Biomedical subjects

David S Dean

Publications and source records attributed to David S Dean.

9 recordsLinked to original sources

PubMed2006

Statistical mechanics of multi-index matching problems with site disorder.

We study the statistical mechanics of multi-index matching problems where the quenched disorder is a geometric site disorder rather than a link disorder. A recently developed functional formalism is exploited that yields exact results in the finite-temperature thermodynamic limit. Particular attention is paid to the zero-temperature limit of maximal matching problems where the method allows us to obtain the average value of the optimal match and also sheds light on the algorithmic heuristics leading to that optimal match.

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PubMed2006

Large deviations of extreme eigenvalues of random matrices.

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.

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PubMed2006

Distance traveled by random walkers before absorption in a random medium.

We consider the penetration length l of random walkers diffusing in a medium of perfect or imperfect absorbers of number density rho. We solve this problem on a lattice and in the continuum in all dimensions D, by means of a mean-field renormalization group. For a homogeneous system in D > 2, we find that l is similar to max(xi,rho(-1/2)), where is the absorber density correlation length. The cases of D = 1 and D = 2 are also treated. In the presence of long-range correlations, we estimate the temporal decay of the density of random walkers not yet absorbed. These results are illustrated by exactly solvable toy models, and extensive numerical simulations on directed percolation, where the absorbers are the active sites. Finally, we discuss the implications of our results for diffusion limited aggregation (DLA), and we propose a more effective method to measure l in DLA clusters.

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PubMed2005

Statistical mechanics of combinatorial optimization problems with site disorder.

We study the statistical mechanics of a class of problems whose phase space is the set of permutations of an ensemble of quenched random positions. Specific examples analyzed are the finite-temperature traveling salesman problem on several different domains and various problems in one dimension such as the so-called descent problem. We first motivate our method by analyzing these problems using the annealed approximation. Then in the limit of a large number of points we develop a formalism to carry out the quenched calculation. This formalism does not require the replica method, and its predictions are found to agree with Monte Carlo simulations. In addition our method reproduces an exact mathematical result for the maximum traveling salesman problem in two dimensions and suggests its generalization to higher dimensions. The general approach may provide an alternative method to study certain systems with quenched disorder.

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PubMed2005

Equilibrium statistics of a slave estimator in Langevin processes.

We analyze the statistics of an estimator, denoted by xi(t) and referred to as the slave, for the equilibrium susceptibility of a one dimensional Langevin process x(t) in a potential phi (x). The susceptibility can be measured by evolving the slave equation in conjunction with the original Langevin process. This procedure yields a direct estimate of the susceptibility and avoids the need, when performing numerical simulations, to include applied external fields explicitly. The success of the method, however, depends on the statistical properties of the slave estimator. The joint probability density function for x(t) and xi(t) is analyzed. In the case where the potential of the system has a concave component the probability density function of the slave acquires a power law tail characterized by a temperature dependent exponent. Thus we show that while the average value of the slave, in the equilibrium state, is always finite and given by the fluctuation-dissipation relation, higher moments and indeed the variance may show divergences. The behavior of the power law exponent is analyzed in a general context and it is calculated explicitly in some specific examples. Our results are confirmed by numerical simulations and we discuss possible measurement discrepancies in the fluctuation dissipation relation which could arise due to this behavior.

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PubMed2003

Possible test of the thermodynamic approach to granular media.

We study the steady state distribution of the energy of the Sherrington-Kirkpatrick model driven by a tapping mechanism which mimics the mechanically driven dynamics of granular media. The dynamics consists of two phases: a zero temperature relaxation phase which leads the system to a metastable state, then a tapping which excites the system thus reactivating the relaxational dynamics. Numerically, we investigate whether the distribution of the energies of the blocked states obtained agrees with a simple canonical form of the Edwards measure. It is found that this canonical measure is in good agreement with the dynamically measured energy distribution. A possible experimental test of the Edwards measure based on the study here is proposed.

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PubMed2002

Slow relaxation in a constrained Ising spin chain: toy model for granular compaction.

We present detailed analytical studies on the zero-temperature coarsening dynamics in an Ising spin chain in the presence of a dynamically induced field that favors locally the "-" phase compared to the "+" phase. We show that the presence of such a local kinetic bias drives the system into a late time state with average magnetization m equal to -1. However the magnetization relaxes into this final value extremely slowly in an inverse logarithmic fashion. We further map this spin model exactly onto a simple lattice model of granular compaction that includes the minimal microscopic moves needed for compaction. This toy model then predicts analytically an inverse logarithmic law for the growth of density of granular particles, as seen in recent experiments and thereby provides a mechanism for the inverse logarithmic relaxation. Our analysis utilizes an independent interval approximation for the particle and the hole clusters and is argued to be exact at late times (supported also by numerical simulations).

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PubMed2002

Exact occupation time distribution in a non-Markovian sequence and its relation to spin glass models.

We compute exactly the distribution of the occupation time in a discrete non-Markovian toy sequence that appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function that is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model; at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.

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PubMed2002

Exact solution of a drop-push model for percolation.

Motivated by a computer science algorithm known as "linear probing with hashing," we study a new type of percolation model whose basic features include a sequential "dropping" of particles on a substrate followed by their transport via a "pushing" mechanism. Our exact solution in one dimension shows that, unlike the ordinary random percolation model, the drop-push model has nontrivial spatial correlations generated by the dynamics itself. The critical exponents in the drop-push model are also different from those of the ordinary percolation. The relevance of our results to computer science is pointed out.

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