Damped harmonic motion energy

The motion of a protein on its PES can be described as anharmonic motions near local minima (i.e. conformations), with rare hops between conformations. While the system executes this motion, we can record, for example, the distance Q(t) between two residues. If the Fourier transform of Q(t) is relatively peaked, then the distance between these residues varied in time like a damped harmonic motion. The quantity Q(t) is not an oscillator with energy levels, that is embedded in the enzyme, rather it is an internal distance of, for example, residues that participate in the equilibrium fluctuations of the enzyme. 

In a damped forced vibration system such as the one shown in Figure 43.14, the motion of the mass M has two parts (1) the damped free vibration at the damped natural frequency and (2) the steady-state harmonic motions at the forcing frequency. The damped natural frequency component decays quickly, but the steady state harmonic associated with the external force remains as long as the energy force is present. 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

Trapped ions possess a secular oscillatory motion whose frequency is given approximately by the well-known SHO equation. This same motion can be excited to larger amplitudes by application of an RF potential to the endcap electrodes at the harmonic frequency. By excitation of this motion, molecular ions can attain sufficient internal energy via coUisions with the buffer gas to dissociate. Since the secular frequency of an ion depends on its mass-to-charge ratio, such excitations can also be used to eject specific m /z ions from the trap. An ion trapped without buffer gas is an undamped oscillator whose motion is described by the SHO equation. Addition of buffer gas dampens this motion somewhat. Addition of an RF potential at the secular frequency drives the ion motion. The result is that the ion can be described as a damped, driven, harmonic oscillator. Such behavior has been extensively studied in RF quadrupole ion traps. 

---