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We study a stochastically driven, damped nonlinear oscillator whose frequency is modulated by a white or coloured noise. Using diagrammatic perturbation theory, we find that in the absence of nonlinearity, parametric modulation by a coloured noise can lead to a Kapitza pendulum-like stabilization of an unstable configuration provided the noise is anti-correlated. Further, we show that for modulation by a white noise of amplitude $\lambda$ and correlation strength $F$, the system will have an extremely large response if the product of $\lambda^{2}F$ equals a specific combination of the frequency and the damping coefficient. This prediction can be experimentally tested., Comment: 18 pages, 5 figures

We consider the trick of balancing a vertical stick on a horizontal plate. It is shown that the horizontal stochastic driving of the point of contact can prevent the stick from falling provided that the stochasticity is that of a coloured noise with a correlation strength stronger than a critical value., Comment: 5 pages, 1 figure

We apply perturbative techniques to a driven undamped sinusoidal oscillator at resonance. The angular displacement, $\theta$, obeys the dynamics $\Ddot{\theta} + \omega^{2} \sin{\theta} = H\cos{\omega t}$. The linearized approximation gives a divergent response (at long times) but the nonlinear terms make the response finite. We address the nonlinearity-induced finiteness in two ways by separately treating the short and long time scales. At long times, we use the traditional perturbative techniques to extract two drive dependent behaviours - one, the amplitude of oscillation scales as $(H/\omega^{2})^{1/3}$ and, two, the time period of the slow mode varies as $(H/\omega^{2})^{-2/3}$. For the early time behaviour, on the other hand, we devise an alternate perturbative expansion where the successive terms get larger with the order of evaluation but have alternating signs. The alternating signs (phase differences) between these terms leads to adestructive interference like effect. A careful consideration of this destructive interference like effect between successive terms leads to a finite response which describes the initial behaviour of the amplitude of the response reasonably correctly. We further note that for larger drive values, the system seems to undergo a first order transitional behaviour with a sudden jump in the largest Lyapunov exponent., Comment: 14 pages, 11 figures; Added references; Adjusted figures and text; added new figures and text

Manipulation of long-range order in two-dimensional (2D) van der Waals (vdW) magnetic materials (e.g., CrI$_3$, CrSiTe$_3$ etc.), exfoliated in few-atomic layer, can be achieved via application of electric field, mechanical-constraint, interface engineering, or even by chemical substitution/doping. Usually, active surface oxidation due to the exposure in the ambient condition and hydrolysis in the presence of water/moisture causes degradation in magnetic nanosheets which, in turn, affects the nanoelectronic/spintronic device performance. Counterintuitively, our current study reveals that exposure to the air at ambient atmosphere results in advent of a stable nonlayered secondary ferromagnetic phase in the form of Cr$_2$Te$_3$ (T$_{C2}$ ~ 160 K) in the parent vdW magnetic semiconductor Cr$_2$Ge$_2$Te$_6$ (T$_{C1}$ ~ 69 K). In addition, the magnetic anisotropy energy (MAE) enhances in the hybrid by an order from the weakly anisotropic pristine Cr$_2$Ge$_2$Te$_6$ crystal, increasing the stability of the FM ground state with time. Comparing with the freshly prepared Cr$_2$Ge$_2$Te$_6$, the coexistence of the two ferromagnetic phases in the time elapsed bulk crystal is confirmed through systematic investigation of crystal structure along with detailed dc/ac magnetic susceptibility, specific heat, and magnetotransport measurement. To capture the concurrence of the two ferromagnetic phases in a single material, Ginzburg-Landau theory with two independent order parameters (as magnetization) with a coupling term can be introduced. In contrast to rather common poor environmental stability of the vdW magnets, our results open possibilities of finding air-stable novel materials having multiple magnetic phases.

We analyse the scaling properties of the energy spectra in fully developed incompressible turbulence in forced, rotating fluids in three dimensions (3D), which are believed to be characterised by universal scaling exponents in the inertial range. To elucidate the scaling regimes, we set up a scaling analysis of the 3D Navier-Stokes equation for a rotating fluid that is driven by large-scale external forces. We use scaling arguments to extract the scaling exponents, which characterise the different scaling regimes of the energy spectra. We speculate on the intriguing possibility of two-dimensionalisation of 3D rotating turbulence within our scaling theory. Our results can be tested in large scale simulations and relevant laboratory-based experiments.

The microcanonical ensemble is a natural starting point of statistical mechanics. However, when it comes to perturbation theory in statistical mechanics, traditionally only the canonical and grand canonical ensembles have been used. In this article we show how the microcanonical ensemble can be directly used to carry out perturbation theory for both non-interacting and interacting systems. We obtain the first non-trivial order answers for the specific heat of anharmonic oscillators and for the virial expansion in real gases. They are in exact agreement with the results obtained from the canonical ensemble. In addition, we show how crossover functions for the specific heat of anharmonic oscillators can be constructed using a microcanonical ensemble and also how the subsequent terms of the virial expansion can be obtained. However, we find that if we consider quantum free particles in a one-dimensional box of extension L, then the two ensembles give strikingly different answers for the first correction to the specific heat in the high temperature limit.

Turbulence in driven stratified active matter is considered. The relevant parameters characterizing the problem are the Reynolds number Re and an active matter Richardson-like number,R. In the mixing limit,Re>>1, R<<1, we show that the standard Kolmogorov energy spectrum 5/3 law is realized. On the other hand, in the stratified limit, Re>>1,R>>1, there is a new turbulence universality class with a 7/5 law. The crossover from one regime to the other is discussed in detail. Experimental predictions and probes are also discussed.

We impose a linearized Eulerian perturbation on a steady, shallow, radial outflow of a liquid (water), whose local pressure function includes both the hydrostatic and the Laplace pressure terms. The resulting wave equation bears the form of a hydrodynamic metric. A dispersion relation, extracted from the wave equation, gives an instability due to surface tension and the cylindrical flow symmetry. Using the dispersion relation, we also derive three known relations that scale the radius of the circular hydraulic jump in the outflow. The first two relations are scaled by viscosity and gravity, with a capillarity-dependent crossover to the third relation, which is scaled by viscosity and surface tension. The perturbation as a high-frequency travelling wave, propagating radially inward against the bulk outflow, is blocked just outside the circular hydraulic jump. The amplitude of the wave also diverges here because of a singularity. The blocking is associated with surface tension, which renders the circular hydraulic jump a hydrodynamic white hole., Comment: 7 pages, 1 figure, ReVTeX double column format

We consider the transmission dynamics of COVID-19 which is characterized by two distinct features. One is the existence of asymptomatic carriers which is a hidden variable in the problem. The other is the issue of latency which means that among the symptomatic carriers there could be a fraction whose symptoms develop after a couple of days. Our modelling is restricted to what we call the Phase -1 of the disease. During this phase the disease sets in and the number of infected people starts growing fast ( the number of new cases per day keeps growing on an average ) and then it slows down ( the number of new cases per day starts decreasing ) with the number of new cases decreasing to about one tenth of its peak value or even smaller). We define Phase-1 to be over when the daily cases start rising once again. We write down a Henon-like map to take various effects into account for this first phase., Comment: (19 pages,10 figures)

We study the linear instabilities and bifurcations in the Selkov model for glycolysis with diffusion. We show that this model has a zero wave-vector, finite frequency Hopf bifurcation to a growing oscillatory but spatially homogeneous state and a saddle-node bifurcation to a growing inhomogeneous state with a steady pattern with a finite wavevector. We further demonstrate that by tuning the relative diffusivity of the two concentrations, it is possible to make both the instabilities to occur at the same point in the parameter space, leading to an unusual type of codimension-two bifurcation. We then show that in the vicinity of this bifurcation the initial conditions decide whether a spatially uniform oscillatory or a spatially periodic steady pattern emerges in the long time limit.

Quasi-periodically driven quantum parametric oscillators have been the subject of several recent investigations. Here we show that for such oscillators, the instability zones of the mean position and variance (alternatively the mean energy) for a time developing wave packet are identical for the strongest resonance in the three-dimensional parameter space of the quasi-periodic modulation as it is for the two-dimensional parameter space of the periodic modulations.

It is necessary to introduce an external forcing to induce turbulence in a stably stratified fluid. The Heisenberg eddy viscosity technique should in this case suffice to calculate a space-time averaged quantity like the global anisotropy parameter as a function of the Richardson number. We find analytically that the anisotropy increases linearly with the Richardson number, with a small quadratic correction. A numerical simulation of the complete equations shows the linear behaviour.

A Lorenz-like model was set up recently, to study the hydrodynamic instabilities in a driven active matter system. This Lorenz model differs from the standard one in that all three equations contain non-linear terms. The additional non-linear term comes from the active matter contribution to the stress tensor. In this work, we investigate the non-linear properties of this Lorenz model both analytically and numerically. The significant feature of the model is the passage to chaos through a complete set of period-doubling bifurcations above the Hopf point for inverse Schmidt numbers above a critical value. Interestingly enough, at these Schmidt numbers a strange attractor and stable fixed points coexist beyond the homoclinic point. At the Hopf point, the strange attractor disappears leaving a high-period periodic orbit. This periodic state becomes the expected limit cycle through a set of bifurcations and then undergoes a sequence of period-doubling bifurcations leading to the formation of a strange attractor. This is the first situation where a Lorenz-like model has shown a set of consecutive period-doubling bifurcations in a physically relevant transition to turbulence., Comment: 12 pages, 11 figures

We approach quantum dynamics in one spatial dimension from a systematic study of moments starting from the dynamics of the mean position. This is complementary to the approach of Brizuela whose starting point was generalized recursion relations between moments. The infinite set of coupled equations is truncated which allows us to use the techniques used in the study of dynamical systems. In particular we predict for what initial variance the purely quartic oscillator will time develop with minimal change in the shape of the initial packet and what the frequency of oscillation of the mean position will be. We show how quantum fluctuations will cause a particle to escape from the well of a volcano potential and how they will cause an oscillation between the two wells of a double well potential. Further, we consider an oscillatory external field in addition to the double well potential and work near the separatrix where the classical system is known to be chaotic. We show how the quantum fluctuations suppresses the chaotic behaviour after a time interval inversely proportional to the strength of the quantum fluctuations., Comment: 12 pages and 20 figures

We set up the scaling theory for stably stratified turbulent fluids. For a system having infinite extent in the horizontal directions, but with a finite width in the vertical direction, this theory predicts that the inertial range can display three possible scaling behaviour, which are essentially parametrised by the buoyancy frequency $N$, or dimensionless horizontal Froude number $F_h$, and the vertical length scale $l_v$ that sets the scale of variation of the velocity field in the vertical direction, for a fixed Reynolds number. For very low $N$ or very high $Re_b$ or $F_h$, and with $l_v$ being of the same order as $l_h$, the typical horizontal length scale, buoyancy forces are irrelevant and hence, unsurprisingly, the kinetic energy spectra shows the well-known K41 scaling in the inertial range. In this regime, the local temperature behaves as a passively advected scalar, without any effect on the flow fields. For intermediate ranges of values of $N,\,F_h\sim {\cal O}(1)$, corresponding to moderate stratification, buoyancy forces are important enough to affect the scaling. This leads to the Bolgiano-Obukhov scaling which is isotropic, when $l_v\sim l_h$. Finally, for very large $N$ or equivalently for vanishingly small $F_h,\,L_o$, corresponding to strong stratification, together with a very small $l_v$, the system effectively two-dimensionalise; the kinetic energy spectrum is predicted to be anisotropic with only the horizontal part of the kinetic energy spectra follows the K41 scaling, suggesting an intriguing {\em re-entrant} K41 scaling, as a function of stratification, for ${\bf v}_\perp$ in this regime. The scaling theory further predicts the scaling of the thermal energy in each of these three scaling regimes. Our theory can be tested in large scale simulations and appropriate laboratory-based experiments.

We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate., Comment: Document contains 8 pages and 15 diagrams

We revisit the scaling properties of the energy spectra in fully developed incompressible homogeneous turbulence in forced magnetofluids (MHD) in three dimensions (3D), which are believed to be characterised by {\em universal scaling exponents} in the inertial range. Enumerating these universal scaling exponents that characterise the energy spectra remains a theoretical challenge. To study this, we set up a scaling analysis of the 3D MHD equations, driven by large-scale external forces and with or without a mean magnetic field. We use scaling arguments to bring out various scaling regimes for the energy spectra. We obtain a variety of scaling in the inertial range, ranging from the well-known Kolmogorov spectra in the isotropic 3D ordinary MHD to more complex scaling in the anisotropic cases that depend on the magnitude of the mean magnetic field. We further dwell on the possibility that the energy spectra scales as $k^{-2}$ in the inertial range, where $k$ is a wavevector belonging to the inertial range, and also speculate on unequal scaling by the kinetic and magnetic energy spectra in the inertial range of isotropic 3D ordinary MHD. We predict the possibilities of {\em scale-dependent anisotropy} and intriguing {\em weak dynamic scaling} in the Hall MHD and electron MHD regimes of anisotropic MHD turbulence. Our results can be tested in large scale simulations and relevant laboratory-based and solar wind experiments., Comment: To appear in PRE (2018)

We use Ehrenfest's theorem to provide a particularly simple derivation of the {\em zitterbewegung} in the dynamics of initial Gaussian wave packets in a two-dimensional electron gas. For initial packets which are very wide in the $y$-direction, the {\em zitterbewegung} is only in the $y$-component of the velocity. We extend our Ehrenfest theorem based calculation to the spin-orbit coupled spinor Bose-Einstein condensate (BEC) to predict that there can be {\em zitterbewegung} in the $x$-component of the velocity in this situation driven by a combination of the nonlinear interaction in the condensate and the splitting due to the spin-orbit coupling.

We consider the dynamics of a particle confined in a double well potential which is subjected to a periodic drive. In the case of deep and well separated wells, we find that by adjusting the parameters of the drive we can generate, to a very good approximation, a volcano potential. The quantum dynamics in this volcano potential is studied by a variation of what can be called a generalized Ehrenfest's theorem. We find that the coupling of the mean position and the width of the wave packet in this dynamics causes the particle to escape from the central well in accordance with the fact that the volcano potential only supports resonance states., Comment: 8 pages, 8 figures. Submitted to EPJ D

Work fluctuations and work probability distributions are fundamentally different in systems with short- ranged versus long-ranged correlations. Specifically, in systems with long-ranged correlations the work distribution is extraordinarily broad compared to systems with shortranged correlations. This difference profoundly affects the possible applicability of fluctuation theorems like the Jarzynski fluctuation theorem. The Heisenberg ferromagnet , well below its Curie temperature, is a system with long-ranged correlations in very low magnetic fields due to the presence of Goldstone modes. As the magnetic field is increased the correlations gradually become short-ranged. Hence, such a ferromagnet is an ideal system for elucidating the changes of the work probability distribution as one goes from a domain with long-ranged correlations to a domain with short-ranged correlations by tuning the magnetic field. A quantitative analysis of this crossover behaviour of the work probability distribution and the associated fluctuations is presented.

We derive Ehrenfest like equations for the coupled Gross Pitaevskii equations (CGPE) which describe the dynamics of the binary Bose-Einstein condensate (BBEC) both in the free particle regime and in the regime where condensate is well trapped. Instead of traditional variational technique, we propose a new Ehrenfest based approach to explore so far unrevealed dynamics for CGPE and illustrate the possibility of almost shape invariant states in both the regimes. In absence of trapping potential, when all the interactions present in the system are attractive, it is possible for an initially mixed Gaussian state to propagate with almost no change in width if the proper initial condition is satisfied. Even for repulsive intra-atomic and attractive inter-atomic interaction ($g_{\alpha\beta}$) one can tune $|g_{\alpha\beta}|$ such that the width of the propagating wave packet remains bounded within almost about $10\%$. We also discuss the dynamics of the initially phase separated condensate and have shown the breakdown of Gaussian nature of the wave packets due to collisions. However, when BEC is trapped in simple harmonic oscillator(SHO) potential, for $g_{\alpha\beta}>0$, it is possible for an initially overlapping state to retain its initial shape if $g_{\alpha\beta}$ is less than a critical value ($g_{\alpha\beta}^c$). If $g_{\alpha\beta}$ exceeds $g_{\alpha\beta}^c$, an overlapping state can become phase separated while keeping its shape unchanged., Comment: 10 pages, 20 figures