7.4 Examples And Model Calculations
7.4.1 Discretely damped 0-D harmonic oscillator
Chapter 7 Particle-Wave Interaction Method
The frictional part γdxdt dissipates energy from the system in a continuous process. We want to model the same system to have a discrete, noncontinuous damping and within an ensemble average both systems should behave the same.
Casting eq. 7.35to an Hamilton equation leads to an equation with an explicit time dependence:
H = p2
2m e−γt+1
2mω02x2 e−γt (7.36)
Typically there are three types of solutions to equation (7.35), under-damped, critical-damped and over-damped. All three solutions show their dissipation via an exponential damping function
∼exp(−γeff·t) withγeff =γeff(ω0, γ) as an efective damping parameter.
Motivation for a noncontinuous damping In classical mechanics, the harmonic oscillator is continuous both in space and momentum. Its energy levels are on a continuum and so are any dissipative processes. Quantum mechanics changed this behavior fundamentally with the introduction of quantization. The quantum energy levels became discrete with a lower limit:
En=ℏω(n+ 1/2) , (7.37)
n= En ℏω −1
2 , (7.38)
so forn≫1:
n∼E . (7.39)
Within this ansatz the damping of a quantum oscillator becomes highly non-trivial, because the energy can now only jump between the energy niveaus En−En+1 = ∆E. In theory this has been a long unsolved problem and was most of the time solved by coupling quantum oscillators to heat-baths or stochastic-classical systems which absorbs the exchanged energy [119Ű122]. The idea of discrete energy jumps in the oscillator was the motivation for this Ąrst example calculation.
Energy of a classical harmonic oscillator In the following the properties of a classical, damped harmonic oscillator are discussed. The equation of motion is
dx(t)2
dt2 +ω02 x(t) + 2γdx(t)
dt = 0 . (7.40)
Depending on the damping coeicient γ, the following solutions exist:
Chapter 7 Particle-Wave Interaction Method
γ > ω0, over-damping : x(t) =C0exp(γ+t) +C1exp(γ−t) , (7.41) γ =ω0, critical-damping :x(t) = (C0+tC1) exp(−ω0t) , (7.42) γ < ω0, under-damping : x(t) = exp(−γt)Csin (wdt+φ) , (7.43)
wd=w0
√1−γ2 . (7.44)
To compare against the quantum oscillator, the energy for the classical oscillator is calculated.
The over-damped and critically damped oscillators both show an exponential decay when higher order terms are ignored:
Ekin(γ>ω0)≈E¯0 e−¯γt , (7.45)
Ekin(γ=ω0)≈E¯0 e−ω0t. (7.46) A more interesting case is the under-damped one
Ekin(γ<ω0)= C2
2 e−2γt[ωdcos (ωdt+φ)− γsin (ωdt+φ)]2 . (7.47) The energy of the under-damped oscillator shows an exponential decay with cosine and sine terms. These terms lead to a modulation of the dissipation because the oscillator only looses energy when it is moving. However, on average, on long timescales or for weak coupling, the energy loss scales roughly as an exponential:
Ekin(γ<1) ≈E¯0e−γω0t (7.48)
In all three cases of the damping, the energy loss of a harmonic oscillator can be approximated by a simple exponential decay.
Modeling Of The Discrete Energy Loss
In Section 7.3.3 the description of the interaction was discussed in terms of a probability distribution functionP(∆E,∆P,∆t). In this section the energy loss distribution function for the discrete damped oscillator will be derived.
To model a discrete damping for the harmonic oscillator with linear damping γx, a deterministic˙ formalism which removes a given quantum of energy at Ąxed times out of the system could be introduced. However a more natural choice is a probabilistic ansatz, which models the systemŠs initial total energy E0 as a sum of small energy quanta ∆E:
E0 =N0·∆E . (7.49)
103
N0 is the initial number of energy quanta and can also be called ŞsteppinessŤ because it deĄnes in which energy steps the system can be damped. The energy of the system can be damped by changing the energy quanta N(t)
E(t) =N(t)·∆E . (7.50)
This change ofN(t) has to be modeled and mapped to the equations of motion. We now explain how to Ąnd an interaction probability function like (7.34) for this system. In this example, we can assume a two-state interaction: for a given ∆t, the oscillator can lose a quantum of energy
∆E, or it can be left undisturbed. For ∆t≪1 we can neglect multiple decays; additionally we assume a Markov process, so the oscillator only depends on its current state and has no ŚmemoryŠ.
Using these constraints, the interaction probability functionP(∆E,∆t) without memory-kernel can be described as:
P(∆E,∆t) = Pr
loss(∆t)δ(∆E−∆E) + Pr
0 (∆t)δ(∆E) (7.51)
with Prloss being the probability to lose an energy quantum ∆E in the time interval ∆t. Pr0 is the probability for the system to stay unchanged. Both probabilities are related by the norm of the probability distribution
∫
P(∆E,∆t) d∆E = Pr
loss(∆t) + Pr
0 (∆t) = 1 (7.52)
To Ąnd the probability for the oscillator to lose a certain amount of energy, we assume that every energy quantum ∆E can decay independently. The deĄnition for the exponential decay is
dN(t)
dt =−γN(t) (7.53)
with each decay event having a constant and independent decay probability in a time step dt of Pr =γdt. With dt→∆t and ∆t≪1 we can write
∆N(t) =−γ∆tN(t) (7.54)
However, we want to calculate the probability of a single energy quantum to decay. The number of energy quanta is given by (7.50)
N(t) =E(t)/∆E (7.55)
which increases the number of energy quanta if ∆E decreased.
The total probability of a decay of a single quantum in a system of many quanta is in Ąrst-order approximation the sum of all single-probabilities
lossPr(∆E) =γ·∆t·N(t) =γ·∆t E(t)
∆E . (7.56)
Chapter 7 Particle-Wave Interaction Method For N0 → ∞ or ∆E → 0 this is the deĄnition of the exponential decay law, while a Ąnite N0 will give a discrete exponential decay for a Ąnite ensemble. For the total probability distribution functionP(∆E,∆t) we obtain
P(∆E,∆t) =δ(∆E−∆E)(γ ·∆tE(t)
∆E )
, +δ(∆E)(1−γ·∆tE(t)
∆E )
.
(7.57)
Simulating P(∆E,∆t) will give the same average energy loss scaling for E(t) as the original harmonic oscillator with continuous damping.
In a numerical realization, the oscillator is propagated with the free equation of motion, d2x
dt2 +ω20x= 0. (7.58)
This equation of motion conserves the total energy. To simulate damping, at every time step the decay probability density P(∆E) is sampled using Monte-Carlo techniques. In case of a decay, the oscillator will lose the given amount ∆E by employing the method described in Section7.3.1.
In case of a oscillator, only the energy equation (7.24) has to be solved. The changeδx onx(t) becomes a simple shift of the oscillator,
xt→xt+δx. (7.59)
For a harmonic potential, this can be done analytically by solving
∆E =E(xt+1)−E(xt)
= 1 2
[ω20x2t+1+ ˙x2t+1−ω02x2t −x˙2t]. (7.60) The derivatives are approximated by the Ąrst-order diference discretization and with dt→∆t:
˙
xt+1= xt+1−xt
∆t , (7.61)
∆E = 1
2∆t2 (xt+1−xn)2−1 2x˙2t− 1
2ω20x2t +1
2ω20x2t+1 . (7.62) Solving (7.60) for xt+1 results in
xt±∆t√2∆E+ 2∆E∆t2ω20+(∆t2ω02+ 1)x˙2t + ∆t2ω4x2t
1 + ∆t2ω20 . (7.63)
By neglecting higher order terms of order O(ω2∆t2)and higher results in the simple Ąrst-order approximation
xt+1 =xt±∆t√2∆E+ ˙x2t (7.64) 105
0 10 20 30 40 50
time (a.u.)
−2 .0
−1 .5
−1.0
−0.5 0.0 0.5 1.0 1
.5 2
.0
arbitraryunits
dynamically damped classical damped
energy (dynamically damped) energy (classical damped)
0 20 40 60 80 100 120 140
time (a.u.)
−2 .0
−1 .5
−1 .0
−0 .5 0.0 0.5 1.0 1.5 2.0
arbitraryunits
dynamically damped classical damped
energy (dynamically damped) energy (classical damped)
Figure 7.4: Simulation of discrete damped harmonic oscillators. The number of simulated particles / energy excitations isN = 150. With every decay of a particle, the oscillator loses a bit of energy, leading to damping of its motion. The discrete and continuous version have the same ensemble average. In a single run statistical deviation occurs.
This can be seen as an addition to the undisturbed equations of motion. With ∆E → 0 equation (7.64) becomes the usual, Ąrst order Euler propagation for a diferential equation:
xt+1 =xt+ ∆t·x˙t. The additional term ∆E is the change of the system given by the interaction-kick, changing the total systemŠs energy with exactly this amount of energy. The sign ±in front of the square root is determined by the direction of ˙xt, a kick with ∆E >0 should always point in the direction of the current velocity ˙xt. Note that ∆E can always be positive while it can only be negative if
˙
x2t ≥2\\\∆E\\\ (7.65)
to have a real solution for the propagation equation. Even if there would be enough potential energy to fulĄll this equation, a larger change of xt+1 would induce a larger kinetic energy than it should be dissipated from the system.
With these equations, a simple harmonic oscillator with discrete damping can be implemented.
Two example calculations are shown in Figure 7.4, where a oscillator is simulated with 150 possible energy steps ∆E/E0 = 150. Every time such an energy mode decays, the energy is taken out of the system. In the limit of ∆E →0, the the oscillator is damped continuously, again. On average, the continuously damped and the discretely damped oscillator have the same ensemble average. However, due to statistical Ćuctuations, both the amplitude and the phase can difer from time to time. In terms of Monte-Carlo simulations, this is highly intentionally.
Chapter 7 Particle-Wave Interaction Method