arXiv subjects

Petar Mali

Publications and source records attributed to Petar Mali.

14 recordsLinked to original sources

arXiv2025

Hamiltonian dynamics of classical spins

We discuss the geometry behind classical Heisenberg model at the level suitable for third or fourth year students who did not have the opportunity to take a course on differential geometry. The arguments presented here rely solely on elementary algebraic concepts such as vectors, dual vectors and tensors, as well as Hamiltonian equations and Poisson brackets in their simplest form. We derive Poisson brackets for classical spins, along with the corresponding equations of motion for classical Heisenberg model, starting from the geometry of two-sphere, thereby demonstrating the relevance of standard canonical procedure in the case of Heisenberg model.

physics.ed-ph↗

arXiv2024

Infinite series involving special functions obtained using simple one-dimensional quantum mechanical problems

In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an infinite potential well. The infinite sums $\sum^{\infty}_{n=0}\frac{2^{2n}}{(2n+1)!}\Gamma^{2}\left(n+\frac{3}{2}\right)\left[\hspace{0.2mm}_2\hspace{-0.03cm}F_1\left(-n,\frac{\nu+2}{2};\frac{3}{2};\frac{1}{2}\right)\right]^{2}$, $\sum^{\infty}_{n=0}\frac{\left[L_{\nu}^{2n+1-\nu}\left(\frac{b^{2}}{2}\right)\right]^{2}b^{4n}}{2^{2n}(2n+1)!}$ and $\sum^{\infty}_{n=1}\frac{\big[J_{\nu+1}(n\pi)\big]^{2}}{n^{2\nu}}$, where $_2\hspace{-0.03cm}F_1\left(-n,\frac{\nu+2}{2};\frac{3}{2};\frac{1}{2}\right)$ is generalized hypergeometric function, $L_{\nu}^{2n+1-\nu}\left(\frac{b^{2}}{2}\right)$ associated Laguerre polynomial and $J_{\nu+1}(n\pi)$ Bessel function of the first kind, are calculated for integer $\nu$. It is also demonstrated that the same procedure can be generalized by application to some classes of functions which are not regular wave functions leading to additional infinite sums, as a consequence of which the series $\sum_{n=1}^{\infty}\frac{\left[\mathsf{H}_{\nu}(n\pi)\right]^{2}}{n^{2\nu}}$ containing Struve functions of the first kind $\mathsf{H}_{\nu}(n\pi)$ are evaluated. Convergence of the evaluated series, additionally verified by the application of different convergence tests, is secured by the properties of the corresponding Hilbert space.

quant-ph↗

arXiv2024

Summation formulas generated by Hilbert space eigenproblem

We demonstrate that certain classes of Schl\" omilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence.

quant-ph↗

arXiv2023

The largest Lyapunov exponent as a tool for detecting relative changes in the particle positions

Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent and Poincaré sections for two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration of both damped Frenkel-Kontorova model with inertial term and its strictly overdamped limit. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime. These jumps are also registered in the sliding regime, where they are a consequence of a more complex structure of largest Lyapunov exponent on the step.

nlin.CD↗

arXiv2021

Using quantum mechanics for calculation of different infinite sums

We demonstrate that certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument $p$ ($p>2$) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.

quant-ph↗

arXiv2021

Random Matrix Ensembles in Hyperchaotic Classical Dissipative Dynamical Systems

We study the statistical fluctuations of Lyapunov exponents in the discrete version of the non-integrable perturbed sine-Gordon equation, the dissipative ac+dc driven Frenkel-Kontorova model. Our analysis shows that the fluctuations of the exponent spacings in the strictly overdamped limit, which is nonchaotic, conforms to the \textit{uncorrelated} Poisson distribution. By studying the spatiotemporal dynamics we relate the emergence of the Poissonian statistics to Middleton's no-passing rule. Next, by scanning over the dc driving and particle mass we identify several parameter regions for which this one-dimensional model exhibits hyperchaotic behavior. Furthermore, in the hyperchaotic regime where roughly fifty percent of exponents are positive, the fluctuations exhibit features of the \textit{correlated} universal statistics of the Gaussian Orthogonal Ensemble (GOE). Due to the dissipative nature of the dynamics, we find that the match, between the Lyapunov spectrum statistics and the universal statistics of GOE, is not complete. Finally, we present evidence supporting the existence of the Tracy-Widom distribution in the fluctuation statistics of the largest Lyapunov exponent.

cond-mat.stat-mech↗

arXiv2020

Influence of anharmonic convex interparticle potential and Shapiro steps in the opposite direction of driving force

The response function and largest Lyapunov exponent analysis were applied to the driven overdamped Frenkel-Kontorova model with two types of anharmonic convex interparticle potentials. In both cases model reduces to a single particle model for integer values of winding number. It is shown that the mirror image of the amplitude dependence of critical depinning force and largest Lyapunov exponent observed recently in the standard Frenkel-Kontorova model [Commun. Nonlinear Sci. Numer. Simul. 47, 100 (2017)] is not retained generally. Behaviour of systems with relatively strong interparticle force was examined and evidence for the appearance of mode-locking phenomenon in both directions of particles' motion is presented.

nlin.CD↗

arXiv2020

Correlation between quantum entanglement and quantum coherence in the case of XY spin chains with the Dzyaloshinskii-Moriya interaction

Recently, there has been an increased interest in studying quantum entanglement and quantum coherence. Since both of these properties are attributed to the existence of quantum superposition, it would be useful to determine if some type of correlation between them exists. Hence, the purpose of this paper is to explore the type of the correlation in several systems with different types of anisotropy. The focus will be on the XY spin chains with the Dzyaloshinskii-Moriya interaction and the type of the mentioned bond will be explored using the quantum renormalization group method.

cond-mat.str-el↗

arXiv2020

Exciton dynamics in different aromatic hydrocarbon systems

The exciton dispersion is examined in the case of four selected prototypical molecular solids: pentacene,tetracene,picene,chrysene. The model parameters are determined by fitting to experimental data obtained by inelastic electron scattering. Within the picture that relies on Frenkel-type excitons we obtain that theoretical dispersion curves along different directions in the Brillouin zone are in good agreement with the experimental data, suggesting that the influence of charge-transfer excitons on exciton dispersion of the analyzed organic solids is not as large as proposed. In reciprocal space directions where Davydov splitting is observed we employ the upgraded version of Hamiltonian used in Materials 11, 2219 (2018).

cond-mat.mtrl-sci↗

arXiv2020

Complexity of Shapiro steps

We demonstrate on the example of the dc+ac driven overdamped Frenkel-Kontorova model that an easily calculable measure of complexity can be used for the examination of Shapiro steps in presence of thermal noise. In real systems, thermal noise causes melting or even disappearance of Shapiro steps, which makes their analysis in the standard way from the response function difficult. Unlike in the conventional approach, here, by calculating the Kolmogorov complexity of certain areas in the response function we were able to detect Shapiro steps, measure their size with desired precision and examine their temperature dependence. The aim of this work is to provide scientists, particularly experimentalists, an unconventional but a practical and easy tool for examination of Shapiro steps in real systems.

nlin.CD↗

arXiv2018

Dynamics of Frenkel excitons in pentacene

The dispersion relation for noninteracting excitons and the influence of perturbative correction is examined in the case of pentacene structure. The values of exchange integrals are determined by the nonlinear fits to the experimental dispersion data obtained by inelastic electron scattering in Phys. Rev. Lett. \textbf{98}, 037402 (2007). We obtain theoretical dispersion curves along four different directions in the Brillouin zone which possess the same periodicity as the experimental data. We also showed that perturbative corrections are negligible since the exciton gap in dispersion relation is huge in comparison to exchange integrals.

cond-mat.str-el↗

arXiv2016

Application of largest Lyapunov exponent analysis on the studies of dynamics under external forces

Dynamics of driven dissipative Frenkel-Kontorova model is examined by using largest Lyapunov exponent computational technique. Obtained results show that besides the usual way where behavior of the system in the presence of external forces is studied by analyzing its dynamical response function, the largest Lyapunov exponent analysis can represent a very convenient tool to examine system dynamics. In the dc driven systems, the critical depinning force for particular structure could be estimated by computing the largest Lyapunov exponent. In the dc+ac driven systems, if the substrate potential is the standard sinusoidal one, calculation of the largest Lyapunov exponent offers a more sensitive way to detect the presence of Shapiro steps. When the amplitude of the ac force is varied the behavior of the largest Lyapunov exponent in the pinned regime completely reflects the behavior of Shapiro steps and the critical depinning force, in particular, it represents the mirror image of the amplitude dependence of critical depinning force. This points out an advantage of this technique since by calculating the largest Lyapunov exponent in the pinned regime we can get an insight into the dynamics of the system when driving forces are applied.

nlin.CD↗

arXiv2015

Farey sequence in the appearance of subharmonic Shapiro steps

Largest Lyapunov exponent has been examined in the dynamical-mode locking phenomena of the ac+dc driven dissipative Frenkel-Kontorova model with deformable substrate potential. Due to deformation, large fractional and higher order subharmonic steps appear in the response function of the system. Computation of the largest Lyapunov exponent as a way to verify their presence led to the observation of the Farey sequence. In the standard regime, between the large harmonic steps, the appearance of halfinteger and subharmonic steps, and their relative sizes follow the Farey construction. In the nonstandard regime, though halfinteger steps are larger than harmonic ones, Farey construction is still present in the appearance of higher order subharmonic steps. The examination of Lyapunov exponents also shows that there is no chaos in the system.

nlin.CD↗

arXiv2014

Thermodynamics of O(3) Classical Heisenberg Model in Multipath Metropolis Simulation

We study the thermodynamics of classical Heisenberg model using the multipath approach to Metropolis algorithm Monte Carlo simulation. This simulation approach produces uncorrelated results with known precision. Also, it can be easily generalized to other classical models of magnetism. Comparing results obtained from multipath and from single--path simulations we demonstrate that these approaches produce equivalent results.

cond-mat.stat-mech↗