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 $dgeq 2$. We also present evidence of coexistence of nematic and disordered phases from Monte Carlo simulations for rods of length $9$ on the square lattice.
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.
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 granu
lar 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.
We consider the steady states of a driven inelastic Maxwell gas consisting of two types of particles with scalar velocities. Motivated by experiments on bilayers where only one layer is driven, we focus on the case when only one of the two types of p
articles are driven externally, with the other species receiving energy only through inter-particle collision. The velocity $v$ of a particle that is driven is modified to $-r_w v+eta$, where $r_w$ parameterises the dissipation upon the driving and the noise $eta$ is taken from a fixed distribution. We characterize the statistics for small velocities by computing exactly the mean energies of the two species, based on the simplifying feature that the correlation functions are seen to form a closed set of equations. The asymptotic behaviour of the velocity distribution for large speeds is determined for both components through a combination of exact analysis for a range of parameters or obtained numerically to a high degree of accuracy from an analysis of the large moments of velocity. We show that the tails of the velocity distribution for both types of particles have similar behaviour, even though they are driven differently. For dissipative driving ($r_w<1$), the tails of the steady state velocity distribution show non-universal features and depend strongly on the noise distribution. On the other hand, the tails of the velocity distribution are exponential for diffusive driving ($r_w=1$) when the noise distribution decays faster than exponential.
We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and s
mall mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses), or non-local (collision between particles of widely different masses give the main contribution to the mass flux). We show that the non-local regime is further divided into two sub-regimes corresponding to weak and strong non-locality. We also observe that at the boundaries between the local and non-local regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some non-rigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the non-local case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the non-locality of the kernel.
Irreversible aggregation is an archetypal example of a system driven far from equilibrium by sources and sinks of a conserved quantity (mass). The source is a steady input of monomers and the evaporation of colliding particles with a small probabilit
y is the sink. Using exact and heuristic analyses, we find a universal regime and two distinct non-universal regimes distinguished by the relative importance of mergers between small and large particles. At the boundary between the regimes we find an analogue of the logarithmic correction conjectured by Kraichnan for two-dimensional turbulence.
We study pressurised self-avoiding ring polymers in two dimensions using Monte Carlo simulations, scaling arguments and Flory-type theories, through models which generalise the model of Leibler, Singh and Fisher [Phys. Rev. Lett. Vol. 59, 1989 (1987)
]. We demonstrate the existence of a thermodynamic phase transition at a non-zero scaled pressure $tilde{p}$, where $tilde{p} = Np/4pi$, with the number of monomers $N rightarrow infty$ and the pressure $p rightarrow 0$, keeping $tilde{p}$ constant, in a class of such models. This transition is driven by bond energetics and can be either continuous or discontinuous. It can be interpreted as a shape transition in which the ring polymer takes the shape, above the critical pressure, of a regular N-gon whose sides scale smoothly with pressure, while staying unfaceted below this critical pressure. In the general case, we argue that the transition is replaced by a sharp crossover. The area, however, scales with $N^2$ for all positive $p$ in all such models, consistent with earlier scaling theories.
We study the Bethe approximation for a system of long rigid rods of fixed length k, with only excluded volume interaction. For large enough k, this system undergoes an isotropic-nematic phase transition as a function of density of the rods. The Bethe
lattice, which is conventionally used to derive the self-consistent equations in the Bethe approximation, is not suitable for studying the hard-rods system, as it does not allow a dense packing of rods. We define a new lattice, called the random locally tree-like layered lattice, which allows a dense packing of rods, and for which the approximation is exact. We find that for a 4-coordinated lattice, k-mers with k>=4 undergo a continuous phase transition. For even coordination number q>=6, the transition exists only for k >= k_{min}(q), and is first order.
This paper gives a derivation for the large time asymptotics of the $n$-point density function of a system of coalescing Brownian motions on $bf{R}$.
