arXiv subjects

Prabodh Shukla

Publications and source records attributed to Prabodh Shukla.

At least 19 recordsLinked to original sources

arXiv2026

Note on Boltzmann's H-Theorem and Detailed Balance Dynamics

The Boltzmann H-theorem states that entropy of an ensemble of thermodynamic states increases until all states become equally probable. The evolution randomly picks one of the states in the ensemble and brings every other state to have the same probability as the selected state. Different realizations of evolution yield a distribution of selected states. We study this distribution numerically and discuss its relevance for equilibrium fluctuations in thermal as well as non thermal systems.

cond-mat.stat-mech↗

arXiv2023

Equilibrium via multi-spin-flip Glauber dynamics in Ising Model

Notwithstanding great strides that statistical mechanics has made in recent decades, an analytic solution of arguably the simplest model of relaxation dynamics, the Ising model in an applied external field remains elusive even in $1d$. Extant studies are based on numerics using single-spin-flip Glauber dynamics. There is no reason why this algorithm should lead to the global minimum energy state of the system. With this in mind, we explore multi-spin-flip parallel and sequential Glauber dynamics of Ising spins in $1d$ and also on a regular random graph of coordination number $z=3$. We view our study as a small initial step to test the generally implied hypothesis that the equilibrium is independent of the relaxational dynamics or if it carries some signature of it.

cond-mat.stat-mech↗

arXiv2019

Surprising variants of Cauchy's formula for mean chord length

We examine isotropic and anisotropic random walks which begin on the surface of linear ($N$), square ($N \times N$), or cubic ($N \times N \times N$) lattices and end upon encountering the surface again. The mean length of walks is equal to $N$ and the distribution of lengths $n$ generally scales as $n^{-1.5}$ for large $n$. Our results are interesting in the context of an old formula due to Cauchy that the mean length of a chord though a convex body of volume $V$ and surface $S$ is proportional to $V/S$. It has been realized in recent years that Cauchy's formula holds surprisingly even if chords are replaced by irregular insect paths or trajectories of colliding gas molecules. The random walk on a lattice offers a simple and transparent understanding of this result in comparison to other formulations based on Boltzmann's transport equation in continuum.

cond-mat.stat-mech↗

arXiv2019

Dependence of persistence exponent on initial state

We examine persistence in one dimensional Ising model under zero temperature Glauber dynamics for random initial states with unequal fraction of up and down spins. We find the persistence exponent varies continuously with the fraction of up spins in the initial state. Apparently this feature has been overlooked in the studies so far.

cond-mat.stat-mech↗

arXiv2019

Critical hysteresis on dilute triangular lattice

Critical hysteresis in the zero-temperature random-field Ising model on a two-dimensional triangular lattice has been studied earlier with site dilution on one sublattice. It was reported that criticality vanishes if less than one third of the sublattice is occupied. This appears at variance with recently obtained exact solutions of the model on dilute Bethe lattices and prompts us to revisit the problem using an alternate numerical method. Contrary to our speculation that criticality may not be exactly zero below one third dilution, the present study indicates it is nearly zero if approximately less than two-thirds of the sublattice is occupied. This suggests that hysteresis on dilute periodic lattices is qualitatively different from that on dilute Bethe lattices. Possible reasons are discussed briefly.

cond-mat.stat-mech↗

arXiv2018

Hysteresis in the zero-temperature random field Ising model on directed random graphs

We use zero-temperature Glauber dynamics to study hysteresis in the random-field Ising model on directed random graphs. The critical behavior of the model depends on the connectivity $z$ of the graph rather differently from that on undirected graphs. Directed graphs and zero-temperature dynamics are relevant to a wide class of social phenomena including opinion dynamics. We discuss the efficacy of increasing external influence in inducing a first-order phase transition in opinion dynamics. The numerical results are supported by an analytic solution of the model.

cond-mat.stat-mech↗

arXiv2018

Hysteresis in the Ising model with Glauber dynamics

We use Glauber dynamics to study frequency and temperature dependence of hysteresis loops in the pure (without quenched disorder) Ising model on cubic, square, honeycomb lattices and random graphs. Results are discussed in the context of more extensive studies of hysteresis in the random field Ising model.

cond-mat.stat-mech↗

arXiv2017

Statistical Mechanics of Avalanches

Statistical mechanics of infinite avalanches is studied in the framework of nonequilibrium random-field Ising model. Critical behavior of the model on a random graph (dilute Bethe lattice) is analyzed in detail. We show that sites with a minimum coordination number 4 play a key role in the occurrence of infinite avalanches. Earlier results which did not seem to fit together very well are explained.

cond-mat.stat-mech↗

arXiv2015

Hysteresis in Random-field Ising model on a Bethe lattice with a mixed coordination number

We study zero-temperature hysteresis in the random-field Ising model on a Bethe lattice where a fraction $c$ of the sites have coordination number $z=4$ while the remaining fraction $1-c$ have $z=3$. Numerical simulations as well as probabilistic methods are used to show the existence of critical hysteresis for all values of $c > 0$. This extends earlier results for $c=0$ and $c=1$ to the entire range $0 \le c \le 1$, and provides new insight in non-equilibrium critical phenomena.

cond-mat.stat-mech↗

arXiv2015

Nonequilibrium random-field Ising model on a diluted triangular lattice

We study critical hysteresis in the random-field Ising model (RFIM) on a two-dimensional periodic lattice with a variable coordination number $z_{eff}$ in the range $3 \le z_{eff} \le 6$. We find that the model supports critical behavior in the range $4 < z_{eff} \le 6$, but the critical exponents are independent of $z_{eff}$. The result is discussed in the context of the universality of nonequilibrium critical phenomena and extant results in the field.

cond-mat.stat-mech↗

arXiv2013

Effect of Coordination Number on Nonequilibrium Critical Point

We study the nonequilibrium critical point of the zero temperature random field Ising model on a triangular lattice and compare it with known results on honeycomb, square, and simple cubic lattices. We suggest that the coordination number of the lattice rather than its dimension plays the key role in determining the universality class of the nonequilibrium critical behavior. This is discussed in the context of numerical evidence that equilibrium and nonequilibrium critical points of the zero-temperature random field Ising model belong to the same universality class. The physics of this curious result is not fully understood.

cond-mat.stat-mech↗

arXiv2011

Analysis of wasp-waisted hysteresis loops in magnetic rocks

The random-field Ising model of hysteresis is generalized to dilute magnets and solved on a Bethe lattice. Exact expressions for the major and minor hysteresis loops are obtained. In the strongly dilute limit the model provides a simple and useful understanding of the shapes of hysteresis loops in magnetic rock samples.

cond-mat.dis-nn↗

arXiv2011

Hysteresis in Anti-Ferromagnetic Random-Field Ising Model at Zero Temperature

We study hysteresis in anti-ferromagnetic random-field Ising model at zero temperature. The external field is cycled adiabatically between -$\infty$ and $\infty$. Two different distributions of the random-field are considered, (i) a uniform distribution of width $2Δ$ centered at the origin, and (ii) a Gaussian distribution with average value zero and standard deviation $σ$. In each case the hysteresis loop is determined exactly in one dimension and compared with numerical simulations of the model.

cond-mat.dis-nn↗

arXiv2011

Critical Hysteresis in Random Field XY and Heisenberg Models

We study zero-temperature hysteresis in random-field XY and Heisenberg models in the zero-frequency limit of a cyclic driving field. We consider three distributions of the random field and present exact solutions in the mean field limit. The results show a strong effect of the form of disorder on critical hysteresis as well as the shape of hysteresis loops. A discrepancy with an earlier study based on the renormalization group is resolved.

cond-mat.dis-nn↗

arXiv2010

Hysteresis in Random Field XY and Heisenberg Models: Mean Field Theory and Simulations at Zero Temperature

We examine zero temperature hysteresis in random field XY and Heisenberg models in the zero frequency limit of a cyclic driving field. Exact expressions for hysteresis loops are obtained in the mean field approximation. These show rather unusual features. We also perform simulations of the two models on a simple cubic lattice and compare them with the predictions of the mean field theory.

cond-mat.stat-mech↗

arXiv2009

Dynamics of bootstrap percolation

Bootstrap percolation transition may be first order or second order, or it may have a mixed character where a first order drop in the order parameter is preceded by critical fluctuations. Recent studies have indicated that the mixed transition is characterized by power law avalanches, while the continuous transition is characterized by truncated avalanches in a related sequential bootstrap process. We explain this behavior on the basis of a through analytical and numerical study of the avalanche distributions on a Bethe lattice.

cond-mat.stat-mech↗

arXiv2006

The magnetization-driven random field Ising model at T=0

We study the hysteretic evolution of the random field Ising model (RFIM) at T=0 when the magnetization M is controlled externally and the magnetic field H becomes the output variable. The dynamics is a simple modification of the single-spin-flip dynamics used in the H-driven situation and consists in flipping successively the spins with the largest local field. This allows to perform a detailed comparison between the microscopic trajectories followed by the system with the two protocols. Simulations are performed on random graphs with connectivity z=4 (Bethe lattice) and on the 3-D cubic lattice. The same internal energy U(M)is found with the two protocols when there is no macroscopic avalanche and it does not depend on whether the microscopic states are stable or not. On the Bethe lattice, the energy inside the macroscopic avalanche also coincides with the one that is computed analytically with the H-driven algorithm along the unstable branch of the hysteresis loop. The output field, defined here as dU/dM, exhibits very large fluctuations with the magnetization and is not self-averaging. Relation to the experimental situation is discussed.

cond-mat.dis-nn↗

arXiv2005

Zero Temperature Hysteresis in Random Field Ising Model on Bethe Lattices: approach to mean field behavior with increasing coordination number z

We consider the analytic solution of the zero temperature hysteresis in the random field Ising model on a Bethe lattice of coordination number $z$, and study how it approaches the mean field solution in the limit z-> \infty. New analytical results concerning the energy of the system along the hysteresis loop and first order reversal curves (FORC diagrams) are also presented.

cond-mat.dis-nn↗