42 research outputs found
On the robustness of random Boolean formulae
Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models
Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables
We study the Langevin dynamics of the adaptive immune system, modelled by a
lymphocyte network in which the B cells are interacting with the T cells and
antigen. We assume that B clones and T clones are evolving in different thermal
noise environments and on different timescales. We derive stationary
distributions and use statistical mechanics to study clonal expansion of B
clones in this model when the B and T clone sizes are assumed to be the slow
and fast variables respectively and vice versa. We derive distributions of B
clone sizes and use general properties of ferromagnetic systems to predict
characteristics of these distributions, such as the average B cell
concentration, in some regimes where T cells can be modelled as binary
variables. This analysis is independent of network topologies and its results
are qualitatively consistent with experimental observations. In order to obtain
full distributions we assume that the network topologies are random and locally
equivalent to trees. The latter allows us to employ the Bethe-Peierls approach
and to develop a theoretical framework which can be used to predict the
distributions of B clone sizes. As an example we use this theory to compute
distributions for the models of immune system defined on random regular
networks.Comment: A more recent version (accepted for publication in Journal of Physics
A: Mathematical and Theoretical) with improved figures, references, et
Generating functional analysis of complex formation and dissociation in large protein interaction networks
We analyze large systems of interacting proteins, using techniques from the
non-equilibrium statistical mechanics of disordered many-particle systems.
Apart from protein production and removal, the most relevant microscopic
processes in the proteome are complex formation and dissociation, and the
microscopic degrees of freedom are the evolving concentrations of unbound
proteins (in multiple post-translational states) and of protein complexes. Here
we only include dimer-complexes, for mathematical simplicity, and we draw the
network that describes which proteins are reaction partners from an ensemble of
random graphs with an arbitrary degree distribution. We show how generating
functional analysis methods can be used successfully to derive closed equations
for dynamical order parameters, representing an exact macroscopic description
of the complex formation and dissociation dynamics in the infinite system
limit. We end this paper with a discussion of the possible routes towards
solving the nontrivial order parameter equations, either exactly (in specific
limits) or approximately.Comment: 14 pages, to be published in Proc of IW-SMI-2009 in Kyoto (Journal of
Phys Conference Series
Universal Robotic Gripper based on the Jamming of Granular Material
Gripping and holding of objects are key tasks for robotic manipulators. The
development of universal grippers able to pick up unfamiliar objects of widely
varying shape and surface properties remains, however, challenging. Most
current designs are based on the multi-fingered hand, but this approach
introduces hardware and software complexities. These include large numbers of
controllable joints, the need for force sensing if objects are to be handled
securely without crushing them, and the computational overhead to decide how
much stress each finger should apply and where. Here we demonstrate a
completely different approach to a universal gripper. Individual fingers are
replaced by a single mass of granular material that, when pressed onto a target
object, flows around it and conforms to its shape. Upon application of a vacuum
the granular material contracts and hardens quickly to pinch and hold the
object without requiring sensory feedback. We find that volume changes of less
than 0.5% suffice to grip objects reliably and hold them with forces exceeding
many times their weight. We show that the operating principle is the ability of
granular materials to transition between an unjammed, deformable state and a
jammed state with solid-like rigidity. We delineate three separate mechanisms,
friction, suction and interlocking, that contribute to the gripping force.
Using a simple model we relate each of them to the mechanical strength of the
jammed state. This opens up new possibilities for the design of simple, yet
highly adaptive systems that excel at fast gripping of complex objects.Comment: 10 pages, 7 figure
Phase transitions and memory effects in the dynamics of Boolean networks
The generating functional method is employed to investigate the synchronous
dynamics of Boolean networks, providing an exact result for the system dynamics
via a set of macroscopic order parameters. The topology of the networks studied
and its constituent Boolean functions represent the system's quenched disorder
and are sampled from a given distribution. The framework accommodates a variety
of topologies and Boolean function distributions and can be used to study both
the noisy and noiseless regimes; it enables one to calculate correlation
functions at different times that are inaccessible via commonly used
approximations. It is also used to determine conditions for the annealed
approximation to be valid, explore phases of the system under different levels
of noise and obtain results for models with strong memory effects, where
existing approximations break down. Links between BN and general Boolean
formulas are identified and common results to both system types are
highlighted
Emergence of equilibriumlike domains within nonequilibrium Ising spin systems
Many natural, technological and social systems are inherently not in equilibrium. We show, by detailed analysis of exemplar models, the emergence of equilibriumlike behavior in localized or nonlocalized domains within nonequilibrium Ising spin systems. Equilibrium domains are shown to emerge either abruptly or gradually depending on the system parameters and disappear, becoming indistinguishable from the remainder of the system for other parameter values
The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph
We use the cavity method to study parallel dynamics of disordered Ising
models on a graph. In particular, we derive a set of recursive equations in
single site probabilities of paths propagating along the edges of the graph.
These equations are analogous to the cavity equations for equilibrium models
and are exact on a tree. On graphs with exclusively directed edges we find an
exact expression for the stationary distribution of the spins. We present the
phase diagrams for an Ising model on an asymmetric Bethe lattice and for a
neural network with Hebbian interactions on an asymmetric scale-free graph. For
graphs with a nonzero fraction of symmetric edges the equations can be solved
for a finite number of time steps. Theoretical predictions are confirmed by
simulation results. Using a heuristic method, the cavity equations are extended
to a set of equations that determine the marginals of the stationary
distribution of Ising models on graphs with a nonzero fraction of symmetric
edges. The results of this method are discussed and compared with simulations
Noisy random Boolean formulae:a statistical physics perspective
Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field
Dynamical replica analysis of processes on finitely connected random graphs II: Dynamics in the Griffiths phase of the diluted Ising ferromagnet
We study the Glauber dynamics of Ising spin models with random bonds, on
finitely connected random graphs. We generalize a recent dynamical replica
theory with which to predict the evolution of the joint spin-field
distribution, to include random graphs with arbitrary degree distributions. The
theory is applied to Ising ferromagnets on randomly diluted Bethe lattices,
where we study the evolution of the magnetization and the internal energy. It
predicts a prominent slowing down of the flow in the Griffiths phase, it
suggests a further dynamical transition at lower temperatures within the
Griffiths phase, and it is verified quantitatively by the results of Monte
Carlo simulations.Comment: 30 pages, 4 figures, submitted to J.Phys.