Analytical expressions for real and complex Fano parameters in a simple classical harmonic oscillator system

We analytically study the Fano resonance in a simple coupled oscillator system. We demonstrate directly from the equation of motion that the resonance profile observed in this system is generally described by the Fano formula with a complex Fano parameter. The analytical expressions are derived for the resonance frequency, resonance width, and Fano parameter, and the conditions under which the Fano parameter becomes a real number are examined. These expressions for the simple system are also expected to be helpful for considering various other physical systems because the Fano resonance is a general wave phenomenon.

Exact calculations of the thermal properties of two-electron GaAs quantum dots with inverse-square interactions

In this study, we theoretically scrutinize the effect of the inverse-square interaction on the thermal properties of two electrons trapped in a parabolic GaAs quantum dot. The analytical energy spectrum was used to calculate the thermal properties of the system using the canonical ensemble formalism. It was found that the thermal energy increased with the increase in temperature, while it remained almost constant for sufficiently low temperatures; it was also demonstrated that the inverse-square interaction increased the thermal mean energy. Moreover, the heat capacity increased sharply within a low-temperature window and saturated to the value of 2kB in the high-temperature limit. As expected, entropy increased linearly with increasing temperature. It was also shown that both entropy and heat capacity decreased rapidly when the confinement strength increased (or the dot size decreased) in the low-temperature limit, regardless of the influence of the interaction between the electrons. We also show that the number of allowed states of the system decreased as the interaction strength increased (Z(λ = 0) > Z(λ ≠ 0)). Finally, the stability of the system was investigated through F–T curves. The three-dimensional surface for the temperature-dependent mean energy and heat capacity was also plotted. It should be noted that, for the thermal mean energy, partition function, and Helmholtz free energy, the normal physical behavior of the two-oscillator system with Fermi statistics is recovered for λ → 0. However, heat capacity and entropy show exact two-fermion oscillator system behavior. The most impressive result found in this work is that the inverse-square interaction does not affect the heat capacity and entropy at all despite its noticeable effects on the thermal mean energy. This, in turn, facilitates theoretical studies related to finding the distinctive parameters of quantum dots without going into the heavy calculations resulting from the effects of interactions.

The Dynamics of Coupled Oscillators

<p>The subject is introduced by considering the treatment of oscillators in Mathematics from the simple Poincar´e oscillator, a single variable dynamical process defined on a circle, to the oscillatory dynamics of systems of differential equations. Some models of real oscillator systems are considered. Noise processes are included in the dynamics of the system. Coupling between oscillators is investigated both in terms of analytical systems and as coupled oscillator models. It is seen that driven oscillators can be used as a model of 2 coupled oscillators in 2 and 3 dimensions due to the dependence of the dynamics on the phase difference of the oscillators. This means that the dynamics are easily able to be modelled by a 1D or 2D map. The analysis of N coupled oscillator systems is also described. The human cardiovascular system is studied as an example of a coupled oscillator system. The heart oscillator system is described by a system of delay differential equations and the dynamics characterised. The mechanics of the coupling with the respiration is described. In particular the model of the heart oscillator includes the baroreceptor reflex with time delay whereby the aortic fluid pressure influences the heart rate and the peripheral resistance. Respiration is modelled as forcing the heart oscillator system. Locking zones caused by respiratory sinus arrhythmia (RSA), the synchronisation of the heart with respiration, are found by plotting the rotation number against respiration frequency. These are seen to be relatively narrow for typical physiological parameters and only occur for low ratios of heart rate to respiration frequency. Plots of the diastolic pressure and heart interval in terms of respiration phase parameterised by respiration frequency illustrate the dynamics of synchronisation in the human cardiovascular system.</p>

The Dynamics of Coupled Oscillators

<p>The subject is introduced by considering the treatment of oscillators in Mathematics from the simple Poincar´e oscillator, a single variable dynamical process defined on a circle, to the oscillatory dynamics of systems of differential equations. Some models of real oscillator systems are considered. Noise processes are included in the dynamics of the system. Coupling between oscillators is investigated both in terms of analytical systems and as coupled oscillator models. It is seen that driven oscillators can be used as a model of 2 coupled oscillators in 2 and 3 dimensions due to the dependence of the dynamics on the phase difference of the oscillators. This means that the dynamics are easily able to be modelled by a 1D or 2D map. The analysis of N coupled oscillator systems is also described. The human cardiovascular system is studied as an example of a coupled oscillator system. The heart oscillator system is described by a system of delay differential equations and the dynamics characterised. The mechanics of the coupling with the respiration is described. In particular the model of the heart oscillator includes the baroreceptor reflex with time delay whereby the aortic fluid pressure influences the heart rate and the peripheral resistance. Respiration is modelled as forcing the heart oscillator system. Locking zones caused by respiratory sinus arrhythmia (RSA), the synchronisation of the heart with respiration, are found by plotting the rotation number against respiration frequency. These are seen to be relatively narrow for typical physiological parameters and only occur for low ratios of heart rate to respiration frequency. Plots of the diastolic pressure and heart interval in terms of respiration phase parameterised by respiration frequency illustrate the dynamics of synchronisation in the human cardiovascular system.</p>

Numerical study of coupled oscillator system using the classical Euler-Lagrange equations

Problems involving vibrations (mechanical or electrical) can be reduced to problems of coupled oscillators. For this, we consider the motion of coupled oscillators system using Lagrangian method. The Lagrangian of the system was initially constructed, and then the Euler-Lagrange equations (i.e., equations of motion of the system) have been obtained. The obtained equations of motion are a homogenous second-order equation. These equations were solved numerically using the ode45 code, which is based on Runge-Kutta method.