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Hard core lattice gas models are minimal models to study entropy driven phase transitions. In the $k$-NN lattice gas, a particle excludes all sites upto the $k$-th next-nearest neighbors from being occupied by another particle. As $k$ increases from one, it extrapolates from nearest neighbor exclusion to the hard sphere gas. In this paper, we study the model on the triangular lattice for $k\leq 7$ using a flat histogram algorithm that includes cluster moves. Earlier studies had focused on $k\leq 3$. We show that for $4\leq k\leq 7$, the system undergoes a single phase transition from a low-density fluid phase to a high-density sublattice-ordered phase. Using partition function zeros and non-convexity properties of the entropy, we show that the transitions are discontinuous. The critical chemical potential, coexistence densities, and critical pressure are determined accurately., Comment: 14 pages and 20 figures in main paper. 5 pages and 4 figures in supplemental material

Collagen fibrils, when subjected to cyclic loading, are known to exhibit hysteretic behaviour with energy dissipation that is partially recovered on relaxation. In this paper, we develop a kinetic model for a collagen fibril incorporating presence of hidden loops and stochastic fragmentation as well as reformation of sacrificial bonds. We show that the model reproduces well the characteristic features of reported experimental data on cyclic response of collagen fibrils, such as moving hysteresis loops, time evolution of residual strains and energy dissipation, recovery on relaxation, etc. We show that the approach to the steady state is controlled by a characteristic cycle number for both residual strain as well as energy dissipation, and is in good agreement with reported existing experimental data., Comment: 12 figures

To update its teaching and departmental review processes, Phillips Academy Andover, an independent boarding school in Massachusetts, commissioned RAND Corporation researchers to review the school's policies, conduct a literature review, interview other independent high schools, develop potential recommendations for revision of Andover's processes, and revise those recommendations after vetting them with Andover's faculty. This report summarizes their work. The report is intended primarily for Andover administrators and teachers to guide their revision of their teaching and departmental review processes, but it also provides relevant research that independent schools can use to inform their design of teaching and departmental review policies. To help readers understand how proposed changes may or may not work in different contexts, the report includes Andover teachers' reactions to initial potential recommendations, the authors' subsequent revisions to the recommendations based on teacher's feedback, and Andover administrators' reactions to the authors' final recommendations. [This study was funded by Phillips Academy Andover.]

The exact solution of the Euler equation, which describes the time evolution of a blast wave created by an intense explosion, is a classic problem in gas dynamics. However, it has been found that the analytical results do not match with results from molecular dynamics simulation of hard spheres in two and three dimensions. In this paper, we show that the mismatch between theory and simulations can be resolved by considering the Navier Stokes equation. From the direct numerical simulation of the Navier Stokes equation in two and three dimensions, we show that the inclusion of heat conduction and viscosity terms is essential to capture the results from molecular dynamics simulations., Comment: 13 pages, 7 figures

A system of hard rigid rods of length $k$ on hypercubic lattices is known to undergo two phases transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. In this paper, we argue that, for large $k$, the second phase transition is a first order transition with a discontinuity in density in all dimensions greater than $1$. We show the chemical potential at the transition is $\approx k \ln [k /\ln k]$ for large $k$, and that the density of uncovered sites drops from a value $ \approx (\ln k)/k^2$ to a value of order $\exp(-ak)$, where $a$ is some constant, across the transition. We conjecture that these results are asymptotically exact, in all dimensions $d\geq 2$. We also present evidence of coexistence of nematic and disordered phases from Monte Carlo simulations for rods of length $9$ on the square lattice., Comment: 14 pages, 19 figures

We obtain the phase diagram of fully-packed hard plates on a cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities $s_\mu$ for `$\mu$ plates', whose normal is the $\mu$ direction ($\mu = x,y,z$). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities $s_{\rm \mu_1}$ and $s_{\mu_2}$ are comparable, and the third fugacity $s_{\mu_{3}}$ is much smaller, we find a spontaneously-layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the $\mu_3$ axis. $\mu_1$ and $\mu_2$ plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. In the opposite limit, with $\mu_3 \gg \mu_1 \sim \mu_2$, we find a phase with long-range columnar order, corresponding to simultaneous $Z_2$ symmetry breaking of lattice translation symmetry in directions $\mu_1$ and $\mu_2$. The spontaneously-layered phases display critical behaviour, with power-law decay of correlations in the $\mu_1$ and $\mu_2$ directions when the slabs are stacked in the $\mu_3$ direction, and represent examples of `floating phases' discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behaviour to the constrained motion of defects in this phase, and develop a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase., Comment: Clarifications and some details (including two new figures) added

We study the phase diagram of a system of $2\times 2\times 1$ hard plates on the three dimensional cubic lattice, {\em i.e.} a lattice gas of plates that each cover an elementary plaquette of the cubic lattice and occupy its four vertices, with the constraint that no two plates occupy the same site of the cubic lattice. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two density-driven phase transitions with increasing density of plates: the first from a disordered fluid to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen cartesian direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. Additionally the symmetry between the three types of plates is spontaneously broken, as plates with normal along the layering direction have a lower density than the other two types of plates. Intriguingly, the occupied slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, inter-slab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is two-fold ($Z_2$) breaking of lattice translation symmetry along all three cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with the three-dimensional $O(3)$ universality class, while the layered to sublattice transition is first-order in nature., Comment: incorporates some corrections and several clarifications

We obtain the phase diagram of the hard core lattice gas with third nearest neighbor exclusion on the triangular lattice using Monte Carlo simulations that are based on a rejection-free flat histogram algorithm. In a recent paper [J. Chem. Phys. 151, 104702 (2019)], it was claimed that the lattice gas with third nearest neighbor exclusion undergoes two phase transitions with increasing density, with the phase at intermediate densities exhibiting hexatic order with continuously varying exponents. Though a hexatic phase is expected when the exclusion range is large, it has not been seen earlier in hard core lattice gases with short range exclusion. In this paper, by numerically determining the entropies for all densities, we show unambiguously that there is only a single phase transition in the system between a low-density fluid phase and a high-density ordered sublattice phase, and that a hexatic phase is absent. The transition is shown to be first order in nature and the critical parameters are determined accurately., Comment: 11 pages, 5 figures, 63 references

We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hard-core lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are excluded from being occupied. It is shown that the algorithm is able to reproduce the known results for k = 1,2,3 both on the square and cubic lattices. We also show that, in comparison, the corresponding flat histogram algorithms with either local moves or unbiased cluster moves are less accurate and do not converge as the system size increases., Comment: 19 Pages, 25 figures

Through an exact analysis, we show the existence of Mpemba effect in an anisotropically driven inelastic Maxwell gas, a simplified model for granular gases, in two dimensions. Mpemba effect refers to the couterintuitive phenomenon of a hotter system relaxing to the steady state faster than a cooler system, when both are quenched to the same lower temperature. The Mpemba effect has been illustrated in earlier studies on isotropically driven granular gases, but its existence requires non-stationary initial states, limiting experimental realisation. In this paper, we demonstrate the existence of the Mpemba effect in anisotropically driven granular gases even when the initial states are non-equilibrium steady states. The precise conditions for the Mpemba effect, its inverse, and the stronger version, where the hotter system cools exponentially faster are derived., Comment: 18 pages, 4 figures. arXiv admin note: text overlap with arXiv:2004.11559

We demonstrate the existence, as well as determine the conditions, of a Mpemba effect - a counterintuitive phenomenon where a hotter system equilibrates faster than a cooler system when quenched to a cold temperature - in anisotropically driven granular gases. In contrast to earlier studies of Mpemba effect in granular systems, the initial states are stationary, making it a suitable system to experimentally study the effect. Our theoretical predictions for the regular, inverse and strong Mpemba effects agree well with results of event-driven molecular dynamics simulations of hard discs., Comment: 5 pages, 4 figures and supplemental material

Multi-phase materials, such as composite materials, exhibit multiple competing failure mechanisms during the growth of a macroscopic defect. For the simulation of the overall fracture process in such materials, we develop a two-phase spring network model that accounts for the architecture between the different components as well as the respective disorders in their failure characteristics. In the specific case of a plain weave architecture, we show that any offset between the layers reduces the delocalization of the stresses at the crack tip and thereby substantially lowers the strength and fracture toughness of the overall laminate. The avalanche statistics of the broken springs do not show a distinguishable dependence on the offsets between layers. The power law exponents are found to be much smaller than that of disordered spring network models in the absence of a crack. A discussion is developed on the possibility of the avalanche statistics being those near breakdown., Comment: 13 pages, 22 figures

We determine the asymptotic behavior of the entropy of full coverings of a $L \times M$ square lattice by rods of size $k\times 1$ and $1\times k$, in the limit of large $k$. We show that full coverage is possible only if at least one of $L$ and $M$ is a multiple of $k$, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a $k \times k$ square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large $k$, we show that the entropy per site $S_2(k)$ tends to $ A k^{-2} \ln k$, with $A=1$. We conjecture, based on a perturbative series expansion, that this large-$k$ behavior of entropy per site is super-universal and continues to hold on all $d$-dimensional hyper-cubic lattices, with $d \geq 2$., Comment: 13 pages, 12 figures