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PubMed2001

Variations of the asset prices.

The empirical established non-Gaussian behavior of asset price fluctuations is studied using an analytical approach. The analysis is based on a nonlinear Fokker-Planck equation with a self-organized feedback-coupling term, devised as a fundamental model for price dynamics. The evidence, and the analytical form of the memory term, are discussed in the context of statistical physics. It will be suggested that the memory term in leading order offers a power law dependence with an exponent straight theta. The stationary solution of the probability density leads asymptotically to a truncated Lévy distribution, the characteristic exponent beta of which is related to the exponent straight theta by beta=3/theta-1. The empirical data can be reproduced by theta approximately 5/4.

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PubMed2001

Average time spent by Lévy flights and walks on an interval with absorbing boundaries.

We consider a Lévy flyer of order alpha that starts from a point x(0) on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of alpha-->2, and the cases of x(0)-->0 and x(0)-->L. For alpha>2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x(0) is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

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PubMed2001

Tighter decoding reliability bound for Gallager's error-correcting code.

Statistical physics is employed to evaluate the performance of error-correcting codes in the case of finite message length for an ensemble of Gallager's error correcting codes. We follow Gallager's approach of upper bounding the average decoding error rate, but invoke the replica method to reproduce the tightest general bound to date, and to improve on the most accurate zero-error noise level threshold reported in the literature. The relation between the methods used and those presented in the information theory literature are explored.

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PubMed2002

Log-periodic route to fractal functions.

Log-periodic oscillations have been found to decorate the usual power-law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance symmetry. For Ising or Potts spins with ferromagnetic interactions on hierarchical systems, the relative magnitude of the log-periodic corrections are usually very small, of order 10(-5). In growth processes [diffusion limited aggregation (DLA)], rupture, earthquake, and financial crashes, log-periodic oscillations with amplitudes of the order of 10% have been reported. We suggest a "technical" explanation for this 4 order-of-magnitude difference based on the property of the "regular function" g(x) embodying the effect of the microscopic degrees of freedom summed over in a renormalization group (RG) approach F(x)=g(x)+mu(-1)F(gamma x) of an observable F as a function of a control parameter x. For systems for which the RG equation has not been derived, the previous equation can be understood as a Jackson q integral, which is the natural tool for describing discrete-scale invariance. We classify the "Weierstrass-type" solutions of the RG into two classes characterized by the amplitudes A(n) of the power-law series expansion. These two classes are separated by a novel "critical" point. Growth processes (DLA), rupture, earthquake, and financial crashes thus seem to be characterized by oscillatory or bounded regular microscopic functions that lead to a slow power-law decay of A(n), giving strong log-periodic amplitudes. If in addition, the phases of A(n) are ergodic and mixing, the observable presents self-affine nondifferentiable properties. In contrast, the regular function of statistical physics models with "ferromagnetic"-type interactions at equilibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables.

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PubMed2002

Combination of improved multibondic method and the Wang-Landau method.

We propose a method for Monte Carlo simulation of statistical physical models with discretized energy. The method is based on several ideas including the cluster algorithm, the multicanonical Monte Carlo method and its acceleration proposed recently by Wang and Landau. As in the multibondic ensemble method proposed by Janke and Kappler, the present algorithm performs a random walk in the space of the bond population to yield the state density as a function of the bond number. A test on the Ising model shows that the number of Monte Carlo sweeps required of the present method for obtaining the density of state with a given accuracy is proportional to the system size, whereas it is proportional to the system size squared for other conventional methods. In addition, the method shows a better performance than the original Wang-Landau method in measurement of physical quantities.

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PubMed2002

Critical noise levels for low-density parity check decoding.

We determine the critical noise level for decoding low-density parity check error-correcting codes based on the magnetization enumerator (M), rather than on the weight enumerator (W) employed in the information theory literature. The interpretation of our method is appealingly simple, and the relation between the different decoding schemes such as typical pairs decoding, MAP, and finite temperature decoding (MPM) becomes clear. In addition, our analysis provides an explanation for the difference in performance between MN and Gallager codes. Our results are more optimistic than those derived using the methods of information theory and are in excellent agreement with recent results from another statistical physics approach.

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PubMed2002

Statistical mechanics of typical set decoding.

The performance of "typical set (pairs) decoding" for ensembles of Gallager's linear code is investigated using statistical physics. In this decoding method, errors occur, either when the information transmission is corrupted by atypical noise, or when multiple typical sequences satisfy the parity check equation as provided by the received corrupted codeword. We show that the average error rate for the second type of error over a given code ensemble can be accurately evaluated using the replica method, including the sensitivity to message length. Our approach generally improves the existing analysis known in the information theory community, which was recently reintroduced in IEEE Trans. Inf. Theory 45, 399 (1999), and is believed to be the most accurate to date.

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PubMed2002

Dynamic phase transition for decoding algorithms.

The state-of-the-art error correcting codes are based on large random constructions (random graphs, random permutations, etc.) and are decoded by linear-time iterative algorithms. Because of these features, they are remarkable examples of diluted mean-field spin glasses, both from the static and dynamic points of view. We analyze the behavior of decoding algorithms by mapping them onto statistical-physics models. This allows us to understand the intrinsic (i.e., algorithm independent) features of this behavior.

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PubMed2003

Slow relaxation due to optimization and restructuring: solution on a hierarchical lattice.

Motivated by the large strain shear of loose granular materials, we introduced a model that consists of consecutive optimization and restructuring steps leading to a self-organization of a density field. The extensive connections to other models of statistical physics are discussed. We investigate our model on a hierarchical lattice that allows an exact asymptotic renormalization treatment. A surprisingly close analogy is observed between the simulation results on the regular and the hierarchical lattices. The dynamics is characterized by the breakdown of ergodicity, by unusual system size effects in the development of the average density, as well as by the age distribution, the latter showing multifractal properties.

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PubMed2003

Statistical mechanics of broadcast channels using low-density parity-check codes.

We investigate the use of Gallager's low-density parity-check (LDPC) codes in a degraded broadcast channel, one of the fundamental models in network information theory. Combining linear codes is a standard technique in practical network communication schemes and is known to provide better performance than simple time sharing methods when algebraic codes are used. The statistical physics based analysis shows that the practical performance of the suggested method, achieved by employing the belief propagation algorithm, is superior to that of LDPC based time sharing codes while the best performance, when received transmissions are optimally decoded, is bounded by the time sharing limit.

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PubMed2003

Hierarchical population model with a carrying capacity distribution for bacterial biofilms.

In order to describe biological colonies with a conspicuous hierarchical structure, a time- and space-discrete model for the growth of a rapidly saturating local biological population N(x,t) is derived from a hierarchical random deposition process previously studied in statistical physics. Two biologically relevant parameters, the probabilities of birth, B, and of death, D, determine the carrying capacity K. Due to the randomness the population depends strongly on position x and there is a distribution of carrying capacities, Pi(K). This distribution has self-similar character owing to the exponential slowing down of the growth, assumed in this hierarchical model. The most probable carrying capacity and its probability are studied as a function of B and D. The effective growth rate decreases with time, roughly as in a Verhulst process. The model is possibly applicable, for example, to bacteria forming a "towering pillar" biofilm, a structure poorly described by standard Eden or diffusion-limited-aggregation models. The bacteria divide on randomly distributed nutrient-rich regions and are exposed to a random local bactericidal agent (antibiotic spray). A gradual overall temperature or chemical change away from optimal growth conditions reduces bacterial reproduction, while biofilm development degrades antimicrobial susceptibility, causing stagnation into a stationary state.

Bacterial Physiological Phenomena↗

PubMed2005

Efficient training of multilayer perceptrons using principal component analysis.

A training algorithm for multilayer perceptrons is discussed and studied in detail, which relates to the technique of principal component analysis. The latter is performed with respect to a correlation matrix computed from the example inputs and their target outputs. Typical properties of the training procedure are investigated by means of a statistical physics analysis in models of learning regression and classification tasks. We demonstrate that the procedure requires by far fewer examples for good generalization than traditional online training. For networks with a large number of hidden units we derive the training prescription which achieves, within our model, the optimal generalization behavior.

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PubMed2005

Generation of complex bipartite graphs by using a preferential rewiring process.

It is important in computer science, sociology, and so on to investigate complex bipartite graphs from a viewpoint of statistical physics. We propose a model to generate complex bipartite graphs without growing; the bipartite graphs are assumed to have two sets of the fixed numbers of nodes and a fixed number of edges between nodes belonging to different sets of nodes. In this model, essential ingredients are a preferential rewiring process and a fitness distribution function. By using the preferential rewiring process, we confirm that a bipartite graph reaches a stationary state after a sufficiently long time has passed. We find that the obtained bipartite graph has a scale-free-like property when a suitable fitness distribution is used. It turns out that a condensation of edges takes place in the cases of certain fitness distributions.

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PubMed2006

Packing dimers on (2p+1) x (2q+1) lattices.

We use computational method to investigate the number of ways to pack dimers on m x n lattices, where both m and n are odd. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on (2k+1) x (2k+1) odd square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on 2k x 2k even square lattices, for which an exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd-by-odd lattice strips with any width n > or =1. This logarithmic term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer-dimer problem might be solvable.

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PubMed2000

Typical performance of gallager-type error-correcting codes

The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly constructed sparse matrices; the number of nonzero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the Thouless-Anderson-Palmer approach which is identical to the commonly used belief-propagation-based decoding.

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PubMed2000

Cryptographical properties of Ising spin systems.

The relation between Ising spin systems and public-key cryptography is investigated using methods of statistical physics. The insight gained from the analysis is used for devising a matrix-based cryptosystem whereby the ciphertext comprises products of the original message bits; these are selected by employing two predetermined randomly constructed sparse matrices. The ciphertext is decrypted using methods of belief propagation. The analyzed properties of the suggested cryptosystem show robustness against various attacks and competitive performance to modern cryptographical methods.

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PubMed2001

Avalanches of popping bubbles in collapsing foams.

We report acoustic measurements of popping bubbles during the collapsing of aqueous foams. The sound pattern is analyzed using classical methods of statistical physics. It is found that membrane rupture concerns a wide variety of situations: small and large membranes at the air/foam interface. Avalanches of popping bubbles are put into evidence. Time durations in between successive pops seem to be distributed on a universal power law.

Cells↗

PubMed2001

Universal learning curves of support vector machines.

Using methods of statistical physics, we investigate the role of model complexity in learning with support vector machines (SVMs), which are an important alternative to neural networks. We show the advantages of using SVMs with kernels of infinite complexity on noisy target rules, which, in contrast to common theoretical beliefs, are found to achieve optimal generalization error although the training error does not converge to the generalization error. Moreover, we find a universal asymptotics of the learning curves which depend only on the target rule but not on the SVM kernel.

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