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2.5 Ponderomotive forces

The plasma droplet model of gravitation makes it possible, massless elementary particles and such with mass to interpret as objects that can vibrate in themselves. Furthermore, it is immediately clear that if the plasma droplet hypothesis should be correct, elementary particles are surrounded by oscillating force fields. This allows to break new ground in quantum mechanics. However, some basics are needed first, which will be explained in the following.

2.5.1 Introduction

The ponderomotive force is a remarkable effect that has received little attention in physics so far. For example, it is not even mentioned in [Schmutzer1988], a quite extensive standard textbook of theoretical physics. In textbooks about experimental physics, such as the [Pfeifer1997], there is no evidence of this either. Despite this low presence in literature, ponderomotive forces are some of the most interesting effects of physics. This section shows how these forces in respect to the plasma droplet model of gravity lead to phenomena that are practically indistinguishable from predictions of quantum mechanics.

Before the mathematical basics of the ponderomotive force are discussed, we will briefly explain what the ponderomotive force is. Animations 2.5.1.1 and 2.5.1.2 show a spatially inhomogeneous alternating field, whereby the electrical force periodically changes its sign in x-direction. The time average of the field strength is zero at each point. However, the simulations 2.5.1.1 and 2.5.1.2 show that electrically charged particles start moving nevertheless, regardless of the sign of the charge (negative: red, positive: blue) and always in the direction in which the amplitude of the field becomes weaker.
Figure 2.5.1.1
Figure 2.5.1.2

The reason for this behavior is quickly explained: The point charge is either moved by the force into an area where the field is even greater or weaker in magnitude. In the first case, after reversing the polarity of the force, the amount of the force pushing back is stronger than the force that previously brought the particle to this place. So the point charge is accelerated a little bit more when pushed back and therefore reaches a bit further into the area where the amplitude of the field is small. In the second case, it is exactly the opposite. The amount of the pushing force is smaller. The starting point is therefore not being reached any more. At the end of both cases, the particle has penetrated a little bit further into an area where the electrical field is smaller in its amplitude. This process is repeated over and over again. In the end, a drift velocity in the direction of the decreasing amplitude, independent of the sign of the charge, is produced.

However, the ponderomotive force does not only act on electrically charged particles, but also on particles that are polarizable but in sum electrically neutral. Such bound particles, for example, form the basis of the plasma droplet model of gravity. The simulation 2.5.1.3 illustrates the mechanism with an example.

Figure 2.5.1.3: Simulation of the behavior of a bound particle with strong coupling in the alternating field.
Figure 2.5.1.4: Simulation of the behavior of a bound particle with weak coupling in the alternating field.

It turns out this time that the particle moves in the direction in which the amplitude of the strength of the oscillating field increases. This is not generally the case, however, but only if the eigenfrequency is greater than the frequency of the electric field. This is especially the case with bound particles in which the electrical charges are coupled to each other by large forces. If, on the other hand, the coupling forces are small, the ponderomotive force points back in the direction in which the amplitude of the alternating field decreases. The simulation 2.5.1.4 provides an example of this.

2.5.2 Ponderomotive force effect on an electrically charged particle

This mechanism will now be investigated mathematically. We model the oscillating and spatially inhomogeneous electric field $\vec{E}$ by
$$\vec{E}(\vec{r},t) = \vec{E}_r(\vec{r})\,\cos(\omega\,t).$$ (2.5.2.1)
The equation of motion is
$$\frac{m}{q}\,\ddot{\vec{r}} = \vec{E}\left(\vec{r},t\right) = \vec{E}_r(\vec{r})\,\cos(\omega\,t).$$ (2.5.2.2)
As animations 2.5.1.1 and 2.5.1.2 have shown, the trajectory of the point charges consists of a fast oscillating part $\vec{r}_o$ and a slow drift part $\vec{r}_d$. Therefore we start with ansatz $\vec{r} = \vec{r}_d + \vec{r}_o$.

If the angular frequency $\omega$ is assumed to be very high, the oscillation amplitude $\vec{r}_o$ is very small. For this reason,
$$\vec{E}_r(\vec{r}_d+\vec{r}_o) \approx \vec{E}_r(\vec{r}_d) + \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\vec{r}_o$$ (2.5.2.3)
is a good approximation, whereby
$$\nabla\otimes\vec{E} = \left(\begin{matrix} \frac{\partial}{\partial\,r_x}E_x & \frac{\partial}{\partial\,r_y}E_x & \frac{\partial}{\partial\,r_z}E_x \\ \frac{\partial}{\partial\,r_x}E_y & \frac{\partial}{\partial\,r_y}E_y & \frac{\partial}{\partial\,r_z}E_y \\ \frac{\partial}{\partial\,r_x}E_z & \frac{\partial}{\partial\,r_y}E_z & \frac{\partial}{\partial\,r_z}E_z \end{matrix}\right)$$ (2.5.2.4)
stands for the Jacobi matrix. If we insert this into the equation of motion (2.5.2.2), we get
$$\frac{m}{q}\,\left(\ddot{\vec{r}}_d + \ddot{\vec{r}}_o\right) \approx \left(\vec{E}_r(\vec{r}_d) + \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\vec{r}_o\right)\,\cos(\omega\,t).$$ (2.5.2.5)
The next step exploits the fact that the acceleration $\ddot{\vec{r}}_d$, which leads to the drift movement, is much smaller than the acceleration $\ddot{\vec{r}}_o$. For this reason and because of the low oscillation amplitude $\vec{r}_o$, we get the approximation
$$\frac{m}{q}\,\ddot{\vec{r}}_o \approx \vec{E}_r(\vec{r}_d)\,\cos(\omega\,t)$$ (2.5.2.6)
from the equation of motion (2.5.2.5). Since $\vec{r}_d$ is only changing very slowly compared to $\vec{r}_o$, $\vec{E}_r(\vec{r}_d)$ is essentially a constant in the considered time period, which makes it possible to solve the differential equation. It follows
$$\vec{r}_o \approx -\frac{q}{m\,w^2}\,\vec{E}_r(\vec{r}_d)\,\cos(\omega\,t).$$ (2.5.2.7)
By inserting this into equation (2.5.2.5) we get
$$\frac{m}{q}\,\left(\ddot{\vec{r}}_d + \frac{q}{m}\,\vec{E}_r(\vec{r}_d)\,\cos(\omega\,t)\right) \approx \left(\vec{E}_r(\vec{r}_d) - \frac{q}{m\,\omega^2} \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\vec{E}_r(\vec{r}_d)\,\cos(\omega\,t)\right)\,\cos(\omega\,t),$$ (2.5.2.8)
which can be simplified by a rearrangement into
$$\ddot{\vec{r}}_d \approx -\frac{q^2}{m^2\,\omega^2} \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\vec{E}_r(\vec{r}_d)\,\cos(\omega\,t)^2.$$ (2.5.2.9)
The term $\cos(\omega\,t)^2$ makes in average a contribution of
$$\overline{\cos(\omega\,t)^2} = \lim\limits_{T\to\infty}\frac{1}{T}\int\limits_0^{T}\,\cos(\omega\,t)^2\,\mathrm{d}t = \lim\limits_{T\to\infty}\left(\frac{1}{2} + \frac{1}{4\,T\,\omega}\,\sin(2\,\omega\,T)\right) = \frac{1}{2}.$$ (2.5.2.10)
Finally, the approximation
$$\ddot{\vec{r}}_d \approx -\frac{q^2}{2\,m^2\,\omega^2} \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\vec{E}_r(\vec{r}_d)$$ (2.5.2.11)
follows and the ponderomotive force $\vec{F}_p$ can be expressed by equation
$$\vec{F}_p = -\frac{q^2}{2\,m\,\omega^2} \nabla\otimes\vec{E}_r\cdot\vec{E}_r.$$ (2.5.2.12)
If $\vec{E}_r$ can be expressed as gradient $-\nabla\varphi_r$ of a scalar potential $\varphi_r$,
$$\nabla\otimes\vec{E}_r\cdot\vec{E}_r = \frac{1}{2}\nabla\left(\vec{E}_r\cdot\vec{E}_r\right) = \frac{1}{2}\,\nabla E_r^2$$ (2.5.2.13)
applies (but only then) because it is
$$\left(\nabla\otimes\nabla\varphi_r\right)\cdot\nabla\varphi_r = \frac{1}{2}\nabla\left(\nabla\varphi_r\cdot\nabla\varphi_r\right).$$ (2.5.2.14)
Thus the formula (2.5.2.12) can be simplified again considerably and we get
$$\vec{F}_p = -\frac{q^2}{4\,m\,\omega^2} \nabla E_r^2\quad(\text{für}\, \nabla\times\vec{E}_r=0).$$ (2.5.2.15)
It also becomes clear that the ponderomotive force itself can also be understood as a gradient field $\vec{F}_p = -\nabla\varphi_p$. The ponderomotive potential is in this case
$$\varphi_p = \frac{q^2\,E_r^2}{4\,m\,\omega^2}\quad(\text{für}\, \nabla\times\vec{E}_r=0).$$ (2.5.2.16)


2.5.3 Ponderomotive force effect on an electrically neutral particle

This time a bound, electrically neutral particle consisting of a negative charge $-Q/2$ and a positive charge $+Q/2$ is considered. In the following calculation, the force between the two charges is not modeled by a Coulomb potential, but by a harmonic oscillator with the coupling constant $k$.

The oscillating, spatially inhomogeneous electric field $\vec{E}$ is again defined by
$$\vec{E}(\vec{r},t) = \vec{E}_r(\vec{r})\,\cos(\omega\,t).$$ (2.5.3.1)
The equations of motion are therefore
$$m_n\,\ddot{\vec{r}}_n = -\frac{Q}{2}\,\vec{E}_r(\vec{r}_n)\,\cos(\omega\,t) + k\,\left(\vec{r}_p - \vec{r}_n\right)$$ (2.5.3.2)
and
$$m_p\,\ddot{\vec{r}}_p = \frac{Q}{2}\,\vec{E}_r(\vec{r}_p)\,\cos(\omega\,t) + k\,\left(\vec{r}_n - \vec{r}_p\right).$$ (2.5.3.3)
Hereby, $\vec{r}_n$ is the trajectory of the negative charge and $\vec{r}_p$ is the trajectory of the positive charge. $m_n$ and $m_p$ are the associated masses, which need not necessarily be equal. An example is the hydrogen atom, where the positively charged proton has significantly higher mass than the electron which carries the negative charge.

Again, the solutions consist of a slow drift part $\vec{r}_d$, which describes the movement of the system center of gravity, and a fast oscillating part $\vec{r}_{o}$. Therefore the ansatzes $\vec{r}_n = \vec{r}_{d} + \vec{r}_{on}$ and $\vec{r}_p = \vec{r}_{d} + \vec{r}_{op}$ are used. The drift component $\vec{r}_d$ for the bound particle is of course identical for both charges and equal to the trajectory of the center of gravity.

By use of the approximation (2.5.2.3), the equations of motion (2.5.3.2) and (2.5.3.3) become
$$m_n\,\left(\ddot{\vec{r}}_{d}+\ddot{\vec{r}}_{on}\right) \approx -\frac{Q}{2}\,\left(\vec{E}_r(\vec{r}_{d}) + \nabla\otimes\vec{E}_r(\vec{r}_{d})\cdot\vec{r}_{on}\right)\,\cos(\omega\,t) + k\,\left(\vec{r}_{op} - \vec{r}_{on}\right)$$ (2.5.3.4)
and
$$m_p\,\left(\ddot{\vec{r}}_{d}+\ddot{\vec{r}}_{op}\right) \approx \frac{Q}{2}\,\left(\vec{E}_r(\vec{r}_{d}) + \nabla\otimes\vec{E}_r(\vec{r}_{d})\cdot\vec{r}_{op}\right)\,\cos(\omega\,t) + k\,\left(\vec{r}_{on} - \vec{r}_{op}\right).$$ (2.5.3.5)
With the approximations $\ddot{\vec{r}}_{d} \ll \ddot{\vec{r}}_{on}$, $\ddot{\vec{r}}_{d} \ll \ddot{\vec{r}}_{op}$, $\vec{r}_{on} \approx 0$ and $\vec{r}_{op} \approx 0$ this can be simplified to
$$m_n\,\ddot{\vec{r}}_{on} \approx -\frac{Q}{2}\,\vec{E}_r(\vec{r}_{d})\,\cos(\omega\,t) + k\,\left(\vec{r}_{op} - \vec{r}_{on}\right)$$ (2.5.3.6)
and
$$m_p\,\ddot{\vec{r}}_{op} \approx \frac{Q}{2}\,\vec{E}_r(\vec{r}_{d})\,\cos(\omega\,t) + k\,\left(\vec{r}_{on} - \vec{r}_{op}\right).$$ (2.5.3.7)
This system of differential equations can be solved by assuming that $\vec{E}_r(\vec{r}_{d})$ is essentially only a time-independent constant at the time of consideration. The solutions are
$$\vec{r}_{on} = \frac{Q\,\vec{E}_r(\vec{r}_{d})\,\left(\cos\left(\omega\,t\right)-\cos\left(\omega_e\,t\right)\right)}{2\,m_n\left(\omega^2-\omega_e^2\right)}$$ (2.5.3.8)
and
$$\vec{r}_{op} = -\frac{Q\,\vec{E}_r(\vec{r}_{d})\,\left(\cos\left(\omega\,t\right)-\cos\left(\omega_e\,t\right)\right)}{2\,m_p\left(\omega^2-\omega_e^2\right)},$$ (2.5.3.9)
whereby $\omega_e$ represents the angular eigenfrequency $\omega_e = \sqrt{\frac{k}{m_{red}}}$ of the bound particle and $m_{red} = \frac{m_n\,m_p}{m_n + m_p}$ the reduced mass.

The center of mass $\vec{r}_d$ is
$$\vec{r}_d = \frac{m_p\,\vec{r}_p + m_n\,\vec{r}_n}{m_p + m_n}.$$ (2.5.3.10)
A double differentiation with respect to time gives
$$\ddot{\vec{r}}_d = \frac{m_p\,\ddot{\vec{r}}_p + m_n\,\ddot{\vec{r}}_n}{m_p + m_n}.$$ (2.5.3.11)
Inserting the right sides of equations (2.5.3.2) and (2.5.3.3) therefore provides
$$\ddot{\vec{r}}_d = \frac{Q}{2\,(m_p + m_n)} \left(\vec{E}_r(\vec{r}_p)-\vec{E}_r(\vec{r}_n)\right)\,\cos(\omega\,t).$$ (2.5.3.12)
With the approximation (2.5.2.3) we get
$$\ddot{\vec{r}}_d = \frac{Q}{2\,(m_p + m_n)} \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\left(\vec{r}_{op}-\vec{r}_{on}\right)\,\cos(\omega\,t).$$ (2.5.3.13)
Here now the solutions (2.5.3.8) and (2.5.3.9) can be used and we obtain
$$\ddot{\vec{r}}_d = -\frac{Q^2\,\left(\cos\left(\omega\,t\right)-\cos\left(\omega_e\,t\right)\right)\,\cos(\omega\,t)}{4\,m_p\,m_n\,\left(\omega^2-\omega_e^2\right)} \nabla\otimes\vec{E}_r(\vec{r}_d)\cdot\vec{E}_r(\vec{r}_{d}).$$ (2.5.3.14)
In the next step, the time is averaged to eliminate the fast oscillations of the two charges, which are insignificant for the movement of the center of gravity. With
$$\lim\limits_{T\to\infty}\frac{1}{T}\int\limits_{0}^{T} \left(\cos\left(\omega\,t\right)-\cos\left(\omega_e\,t\right)\right)\,\cos(\omega\,t)\,\mathrm{d}t = \frac{1}{2}$$ (2.5.3.15)
we then get the ponderomotive force of a bound particle. The general formula is
$$\vec{F}_p = -\frac{Q^2}{8\,m_{red}\,\left(\omega^2-\omega_e^2\right)} \nabla\otimes\vec{E}_r\cdot\vec{E}_r$$ (2.5.3.16)
with
$$m_{red} = \frac{m_n\,m_p}{m_n + m_p} \quad \text{und}\quad \omega_e = \sqrt{\frac{k}{m_{red}}}.$$ (2.5.3.17)
If $\vec{E}_r=-\nabla\varphi_r$ can be can be expressed as the gradient of a potential $\varphi_r$, the ponderomotive force can be simplified because of formula (2.5.2.13) to
$$\vec{F}_p = -\frac{Q^2\,\nabla\,E_r^2}{16\,m_{red}\,\left(\omega^2-\omega_e^2\right)}\quad(\text{for}\, \nabla\times\vec{E}_r=0)$$ (2.5.3.18)
with the ponderomotive potential
$$\varphi_p = \frac{Q^2\,E_r^2}{16\,m_{red}\,\left(\omega^2-\omega_e^2\right)}\quad(\text{for}\, \nabla\times\vec{E}_r=0).$$ (2.5.3.19)

It is important to point out the meaning of the term $\omega^2-\omega_e^2$ below the fraction line in the pre-factor. This namely seems to imply that for $\omega^2 = \omega_e^2$ an infinitely large ponderomotive force occurs. This is not correct in practice, since under these circumstances the assumptions that led to formula (2.5.3.18) are not valid. Numerical simulations show that the bound particle is in resonance and begins to oscillate violently until algorithmic instabilities occur. However, the center of gravity of the bound particle does not change its position. It is true, however, that the ponderomotive force for angular frequencies $\omega$ near the eigenfrequency $\omega_e$ is particularly strong. The direction of the force depends on whether $\omega^2 < \omega_e^2$ or $\omega^2 > \omega_e^2$ applies. For frequencies above the eigenfrequency, the ponderomotive force acts in such a way that bound particles are dragged to where the amplitude increases. In the other case, the areas of high amplitude or intensity have a repulsive effect. The animations 2.5.1.3 and 2.5.1.4 provide illustrative examples for both cases.