.

Generalized thermostatistics and mean-field theory

The present paper studies a large class of temperature-dependent probability distributions and shows that entropy and energy can be defined in such a way that these probability distributions are the equilibrium states of a generalized thermostatistics, which is obtained from the standard formalism by deformation of exponential and logarithmic functions. Since this procedure is non-unique, specific choices are motivated by showing that the resulting theory is well-behaved. In particular, with the choices made in the present paper, the equilibrium state of any system with a finite number of degrees of freedom is, automatically, thermodynamically stable and satisfies the variational principle. The equilibrium probability distribution of open systems deviates generically from the Boltzmann–Gibbs distribution. If the interaction with the environment is not too strong then one can expect that a slight deformation of the exponential function, appearing in the Boltzmann–Gibbs distribution, can reproduce the observed distribution with its temperature dependence. An example of a system, where this statement holds, is a single spin of the Ising chain. However, because all systems of the present generalized thermostatistics are automatically stable, one must not expect that all open systems can be described in this way. Indeed, systems exhibiting a phase transition in their thermodynamic limit, can be unstable even when the interaction with the environment is weak. Therefore, their equilibrium probability distribution cannot be described by a simple deformation of the Boltzmann–Gibbs distribution. In order to be able to handle such systems as well the second part of the paper discusses a further extension of the class of probability distributions using mean-field techniques. Connections are discussed that exist between the present formalism, Tsallis’ thermostatistics, and the superstatistics formalism of Beck and Cohen. In particular, the present generalization sheds some light onto the historical development of the Tsallis formalism. It is pointed out that temperature dependence of the Tsallis distribution has hardly been verified experimentally.

Germanium Island Size Distribution by Atomistic Simulation

ABSTRACTStrained epitaxial growth of Ge on Si(001) produces self-assembled, nanometer scale islands, or quantum dots. We study this growth by atomistic simulation, computing the energy of island structures to determine when and how islanding occurs. The distribution of island sizes on a surface is determined by the relation of island energy to size. Applying the calculated chemical potential to the Boltzmann-Gibbs distribution, we predict size distributions as functions of coverage and temperature. The peak populations around 80 000 atoms (35 nm wide) compare favorably with experiment.

A recurrent stochastic binary network

Stochastic neural networks are usually built by introducing random fluctuations into the network. A natural method is to use stochastic connections rather than stochastic activation functions. We propose a new model in which each neuron has very simple functionality but all the connections are stochastic. It is shown that the stationary distribution of the network uniquely exists and it is approximately a Boltzmann-Gibbs distribution. The relationship between the model and the Markov random field is discussed. New techniques to implement simulated annealing and Boltzmann learning are proposed. Simulation results on the graph bisection problem and image recognition show that the network is powerful enough to solve real world problems.

Statistical mechanics and the description of the early universe. (I). Foundations for a slightly non-extensive cosmology

We analyze how the thermal history of the universe is influenced by the statistical description, assuming a deviation from the usual Bose–Einstein, Fermi–Dirac and Boltzmann–Gibbs distribution functions. These deviations represent the possible appearance of non-extensive effects related with the existence of long-range forces, memory effects, or evolution in fractal or multi-fractal space. In the early universe, it is usually assumed that the distribution functions are the standard ones. Then, considering the evolution in a larger theoretical framework will allow to test this assumption and to place limits to the range of its validity. The corrections obtained will change with temperature, and consequently, the bounds on the possible amount of non-extensivity will also change with time. We generalize results which can be used in other contexts as well, as the Boltzmann equation and the Saha law, and provide an estimate on how known cosmological bounds on the masses of neutrinos are modified by a change in the statistics. We particularly analyze here the recombination epoch, making explicit use of the chemical potentials involved in order to attain the necessary corrections. All these results constitute the basic tools needed for placing bounds on the amount of non-extensivity that could be present at different eras and will be later used to study primordial nucleosynthesis.

RENORMALIZATION GROUP FOR SPIN GLASSES

A renormalization group transformation suitable for spin glasses, and more generally for disordered systems, is presented. The approach is non-standard for two reasons. 1. The coarse graining transformation is performed on the distribution of site overlaps, which clearly is not a Boltzmann-Gibbs distribution. 2. The additional interactions are realized introducing a gauge-field disorder distribution, while the Hamiltonian is kept fixed. Universality classes are thus naturally defined on a large set of models, going from ℤ2 and Gaussian spin glasses to Ising and fully frustrated models, and others. The role of frustration and of the deconfinement transition can also be analysed in terms of fixed points. The proposed analysis is tested numerically on the ℤ2 Edwards-Anderson model in d=4. Good estimates of the critical index ν and of Tc are obtained, and an RG flow diagram is sketched for the first time.

On the classical energy equipartition theorem

A general proof of the energy equipartition theorem is given. Our derivation holds for any distribution function depending on the phase space variables only through the Hamiltonian of the system. This approach generalizes the standard theorem in two main directions. On the one hand, it considers the contribution to the total mean energy of homogeneous functions having a more general type than the ones usually discussed in the literature. On the other hand, our proof does not rely on the assumption of a Boltzmann-Gibbs exponential distribution.

Optimization with the Hopfield network based on correlated noises: Experimental approach

This paper presents two simple optimization techniques based on combining the Langevin Equation with the Hopfield Model. Proposed models – referred as stochastic model (SM) and pulsed noise model (PNM) – can be regarded as straightforward stochastic extensions of the Hopfield optimization network. Both models follow the idea of stochastic neural network (Levy and Adams, IEEE Conference on Neural Networks, vol. III, San Diego, USA, 1987, pp. 681–689) and diffusion machine (Wong, Algorithmica 6 (1991) 466–478). They differ form the referred approaches by the nature of noises and the way of their injection. Optimization with stochastic model, unlike in the previous works, in which δ-correlated Gaussian noises were considered, is based on Gaussian noises with positive autocorrelation times. This is a reasonable assumption from a hardware implementation point of view. In the other model – pulsed noise model, Gaussian noises are injected to the system only at certain time instances, as opposed to continuously maintained δ-correlated noises used in the previous related works. In both models (SM and PNM) intensities of injected noises are independent of neurons’ potentials. Moreover, instead of impractically long inverse logarithmic cooling schedules, the linear cooling is tested. With the above strong simplifications neither SM nor PNM is expected to rigorously maintain thermal equilibrium (TE). However, numerical tests based on the canonical Gibbs–Boltzmann distribution show, that differences between rigorous and estimated values of TE parameters are relatively low (within a few percent). In this sense both models are said to perform quasithermal equilibrium. Optimization performance and quasithermal equilibrium properties of both models are presented based on the travelling salesman problem (TSP).

On the multinomial logit model

Abstract We show that the Multinomial Logit model of bounded rational choice can be derived in the same way as the Gibbs–Boltzmann distribution in statistical physics. In particular, this model describes the behavior of a thermodynamic agent (which is an agent whose utility function depends on a very large number of variables) with respect to a small subset of variables “weakly interacting” with the others. We also show that the same model is obtained if entropic control costs or information costs are introduced, in which case the temperature like parameter can be considered as the price of (negative) entropy.

Thermal properties of slow dynamics

The limit of small entropy production is reached in relaxing systems long after preparation, and in stationary driven systems in the limit of small driving power. Surprisingly, for extended systems this limit is not in general the Gibbs-Boltzmann distribution, or a small departure from it. Interesting cases in which it is not are glasses, phase-separation, and certain driven complex fluids. We describe a scenario with several coexisting temperatures acting on different time scales, and partial equilibrations at each time scale. This scenario entails strong modifications of the fluctuation-dissipation equalities and the existence of some unexpected reciprocity relations. Both predictions are open to experimental verification, particularly the latter. The construction is consistent in general, since it can be viewed as the breaking of a symmetry down to a residual group. It does not assume the presence of quenched disorder. It can be — and to a certain extent has been — tested numerically, while some experiments are on their way. There is furthermore the perspective that analytic arguments may be constructed to prove or disprove its generality.

Possible connection of the generalized thermostatistics with a scale invariant statistical thermodynamics

A conjecture is presented on the possibility that the q-generalized statistical thermodynamics proposed by Tsallis may refer to scale invariant statistical mechanics. This scale symmetry is introduced by means of a fractal canonical ensemble, constructed by replicas of the same system at different scales. In this way, the Tsallis distribution of energy is derived from a renormalization of the Boltzmann-Gibbs distribution. Such an ensemble could be an appropriate description of systems with hierarchical structures. In particular, it might be useful in studying thermodynamics systems with a fractal structure in the phase-space.

MACHINE DESIGN OPTIMIZATION BY THE MONTE CARLO ANNEALING METHOD

The paper presents the application of the Monte Carlo annealing method for machine design optimization, The method generates the finite inhomogeneous Markov chain in the space of discrete design parameters with the transition probability dependent on the objective function and given by the Boltzmann-Gibbs distribution of equilibrium statistical mechanics. In this approach the search for the global minimum is simulated by a relaxation towards an equilibrium of the thermodynamic system with the energy proportional to an objective function. The method is applied for the cost optimization of a speed reductor. The performance and prospects of the method for machine design optimization are discussed.

Monte Carlo structural optimization in discrete variables with annealing algorithm

AbstractThe paper describes the basic ideas of Monte Carlo annealing algorithms for structural optimization with discrete design parameters. The algorithm generates randomly a set of design parameters, with probability depending on the objective function and given by the Boltzmann–Gibbs distribution. In this model the search for the global minimum is simulated by a relaxation process of the statistical mechanical system with the Hamiltonian proportional to the objective function. The rate of the convergence of the method and its dependence upon the annealing probability are discussed. Numerical implementation of the method for the weight optimization of the ten‐bar planar cantilever truss is presented. The results of numerical simulation are compared with those obtained by the dual methods. The principal conjecture is that the method is fairly efficient and has great potential for applicaton in engineering design.

Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm

We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprisingly enough, using this simple algorithm, one can get very close to the optimal solution of the problem or even find the true optimum. We demonstrate this on several examples. We conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.