Searcharxiv⌕ Search

arXiv subjects

R. Rajesh

Publications and source records attributed to R. Rajesh.

At least 109 records · Page 6Linked to original sources
arXiv2007

Constant flux relation for aggregation models with desorption and fragmentation

We study mass fluxes in aggregation models where mass transfer to large scales by aggregation occurs alongside desorption or fragmentation. Two models are considered. (1) A system of diffusing, aggregating particles with influx and outflux of particles (in-out model) (2) A system of diffusing aggregating particles with fragmentation (chipping model). Both these models can exist in phases where probability distributions are power laws. In these power law phases, we argue that the two point correlation function should have a certain homogeneity exponent. These arguments are based on the exact constant flux scaling valid for simple aggregation with input. Predictions are compared with Monte Carlo simulations.

cond-mat.stat-mech↗
arXiv2007

Violation of Porod law in a freely cooling granular gas in one dimension

We study a model of freely cooling inelastic granular gas in one dimension, with a restitution coefficient which approaches the elastic limit below a relative velocity scale v. While at early times (t << 1/v) the gas behaves as a completely inelastic sticky gas conforming to predictions of earlier studies, at late times (t >> 1/v) it exhibits a new fluctuation dominated phase ordering state. We find distinct scaling behavior for the (i) density distribution function, (ii) occupied and empty gap distribution functions, (iii) the density structure function and (iv) the velocity structure function, as compared to the completely inelastic sticky gas. The spatial structure functions (iii) and (iv) violate the Porod law. Within a mean-field approximation, the exponents describing the structure functions are related to those describing the spatial gap distribution functions.

cond-mat.stat-mech↗
arXiv2007

Constant flux relation for driven dissipative systems

Conservation laws constrain the stationary state statistics of driven dissipative systems because the average flux of a conserved quantity between driving and dissipation scales should be constant. This requirement leads to a universal scaling law for flux-measuring correlation functions, which generalizes the 4/5-th law of Navier-Stokes turbulence. We demonstrate the utility of this simple idea by deriving new exact scaling relations for models of aggregating particle systems in the fluctuation-dominated regime and for energy and wave action cascades in models of strong wave turbulence.

cond-mat.stat-mech↗
arXiv2005

Cluster-Cluster Aggregation as an Analogue of a Turbulent Cascade : Kolmogorov Phenomenology, Scaling Laws and the Breakdown of self-similarity

We present a detailed study of the statistics of a system of diffusing aggregating particles with a steady monomer source. We emphasise the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean-field description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterised by a constant flux of mass directed from small to large masses. It admits a phenomenological description based on the assumption of self-similarity and constant mass flux analogous to the Kolmogorov's 1941 theory of turbulence. Unlike turbulence, the aggregation problem is analytically tractable using powerful methods of statistical field theory. We explain in detail how these methods should be adapted to study the far-from-equilibrium, flux-dominated states characteristic of turbulent systems. We consider multipoint correlation functions of the mass density. By an exact evaluation of the scaling exponents for the one and two-point correlation functions, we show that the assumption of self-similiarity breaks down at large masses for spatial dimensions, $d\leq2$. We calculate non-rigourously the exponents of the higher order correlation functions as an $ε$-expansion where $ε=2-d$. We show that the mass distribution exhibits non-trivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The origin of this multiscaling is traced to strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in $d\leq2$.

cond-mat.stat-mech↗
arXiv2005

Multi-Scaling of Correlation Functions in Single Species Reaction-Diffusion Systems

We derive the multi-fractal scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system $A+A \to \emptyset$ in $d \leq 2$ and for the ternary system $3A \to \emptyset$ in $d=1$. For the binary reaction we find that the probability $P_{t}(N, ΔV)$ of finding $N$ particles in a fixed volume element $ΔV$ at time $t$ decays in the limit of large time as $(\frac{\ln t}{t})^{N}(\ln t)^{-\frac{N(N-1)}{2}}$ for $d=2$ and $t^{-Nd/2}t^{-\frac{N(N-1)ε}{4}+\mathcal{O}(\ep^2)}$ for $d<2$. Here $\ep=2-d$. For the ternary reaction in one dimension we find that $P_{t}(N,ΔV) \sim (\frac{\ln t}{t})^{N/2}(\ln t)^{-\frac{N(N-1)(N-2)}{6}}$. The principal tool of our study is the dynamical renormalization group. We compare predictions of $\ep$-expansions for $P_{t}(N,ΔV)$ for binary reaction in one dimension against exact known results. We conclude that the $\ep$-corrections of order two and higher are absent in the above answer for $P_{t}(N, ΔV)$ for $N=1,2,3,4$. Furthermore we conjecture the absence of $\ep^2$-corrections for all values of $N$.

cond-mat.stat-mech↗
arXiv2005

Cosmological mass functions and moving barrier models

In the ellipsoidal collapse model, the critical density for the collapse of a gravitationally bound object is a function of its mass. In the excursion set formalism, this translates into a moving barrier problem such that the mass function of dark matter haloes is given by the first crossing probability of a random walker across the barrier. In this paper, we study this first crossing probability analytically. Complete solutions are obtained for barriers that vary as square root of time and square of time. Large and small time asymptotic behaviour is derived. For arbitrary power-law barriers, the large time behaviour is determined. The solutions allow us to derive useful inferences about the scaling of the conditional mass function in terms of present day halo masses and look back redshifts. As an application of our results, we compare the estimates of major merger rates of haloes in constant and moving barrier models and find that for massive haloes ($10^{12-13} M_\odot$) the latter predicts significantly higher merger rates towards high redshifts ($z \gtrsim 4$)

astro-ph↗
arXiv2004

Breakdown of Kolmogorov scaling in models of cluster aggregation with deposition

The steady state of the model of cluster aggregation with deposition is characterized by a constant flux of mass directed from small masses towards large masses. It can therefore be studied using phenomenological theories of turbulence, such as Kolmogorov's 1941 theory. On the other hand, the large scale behavior of the aggregation model in dimensions lower than or equal to two is governed by a perturbative fixed point of the renormalization group flow, which enables an analytic study of the scaling properties of correlation functions in the steady state. In this paper, we show that the correlation functions have multifractal scaling, which violates linear Kolmogorov scaling. The analytical results are verified by Monte Carlo simulations.

cond-mat.stat-mech↗
arXiv2004

Stationary Kolmogorov Solutions of the Smoluchowski Aggregation Equation with a Source Term

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels,$K_ζ(m_1,m_2)=(m_1m_2)^{ζ/2}$ which includes both gelling and non-gelling cases, reproducing the known solution in the case $ζ=0$. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.

cond-mat.stat-mech↗
arXiv2004

Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers

We calculate the survival probability of a diffusing test particle in an environment of diffusing particles that undergo coagulation at rate lambda_c and annihilation at rate lambda_a. The test particle dies at rate lambda' on coming into contact with the other particles. The survival probability decays algebraically with time as t^{-theta}. The exponent theta in d<2 is calculated using the perturbative renormalization group formalism as an expansion in epsilon=2-d. It is shown to be universal, independent of lambda', and to depend only on delta, the ratio of the diffusion constant of test particles to that of the other particles, and on the ratio lambda_a/lambda_c. In two dimensions we calculate the logarithmic corrections to the power law decay of the survival probability. Surprisingly, the log corrections are non-universal. The one loop answer for theta in one dimension obtained by setting epsilon=1 is compared with existing exact solutions for special values of delta and lambda_a/lambda_c. The analytical results for the logarithmic corrections are verified by Monte Carlo simulations.

cond-mat.stat-mech↗
arXiv2004

Nonequilibrium phase transitions in models of adsorption and desorption

The nonequilibrium phase transition in a system of diffusing, coagulating particles in the presence of a steady input and evaporation of particles is studied. The system undergoes a transition from a phase in which the average number of particles is finite to one in which it grows linearly in time. The exponents characterizing the mass distribution near the critical point are calculated in all dimensions.

cond-mat.stat-mech↗
arXiv2003

Persistence properties of a system of coagulating and annihilating random walkers

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In 1-dimension, the system models the zero temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d < 2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) ~ m^(z/d) t^(-theta), with theta=d Q + Q(Q-1/2) epsilon + O(epsilon^2), z=(2Q-1) epsilon + (2 Q-1) (Q-1)(1/2+A Q) epsilon^2 +O(epsilon^3), where Q=(q-1)/q, epsilon =2-d and A =-0.006. M(t) is shown to scale as t^(d/2-delta), where delta = d (1 -Q)+(Q-1)(Q-1/2) epsilon+ O(epsilon^2). In two dimensions, we show that P(m,t) ~ ln(t)^(Q(3-2Q)) ln(m)^((2Q-1)^2) t^(-2Q) for m << t^(2 Q-1). The 1-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.

cond-mat.stat-mech↗
arXiv2003

Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.

cond-mat.stat-mech↗
arXiv2003

Convex lattice polygons of fixed area with perimeter dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight t^m to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s^{-theta_{conv}} exp[K s^(1/2)] for large s and t less than a critical threshold t_c, where K is a t-dependent constant, and theta_{conv} is a critical exponent which does not change with t. We find theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected non-universality of theta_{conv} is traced to existence of sharp corners in the asymptotic shape of these polygons.

cond-mat.stat-mech↗
arXiv2002

Kang-Redner Anomaly in Cluster-Cluster Aggregation

The large time, small mass, asymptotic behavior of the average mass distribution $\pb$ is studied in a $d$-dimensional system of diffusing aggregating particles for $1\leq d \leq 2$. By means of both a renormalization group computation as well as a direct re-summation of leading terms in the small reaction-rate expansion of the average mass distribution, it is shown that $\pb \sim \frac{1}{t^d} (\frac{m^{1/d}}{\sqrt{t}})^{e_{KR}}$ for $m \ll t^{d/2}$, where $e_{KR}=ε+O(ε^2)$ and $ε=2-d$. In two dimensions, it is shown that $\pb \sim \frac{\ln(m) \ln(t)}{t^2}$ for $ m \ll t/ \ln(t)$. Numerical simulations in two dimensions supporting the analytical results are also presented.

cond-mat.stat-mech↗
arXiv2002

Aggregate formation in a system of coagulating and fragmenting particles with mass-dependent diffusion rates

The effect of introducing a mass dependent diffusion rate ~ m^{-alpha} in a model of coagulation with single-particle break up is studied both analytically and numerically. The model with alpha=0 is known to undergo a nonequilibrium phase transition as the mass density in the system is varied from a phase with an exponential distribution of mass to a phase with a power-law distribution of masses in addition to a single infinite aggregate. This transition is shown to be curbed, at finite densities, for all alpha > 0 in any dimension. However, a signature of this transition is seen in finite systems in the form of a large aggregate and the finite size scaling implications of this are characterized. The exponents characterizing the steady state probability that a randomly chosen site has mass m are calculated using scaling arguments. The full probability distribution is obtained within a mean field approximation and found to compare well with the results from numerical simulations in one dimension.

cond-mat.stat-mech↗
arXiv2002

Effect of spatial bias on the nonequilibrium phase transition in a system of coagulating and fragmenting particles

We examine the effect of spatial bias on a nonequilibrium system in which masses on a lattice evolve through the elementary moves of diffusion, coagulation and fragmentation. When there is no preferred directionality in the motion of the masses, the model is known to exhibit a nonequilibrium phase transition between two different types of steady states, in all dimensions. We show analytically that introducing a preferred direction in the motion of the masses inhibits the occurrence of the phase transition in one dimension, in the thermodynamic limit. A finite size system, however, continues to show a signature of the original transition, and we characterize the finite size scaling implications of this. Our analysis is supported by numerical simulations. In two dimensions, bias is shown to be irrelevant.

cond-mat.stat-mech↗
arXiv2001

The Adsorption and Collapse Transitions in a Linear Polymer Chain near an Attractive Wall

We deduce the qualitative phase diagram of a long flexible neutral polymer chain immersed in a poor solvent near an attracting surface using phenomenological arguments. The actual positions of the phase boundaries are estimated numerically from series expansion up to 19 sites of a self-attracting self avoiding walk in three dimensions. In two dimensions, we calculate analytically phase boundaries in some cases for a partially directed model. Both the numerical as well as analytical results corroborate the proposed qualitative phase diagram.

cond-mat.stat-mech↗