Consider a harmonic oscillator (e.g., a pendulum), driven by bombardment with air molecules. Explain why the oscillator’s position x(t) and velocity ν(t) = dx/dt are random processes. Is x(t)Markov? Why? Is ν(t)Markov? Why? Is the pair {x(t), v(t)} a 2-dimensionalMarkov process? Why? We study this 2-dimensional random process in Ex. 6.23.

Data from Exercises 6.23

Consider a classical simple harmonic oscillator (e.g., the nanomechanical oscillator, LIGO mass on an optical spring, L-C-R circuit, or optical resonator briefly discussed in Ex. 6.17). Let the oscillator be coupled weakly to a dissipative heat bath with temperature T. The Langevin equation for the oscillator’s generalized coordinate x is Eq. (6.79). The oscillator’s coordinate x(t) and momentum p(t) ≡ mẋ together form a 2-dimensional Gaussian-Markov process and thus obey the 2-dimensional Fokker- Planck equation (6.106a). As an aid to solving this Fokker-Planck equation, change variables from {x, p} to the real and imaginary parts X₁ and X₂ of the oscillator’s complex amplitude:

Then {X₁, X₂} is a Gaussian-Markov process that evolves on a timescale ∼τr .

Transcribed Image Text:

(6.107) (1) (1)¹ X=[-(²x! + ¹x)] = x