Scaling to zero of compressive modulus in disordered isostatic cubic networks.

Networks with as many mechanical constraints as degrees of freedom and no redundant constraints are minimally rigid or isostatic. Isostatic networks are relevant in the study of network glasses, soft matter, and sphere packings. Because of being at the verge of mechanical collapse, they have anomalous elastic and dynamical properties not found in the more commonly occurring hyperstatic networks. In particular, while hyperstatic networks are only slightly affected by geometric disorder, the elastic properties of isostatic networks are dramatically altered by it. In this paper, we show how disorder and system size strongly affect the ability of isostatic networks to sustain a compressive load. We develop an analytic method to calculate the bulk compressive modulus B for various boundary conditions as a function of disorder strength and system size. For simplicity, we consider square and cubic lattices with L^{d} sites, each having d mechanical degrees of freedom, and dL^{d} rotatable springs in the presence of hot-solid disorder of magnitude ε. Additionally, ∼L^{θ} sites may be fixed, thus introducing a nonextensive number of redundancies, either in the bulk or on the boundaries of the system. In all cases, B is analytically and numerically shown to decay as L^{-µ} with µ_{large}=d-θ for large disorder and µ_{small}=max{(d-θ-1),0} for small disorder. Furthermore B(L,ε)L^{µ_{small}}=g(λ) with λ=L^{(µ_{large}-µ_{small})}ε^{2} a scaling variable such that λ<<1 is small disorder and λ>1 is large disorder. The faster decay to zero of B in the large disorder regime results from a broad distribution of spring tensions, including tensions of both signs in equal proportions, which is remarkable since the system is under a purely compressive load. Notably, the bulk modulus is discontinuous at ε=0, a consequence of the fact that the regular network sits at an unstable degenerate configuration.

Phase transition in liquid drop fragmentation.

A liquid droplet is fragmented by a sudden pressurized-gas blow, and the resulting droplets, adhered to the window of a flatbed scanner, are counted and sized by computerized means. The use of a scanner plus image recognition software enables us to automatically count and size up to tens of thousands of tiny droplets with a smallest detectable volume of approximately 0.02 nl . Upon varying the gas pressure, a critical value is found where the size distribution becomes a pure power law, a fact that is indicative of a phase transition. Away from this transition, the resulting size distributions are well described by Fisher's model at coexistence. It is found that the sign of the surface correction term changes sign, and the apparent power-law exponent tau has a steep minimum, at criticality, as previously reported in nuclear multifragmentation studies. We argue that the observed transition is not percolative, and introduce the concept of dominance in order to characterize it. The dominance probability is found to go to zero sharply at the transition. Simple arguments suggest that the correlation length exponent is nu=1/2 . The sizes of the largest and average fragments, on the other hand, do not go to zero abruptly but behave in a way that appears to be consistent with recent predictions of Ashurst and Holian.

Culling avalanches in bootstrap percolation.

We study the culling avalanches which occur after the "death" of a single randomly chosen site in a network where sites are unstable, and are culled, if they have coordination less than an integer parameter m. Avalanche distributions are presented for triangular and cubic lattices for values of m where the associated bootstrap transitions are either first or second order. In second order cases, the culling avalanche distribution is found to be exponential, while in first order cases it follows a power law. We present an exact relation between culling avalanches and conventional bootstrap percolation and show that a relation proposed by Manna [Physica A 261, 351 (1998)] can be a good approximation for strongly first order bootstrap transitions but not for continuous bootstrap transitions.

Rigidity percolation in a field.

Rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted Erdös-Renyi graphs, with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n(F) of uncanceled degrees of freedom and gamma(r) of redundant bonds. Upon crossing the coexistence line, gamma(r) and n(F) are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely, that the density of uncanceled degrees of freedom is a "free energy" for rigidity percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1) core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma(c), Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1) rigid core, i.e., the average coordination of these subgraphs is exactly 2g, although gamma(c), the average coordination of the whole system, is smaller than 2g. gamma(c) is found to converge to 2g for large g, i.e., in this limit Maxwell counting is exact globally as well.

Transport on percolation clusters with power-law distributed bond strengths.

The simplest transport problem, namely finding the maximum flow of current, or maxflow, is investigated on critical percolation clusters in two and three dimensions, using a combination of extremal statistics arguments and exact numerical computations, for power-law distributed bond strengths of the type P(sigma) approximately sigma(-alpha). Assuming that only cutting bonds determine the flow, the maxflow critical exponent v is found to be v(alpha)=(d-1)nu+1/(1-alpha). This prediction is confirmed with excellent accuracy using large-scale numerical simulation in two and three dimensions. However, in the region of anomalous bond capacity distributions (0< or =alpha< or =1) we demonstrate that, due to cluster-structure fluctuations, it is not the cutting bonds but the blobs that set the transport properties of the backbone. This "blob dominance" avoids a crossover to a regime where structural details, the distribution of the number of red or cutting bonds, would set the scaling. The restored scaling exponents, however, still follow the simplistic red bond estimate. This is argued to be due to the existence of a hierarchy of so-called minimum cut configurations, for which cutting bonds form the lowest level, and whose transport properties scale all in the same way. We point out the relevance of our findings to other scalar transport problems (i.e., conductivity).

Two rigidity-percolation transitions on binary Bethe networks and the intermediate phase in glass.

Rigidity percolation is studied analytically on randomly bonded networks with two types of nodes, respectively, with coordination numbers z(1) and z(2), and with g(1) and g(2) degrees of freedom each. For certain cases that model chalcogenide glass networks, two transitions, both of first order, are found, with the first transition usually rather weak. The ensuing intermediate pase, although not isostatic in its entirety, has very low self-stress. Our results suggest a possible mechanism for the appearance of intermediate phases in glass that does not depend on a self-organization principle.