arXiv subjects

K. Razi Naqvi

Publications and source records attributed to K. Razi Naqvi.

10 recordsLinked to original sources

arXiv2024

Confusion concerning the extrapolated endpoint. When will it ever end?

In a paper on ``the Brownian motion analog of the well-known Milne problem in radiative transfer theory'' [\textit{J Stat Phys} 25 (1981) 569--82], Burschka and Titulaer reported: ``The value we find for this `Milne extrapolation length' is, in the appropriate dimensionless units, approximately twice the value found in the radiative transfer problem.'' A study by Ziff [\textit{J Stat Phys} 65 (1991) 1217--33], concerned with the absorption of particles executing a Rayleigh flight (randomly directed displacements of equal length $l$) by a black sphere of radius $R$, led to a value for the extrapolation length $γ$ about half as small as the benchmark result (for $R\gg l)$. The first discrepancy is shown to result from the disparity of the two length scales; the second, from the zero variance of the jump lengths. Ziff's finding that $γ$ is independent of $R$ for $0<l\leq 2R$, cannot be reconciled with studies based on the Lorentz-Boltzmann equation and the Klein-Kramers equation.

cond-mat.stat-mech↗

arXiv2024

100+ years of colossal confusion on colloidal coagulation. Part I: Smoluchowski's work on absorbing boundaries

A report by Brillouin (from Perrin's laboratory) on the rate of adsorption of `granules' to a glass plate [\textit{Ann. Chim. Phys.} 27 (1912) 412--23] prompted Marian von Smoluchowski (MvS) to interpret the data in terms of his newly developed theory of restricted Brownian motion. Placing an adsorbing wall at $x=0$, he modelled the particle concentration $n(x,t)$ as that solution of the diffusion equation which vanished at the wall, a boundary condition (BC) hereafter called SBC. A gaping discrepancy between his theory and Brillouin's data elicited a suggestion from MvS (that a particle might not adhere to the wall on every impact), but no further action -- other than that of applying his theory to spherically symmtric systems. In a paper written before, but published shortly after MvS's untimely death [\textit{Proc. Roy. Acad. Amst.} 20 (1918) 642--58], H. C. Burger erected a new and sturdier framework, which led him to an alternative BC, $D(\partial n/\partial x)_{x=0}=\varkappa n(0,t)$, applicable to a surface with an arbitrary absorption probability ($1\leq\varepsilon\leq 0$); a fallacy (that subsequently claimed more victims, including the present author) prevented him from deducing the correct expression for $\varkappa$. Burger's approach became ``The Road Not Taken'', while the SBC became the cornerstone of colloidal coagulation and bimoleculer reaction kinetics. Burger's approach (but not the ABC) was partly rediscovered by Kolmogorov, and used by Sveshnikov and Fuchs.The emended version of Burger's BC is shown here to coincide with that deduced from the Klein-Kramers equation [\textit{Phys. Rev. Lett.} \textbf{49} (1982) 304--07; \textit{J. Chem. Phys.} \textbf{78} (1983) 2710--12] and the Lorentz model of random flights [\textit{J. Phys. Chem.} \textbf{86} (1982) 4750--56].

cond-mat.stat-mech↗

arXiv2021

Translation of Newton's \textit{Principia} into Arabic Under the Aegis of the East India Company: A Rumour Turning into a Myth?

Tafazzul Husain Khan (1727?--1800?), who began his career in the court of Awadh, spent the last two decades of his life as a trusted ally of the East India Company. What set him apart from other court officials was not so much his erudition, political acumen and negotiating prowess, as his favourite pastime: delving into mathematics and astronomy. Contact with the Company personnel, some of whom were conversant with oriental languages and/or contemporary scientific advances, provided him with the opportunity to brush up his mathematical knowledge, and induced him to embark upon -- and, according to some, bring to fruition -- the task of translating a few important mathematical treatises, among them Newton's \textit{Principia}. According to Campbell, the author of an obituary notice (published in 1804), "he translated the Principia from the original Latin, into Arabic". The evidence gathered by Campbell is examined, and found insufficient to warrant this astounding and oft-repeated claim. Of the three tracts authored by Tafazzul (all published posthumously in abridged versions), none can be described as a translation of Newton's \textit{Principia}. Until the emergence of some tangible evidence, any talk of his translations of the \textit{Principia} and other western treatises can only be characterised as rumour, a process in which recall is often accompanied by distortion.

physics.hist-ph↗

arXiv2017

A Superior but Equally Convenient Alternative to the Steady-State Approximation and Secular Equilibrium

The steady-state approximation (hereafter abbreviated as SSA) consists in setting $dy/dt=0$, where $y$ denotes the concentration of a short-lived intermediate subject to first-order decay with a rate constant $k$. The sole reason for enforcing SSA is to convert the rate equation for $y$ into an algebraic equation. The conditions under which SSA becomes trustworthy are now well understood, but a firm grasp of the physical content of the approximation requires more maturity than few teachers, let alone their students, may be expected to possess. Furthermore, there is no simple way to gauge the accuracy of the results derived by imposing SSA. The purpose of this note is to demonstrate that a better, but equally simple, approximation results if, instead of setting $dy/dt$ to zero, one substitutes $y(t+τ)$ for $y+τdy/dt$, where $τ=1/k$; SSA is a cruder approximation because it neglects the second term. For systems modelled as damped harmonic oscillators, the "reverse Taylor approximation" can be extended by retaining one more term in the Taylor expansion. The utility of the approximation (or its extension) is demonstrated by examining the following systems: radioactive equilibria, Brownian motion, dynamic response of linear first- and second-order systems.

physics.gen-ph↗

arXiv2015

Extrapolating from a homologous series of oligomers to the infinite-mer: it's a long long way to infinity

The usual strategy for deducing the $π\mbox{--}π^\ast$ electronic energy (or optical bandgap) in a molecule with an "infinite" number of conjugated double bonds consists in fitting a function with some adjustable parameters to the relevant data for a set of homologous molecules with increasing number of repeat units ($N$), and assuming that, after its parameters have been optimized according to the least-squares criterion, the function can be extended indefinitely, and its output will coincide with, or come close to, the correct limit. Since more than ten homologues are seldom available, one might wonder whether extrapolation to the infinite-mer upon such slender basis is an instance of sound inductive reasoning or a mere leap of faith. The present article argues that the shape of the fitting function is an equally important criterion, and points out that the expressions proposed by Hirayama and by Meier and coworkers are flat functions of $1/N$, and that such functions are incongruent with the currently available evidence. Formulas derived by Davydov and by W. Kuhn are shown to be special cases of a new equation that outperforms both.

cond-mat.mtrl-sci↗

arXiv2015

The finite square well: whatever is worth teaching at all is worth teaching well

A contemporary physicist would be hard put to agree entirely with the author of a 1959 textbook on quantum mechanics, who wrote: "A second simple, one-dimensional system, somewhat divorced from reality but illustrative of the principles of the theory, is a particle in a box with finite walls." Interest in this prototypic system has not diminished over the years, and is not likely to do so in the near future, because it has now been wedded to reality and seen as a paradigm for quantum wells and other nanosystems. The treatment of this topic in standard textbooks has changed little over the last few decades, and such changes as have come to our notice leave, in our opinion, much to be desired. The purpose of this article is to enable, even encourage, other teachers to give the finite square well the attention that it deserves; to this end, the authors provide a tutorial review that is more instructive and comprehensive than the accounts presented in numerous textbooks and dispensed in still more numerous online resources, but does not make any additional demands on the mathematical abilities of the student. Given the elemental nature of the topic, the authors claim no originality; with nothing more than the winnowing fork in their hands, they can only clear the threshing floor, gather the wheat into the barn, and let others decide whether or no the chaff should be burnt with unquenchable fire.

physics.ed-ph↗

arXiv2014

A Note on the Kinetics of Diffusion-mediated Reactions

The prevalent scheme of a diffusion-mediated bimolecular reaction $A+B\rightarrow P$ is an adaptation of that proposed by Briggs and Haldane for enzyme action [{\em Biochem J.\/}, 19:338--339, 1925]. The purpose of this Note is to explain, {\em by using an argument involving no mathematics\/}, why the breakup of the encounter complex cannot be described, except in special circumstances, in terms of a first-order process $\{AB\}\rightarrow A+B$. Briefly, such a description neglects the occurrence of re-encounters, which lie at the heart of Noyes's theory of diffusion-mediated reactions. The relation $k=αk_{\mbox{\scriptsize e}}$ becomes valid only when $α$ (the reaction probability per encounter) is very much smaller than unity (activation-controlled reactions), or when $β$ (the re-encounter probability) is negligible (as happens in a gas-phase reaction). References to some works (by the author and his collaborators) which propound the correct approach for finding $k$ are also supplied.

physics.chem-ph↗

arXiv2005

On the diffusion coefficient of a photon migrating through a turbid medium: a fresh look from a broader perspective

Does the diffusion coefficient of a photon depend on time $t$ or the probability of absorption $k$? To find an answer to the question, photon transport in a medium of infinite extent is analyzed using the method of moments. It is pointed out that if $D$ is defined so as to make it depend on $t$ or $k$, it will also depend on the experimental conditions; that the parameter $k$ which enters the stationary diffusion equation is in general different from that entering the transient version; and that a hitherto unused non-Markovian partial differential equation may be used for treating photon transport.

cond-mat.other↗

arXiv2005

A new equation for describing heat conduction by phonons

A new equation, rooted in the theory of Brownian motion, is proposed for describing heat conduction by phonons. Though a finite speed of propagation is a built-in feature of the equation, it does not give rise to an inauthentic wave front that results from the application of the hyperbolic heat equation (of Cattaneo). Even a simplified, analytically tractable version of the equation yields results close to those found by solving, through more elaborate means, the equation of phonon radiative transfer. An explanation is given as to why both Brownian motion and its inverse (radiative transfer) provide equally serviceable paradigms for phonon-mediated heat conduction.

cond-mat.other↗

arXiv2005

The origin of the Langevin equation and the calculation of the mean squared displacement: Let's set the record straight

Ornstein and his coauthors, who constructed a dynamical theory of Brownian motion, taking the equation $mdv/dt =-ζv+X$ as their starting point, usually named the equation after Einstein alone or after both Einstein and Langevin; furthermore, Ornstein, who was the first to extract from this equation the correct expression for $\bar{Δ^2}$, the mean-squared distance covered by a Brownian particle, credited de Haas-Lorentz, rather than Langevin, for finding the stationary limit of $\bar{Δ^2}$. A glance at Einstein's 1907 paper, titled ``Theoretical remarks on Brownian motion'', should suffice to convince one that it is not unfair to attribute the {\it conception} of the above equation, now universally known as the Langevin equation, to Einstein. Langevin's avowed aim in his 1908 article was to recover, through a route that was `infinitely more simple', Einstein's 1905 expression for the diffusion coefficient, but a careful reading of Langevin's paper shows that--depending on how one interprets his description of the statistical behavior of the random force $X$ appearing in the above equation--his analysis is at best incomplete, and at worst a mere tautology. Since textbook accounts are based on the interpretation that renders the proof fallacious, alternative derivations, which are adaptations of those given by de Haas-Lorentz and Ornstein, are presented here. Some neglected aspects of the contents of Ornstein's early papers on Brownian motion are also brought to light.

physics.chem-ph↗