arXiv subjects

Shubho Banerjee

Publications and source records attributed to Shubho Banerjee.

6 recordsLinked to original sources

arXiv2020

Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres $\textit{increases}$ when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximised though an optimal $\textit{lowering}$ of the voltage on the larger sphere at an optimal sphere separation.

physics.class-ph↗

arXiv2018

Rapidly converging formulae for $ζ(4k\pm 1)$

We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\mathscr{L}_q(s) = \sum_{n=1}^\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\pm 1)$. Our main formula for $ζ(4k-1)$ converges at rate of about $e^{-\sqrt{15}π}$ per term, and the formula for $ζ(4k+1)$, at the rate of $e^{-4π}$ per term. For example, the first order approximation yields $ζ(3)\approx\frac{π^3 \sqrt{15}}{100} +e^{-\sqrt{15} π}\left[\frac{9}{4}+\frac{4}{\sqrt{15}}\sinh (\frac{\sqrt{15} π}{2})\right]$ which has an error only of order $10^{-10}$.

math.NT↗

arXiv2016

Lambert series and q-functions near q=1

We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields the asymptotic forms for several related q-functions: the q-gamma and q-polygamma functions, the q-Pochhammer symbol, and, in closed form, the Jacobi theta functions. Some typical results include $Γ_2(\frac{1}{4}) Γ_2(\frac{3}{4}) \simeq \frac{2^{13/32} π}{\log 2}$ and $\vartheta_4 (0,e^{-1/π}) \simeq 2 πe^{-π^3\!/4}$, with relative errors of order $10^{-25}$ and $10^{-27}$ respectively.

math.NT↗

arXiv2009

On the Stability of Electrostatic Orbits

We analyze the stability of two charged conducting spheres orbiting each other. Due to charge polarization, the electrostatic force between the two spheres deviates significantly from $1/r^2$ as they come close to each other. As a consequence, there exists a critical angular momentum, $L_c$, with a corresponding critical radius $r_c$. For $L > L_c$ two circular orbits are possible: one at $r > r_c$ that is stable and the other at $r < r_c$ that is unstable. This critical behavior is analyzed as a function of the charge and the size ratios of the two spheres.

physics.class-ph↗

arXiv2005

Criticality in Charge-asymmetric Hard-sphere Ionic Fluids

Phase separation and criticality are analyzed in $z$:1 charge-asymmetric ionic fluids of equisized hard spheres by generalizing the Debye-Hückel approach combined with ionic association, cluster solvation by charged ions, and hard-core interactions, following lines developed by Fisher and Levin (1993, 1996) for the 1:1 case (i.e., the restricted primitive model). Explicit analytical calculations for 2:1 and 3:1 systems account for ionic association into dimers, trimers, and tetramers and subsequent multipolar cluster solvation. The reduced critical temperatures, $T_c^*$ (normalized by $z$), \textit{decrease} with charge asymmetry, while the critical densities \textit{increase} rapidly with $z$. The results compare favorably with simulations and represent a distinct improvement over all current theories such as the MSA, SPB, etc. For $z$$\ne$1, the interphase Galvani (or absolute electrostatic) potential difference, $Δϕ(T)$, between coexisting liquid and vapor phases is calculated and found to vanish as $|T-T_c|^β$ when $T\to T_c-$ with, since our approximations are classical, $β={1/2}$. Above $T_c$, the compressibility maxima and so-called $k$-inflection loci (which aid the fast and accurate determination of the critical parameters) are found to exhibit a strong $z$-dependence.

cond-mat.stat-mech↗

arXiv2005

How Multivalency controls Ionic Criticality

To understand how multivalency influences the reduced critical temperatures, Tce (z), and densities, roce (z), of z : 1 ionic fluids, we study equisized hard-sphere models with z = 1-3. Following Debye, Hueckel and Bjerrum, association into ion clusters is treated with, also, ionic solvation and excluded volume. In good accord with simulations but contradicting integral-equation and field theories, Tce falls when z increases while roce rises steeply: that 80-90% of the ions are bound in clusters near T_c serves to explain these trends. For z \neq 1 interphase Galvani potentials arise and are evaluated.

cond-mat.stat-mech↗