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We develop an action formalism to calculate probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels and establish a pathwise large deviation principle with total mass being the rate. As an application, the rate function for the number of surviving particles as well as the optimal evolution trajectory are calculated exactly for the constant, sum and product kernels. For the product kernel, we argue that the second derivative of the rate function has a discontinuity. The theoretical results agree with simulations tailored to the calculation of rare events.

Acoustic emission (AE) activity data resulting from the fracture processes of brittle materials is valuable real time information regarding the evolving state of damage in the material. Here, through a combined experimental and computational study we explore the possibility of utilising the statistical signatures of AE activity data for characterisation of disorder parameter in simulation of tensile fracture of epoxy based polymer. For simulations we use a square random spring network model with quasi-brittle spring behaviour and a normally distributed failure strain threshold. We show that the disorder characteristics while have marginal effect on the power law exponent of the avalanche size distribution, are strongly correlated with the waiting time interval between consecutive record breaking avalanches as well as the total number of records. This sensitivity to disorder is exploited in estimating the disorder parameter suitable for the experiments on tensile failure of epoxy based polymer. The disorder parameter is estimated assuming equivalence between the amplitude distribution of AE data and avalanche size distribution of the simulations. The chosen disorder parameter is shown to well reproduce the failure characteristics in terms of the peak load of the macroscopic response, the power-law behaviour with avalanche dominated fracture type as well as realistic fracture paths.

We study the role of record statistics of damage avalanches in predicting the fracture of a heterogeneous material under tensile loading. The material is modeled using a two-dimensional random spring network where disorder is introduced through randomness in the breakage threshold strains of the springs. It is shown that the waiting time between successive records of avalanches has a maximum for moderate disorder, thus showing an acceleration of records with impending fracture. Such a signature is absent for low disorder strength when the fracture is nucleation-dominated, and high disorder strength when the fracture is percolation type. We examine the correlation between the record with the maximum waiting time and the crossover record at which the avalanche statistics change from off-critical to critical. Compared to the avalanche based predictor for failure, we show that the record statistics have the advantage of both being real-time as well as able to predict final fracture at much smaller strains. We also show that in the avalanche-dominated regime, the failure strain is shown to have a linear relation with the strain at the maximum waiting time, making possible a quantitative prediction., Comment: 9 pages, 7 figures

Using extensive molecular dynamics simulations, we obtain the conformational phase diagram of a charged polymer in the presence of oppositely charged counterions and neutral attractive crowders for monovalent, divalent and trivalent counterion valencies. We demonstrate that the charged polymer can exist in three phases: (1) an extended phase for low charge densities and weak polymer-crowder attractive interactions ($CE$), (2) a collapsed phase for high charge densities and weak polymer-crowder attractive interactions, primarily driven by counterion condensation ($CCI$), and (3) a collapsed phase for strong polymer-crowder attractive interactions, irrespective of the charge density, driven by crowders acting as bridges or crosslinks ($CCB$). Importantly, the simulations reveal that the interaction with crowders can induce collapse, despite the presence of strong repulsive electrostatic interactions, and can replace condensed counterions to facilitate a direct transition from the $CCI$ and $CE$ phases to the $CCB$ phase., Comment: 12 pages, 10 figures

The hysteretic behavior exhibited by collagen fibrils, when subjected to cyclic loading, is known to result in both dissipation as well as accumulation of residual strain. On subsequent relaxation, partial recovery has also been reported. Cross-links have been considered to play a key role in overall mechanical properties. Here, we modify an existing coarse grained molecular dynamics model for collagen fibril with initially cross-linked collagen molecules, which is known to reproduce the response to uniaxial strain, by incorporating reformation of cross-links to allow for possible recovery of the fibril. Using molecular dynamics simulations, we show that our model successfully replicates the key features observed in experimental data, including the movement of hysteresis loops, the time evolution of residual strains and energy dissipation, as well as the recovery observed during relaxation. We also show that the characteristic cycle number, describing the approach towards steady state, has a value similar to that in experiments. We also emphasize the vital role of the degree of cross-linking on the key features of the macroscopic response to cyclic loading.

This article contains our comments and views on the status of the current understanding of phase transitions in systems with only hard-core interactions, based on our work in this area. The equation of state for the hard sphere gas in $d$-dimensions is discussed. The universal repulsive Lee-Yang singularities in the complex activity plane, and its relation to the directed and undirected polymer models are outlined. We also discuss orientationally disordered crystalline mesophases, and some of their models.

Mpemba effect refers to the counterintuitive phenomenon of a hotter system equilibrating faster than a colder system when both are quenched to the same low temperature. For a Brownian particle trapped in a piece-wise linear single well potential that is devoid of any other metastable minima we show the existence of Mpemba effect for a wide range of parameters through an exact solution. This result challenges the prevalent explanation of the Mpemba effect that requires the energy landscape to be rugged with multiple minima. We also demonstrate the existence of inverse and strong Mpemba effects., Comment: 5 pages, 4 figures

The Mpemba effect is a fingerprint of the anomalous relaxation phenomenon wherein an initially hotter system equilibrates faster than an initially colder system when both are quenched to the same low temperature. Experiments on a single colloidal particle trapped in a carefully shaped double well potential have demonstrated this effect recently [Nature 584, 64 (2020)]. In a similar vein, here, we consider a piece-wise linear double well potential that allows us to demonstrate the Mpemba effect using an exact analysis based on the spectral decomposition of the corresponding Fokker-Planck equation. We elucidate the role of the metastable states in the energy landscape as well as the initial population statistics of the particles in showcasing the Mpemba effect. Crucially, our findings indicate that neither the metastability nor the asymmetry in the potential is a necessary or a sufficient condition for the Mpemba effect to be observed., Comment: 14 pages, 9 figures, Included minor changes as compared to its previous version

Mpemba effect refers to the counterintuitive effect where a system which is initially further from the final steady state equilibrates faster than an identical system that is initially closer. The closeness to the final state is defined in terms of a distance measure. For driven granular systems, the Mpemba effect has been illustrated in terms of an ad-hoc measure of mean kinetic energy as the distance function. In this paper, by studying four different distance measures based on the mean kinetic energies as well as velocity distribution, we show that the Mpemba effect depends on the definition of the measures., Comment: 16 pages, 8 figures

We study ballistic aggregation on a two dimensional square lattice, where particles move ballistically in between momentum and mass conserving coalescing collisions. Three models are studied based on the shapes of the aggregates: in the first the aggregates remain point particles, in the second they retain the fractal shape at the time of collision, and in the third they assume a spherical shape. The exponents describing the power law temporal decay of number of particles and energy as well as dependence of velocity correlations on mass are determined using large scale Monte Carlo simulations. It is shown that the exponents are universal only for the point particle model. In the other two cases, the exponents are dependent on the initial number density and correlations vanish at high number densities. The fractal dimension for the second model is close to 1.49., Comment: 14 pages, 23 figures

Extensive coarse grained molecular dynamics simulations are performed to investigate the conformational phase diagram of a neutral polymer in the presence of attractive crowders. We show that, for low crowded densities, the polymer predominantly shows three phases as a function of both intra polymer and polymer-crowder interactions: (1) weak intra polymer and weak polymer-crowder attractive interactions induce extended or coil polymer conformations (phase E) (2) strong intra polymer and relatively weak polymer-crowder attractive interactions induce collapsed or globular conformations (phase CI) and (3) strong polymer-crowder attractive interactions, regardless of intra polymer interactions, induce a second collapsed or globular conformation that encloses bridging crowders (phase CB). The detailed phase diagram is obtained by determining the phase boundaries delineating the different phases based on an analysis of the radius of gyration as well as bridging crowders. The dependence of the phase diagram on strength of crowder-crowder attractive interactions and crowder density is clarified. We also show that when the crowder density is increased, a third collapsed phase of the polymer emerges for weak intra polymer attractive interactions. This crowder density induced compaction is shown to be enhanced by stronger crowder-crowder attraction and is different from the depletion induced collapse mechanism which is primarily driven by repulsive interactions. We also provide a unified explanation of the observed reentrant swollen/extended conformations of earlier simulations of weak and strongly self interacting polymers in terms of crowder-crowder attractive interactions.

We develop a biased Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels. Given a trajectory with a fixed number of collisions, the algorithm modifies both the waiting times between collisions, as well as the sequence of collisions, using local moves. We show that the algorithm is ergodic by giving a protocol that transforms an arbitrary trajectory to a standard trajectory using valid Monte Carlo moves. The algorithm can sample rare events with probabilities of the order of $10^{-40}$ and lower. The algorithm's effectiveness in sampling low-probability events is established by showing that the numerical results for the large deviation function of constant-kernel aggregation reproduce the exact results. It is shown that the algorithm can obtain the large deviation functions for other kernels, including gelling ones, as well as the instanton trajectories for atypical times. The dependence of the autocorrelation times, both temporal and configurational, on the different parameters of the algorithm is also characterized., Comment: 26 pages, 10 figures