text:
Subsections

### D.45 Motion through crystals

This note derives the semi-classical motion of noninteracting electrons in crystals. The derivations will be one-dimensional, but the generalization to three dimensions is straightforward.

#### D.45.1 Propagation speed

The first question is the speed with which a more or less localized electron moves. An electron in free space moves with a speed found by dividing its linear momentum by its mass. However, in a solid, the energy eigenfunctions are Bloch waves and these do not have definite momentum.

Fortunately, the analysis for the wave packet of a free particle is virtually unchanged for a particle whose energy eigenfunctions are Bloch waves instead of simple exponentials. In the Fourier integral (7.64), simply add the periodic factor . Since this factor is periodic, it is bounded, and it plays no part in limit process of infinite time. (You can restrict the times in the limit process to those at which is always at the same position in the period.)

As a result the group velocity is again . Since the energy is and the crystal momentum , the velocity of a localized electron can be written as

In the absence of external forces, the electron will keep moving with the same velocity for all time. The large time wave function is

where is the wave number at which the group speed equals . Note that the wave function looks locally just like a single Bloch wave for large time.

#### D.45.2 Motion under an external force

The acceleration due to an external force on an electrons is not that straightforward. First of all note that you cannot just add a constant external force. A constant force would produce an external potential of the form and that becomes infinite at infinite . However, it can be assumed that the force is constant over the nonzero range of the wave packet.

Next there is a trick. Consider the expectation value of the translation operator that translates the wave function over one atomic cell size . If the wave packet consisted of just a single Bloch wave with wave number , the expectation value of would be . A wave packet must however include a small range of values. Then will be an average of values over the values of the wave packet. Still, if the range of values is small enough, you can write

where is a value somewhere in the middle of the wave packet and is a real number close to one. So still gives the typical value in the wave packet.

Moreover, its magnitude is always less than one and the closer it is to one, the more compact the wave packet. That is because is an average of values. These are all located on the unit circle in the complex plane, the plane with as the horizontal axis and as the vertical axis. If the wave packet would consist of just a single value , then the average of would be exactly , and be on the unit circle. If however the wave numbers spread out a bit around , then the average moves inside the unit circle: if you average positions on a circle, the average is always inside the circle. In the extreme case that the values get uniformly distributed over the entire circle, the average position is at the origin. That would make zero. Conversely, as long as stays very close to one, the wave packet must be very compact in terms of .

The time evolution of can be found using chapter 7.2:

 (D.28)

where is the Hamiltonian for the electron in the crystal, and the additional external potential. Now the commutator of and is zero; the crystal Hamiltonian acts exactly the same on the wave function whether it is shifted one cell over or not. The remainder of the commutator gives, when applied on an arbitrary wave function,

Writing this out with the arguments of the functions explicitly shown gives:

Now assume that the external force is constant over the extent of the wave packet. In that case the difference in the potentials is just , and that is a constant that can be taken out of the expectation value of the commutator. So:

 (D.29)

The solution to this equation is:

where is the value of at the starting time 0.

It follows that the magnitude of the does not change with time. In view of the earlier discussion, this means that the wave packet maintains its compactness in terms of . (In physical space the wave packet will gradually spread out, as can be seen from the form of the large-time wave function given earlier.)

It further follows that the average wave number in the wave packet evolves as:

Since the packet remains compact, all wave numbers in the wave packet change the same way. This is Newton’s second law in terms of crystal momentum.

#### D.45.3 Free-electron gas with constant electric field

This book discussed the effect of an applied electric field on free electrons in a periodic box in chapter 6.20. The effect was described as a change of the velocity of the electrons. Since the velocity is proportional to the wave number for free electrons, the velocity change corresponds to a change in the wave number. In this subsection the effect of the electric field will be examined in more detail. The solution will again be taken to be one-dimensional, but the extension to three dimensions is trivial.

Assume that a constant electric field is applied, so that the electrons experience a constant force . The time-dependent Schrödinger equation is

Assume the initial condition to be

in which a subscript 0 indicates the initial time.

The exact solution to this problem is

where the magnitude of the coefficients is independent of time. This exact solution is in terms of states that change in time. The probability of the particle being in those states does not change.

Unfortunately, this solution is only periodic with period equal to the length of the box for times at which happens to be a whole multiple of the wave number spacing. At those times the Fermi sphere of occupied states has shifted the same whole multiple of wave number spacings to the right.

At intermediate times, the solution is not periodic, so it cannot be correctly described using the periodic box modes. The magnitude of the wave function is still periodic. However, the momentum has values inconsistent with the periodic box. The problem is that even though a constant force is periodic, the corresponding potential is not. Since quantum mechanics uses the potential instead of the force, the quantum solution is no longer periodic.

The problem goes away by letting the periodic box size become infinite. But that brings back the ugly normalization problems. For a periodic box, the periodic boundary conditions will need to be relaxed during the application of the electric field. In particular, a factor difference in wave function and its -derivative must be allowed between the ends of the box. Since the periodic boundary conditions are artificial anyway for modeling a piece of electrical wire, this may not be a big concern. In any case, for a big-enough periodic box, the times at which the solution returns to its original periodicity become spaced very close together.


text:
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## anonymous 4 years ago Graph the function

• This Question is Closed
1. anonymous

|dw:1343429638361:dw|

2. anonymous

$y=\frac{14}{5}x$or $y=\frac{14}{5x}$

3. anonymous

second one

4. anonymous

Use a graphing device to see how it looks like. Then get some points by plugging in x and graph accordingly

5. anonymous
6. anonymous

thank you

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# Inconsistent spacing of suffix in siunitx formatted table

I want to produce a table with two prices in a row, using siunitx to achieve a pleasant and consistent display of the numbers. Trying not to repeat the currency symbol in the data rows, I insert it as a suffix into the relevant columns. This works well with one price tag per row; with a second one, however, siunitx seems to increase the space reserved for the suffix with each occurrence per row. Here is a pretty minimal example:

\documentclass{scrartcl}
\usepackage{polyglossia}
\usepackage{siunitx}
\sisetup{table-number-alignment=right}
\newcolumntype{E}[0]{S[
table-figures-decimal=0,
table-align-text-post=true,
table-space-text-post={€}]<{€}}
\begin{document}
\begin{tabular}{EE}
63 & 126 \\
49 & 49 \\
\end{tabular}
\end{document}

If I get the format of the first column right, there is too much whitespace in the second one. Getting the second one right, moves the currency symbol into the figures of the first column.

(In case it matters: I'm using lualatex from the Debian texlive packages, version 2015.20160117-1.)

Any suggestions how I can get this working?

• Manual, p. 71 "Adding items after the last column of a tabular" – Joseph Wright Feb 10 '16 at 15:53
• Thank you! The \cr helps. Since this didn't hit me as long as I only had one S column per row (even if it was the last one), I must confess I ignored this hint. – DetlefL Feb 10 '16 at 16:03

% arara: lualatex

\documentclass{scrartcl}
\usepackage{polyglossia}
\usepackage{siunitx}
\newcolumntype{E}[1]{S[%
,table-format=#1
,table-space-text-post={\,€}]<{\,€}
}

\begin{document}
% Quote taken from the manual page 71
When processing tables, \verb|siunitx| will expand anything stored
inside a macro, unless it is long or protected. \LaTeXe{} robust
commands are also detected and are not expanded.
Values which would otherwise be expanded
can be protected by wrapping them in a set of braces. As \TeX{} itself
will expand the first token in a table cell before \verb|siunitx|
can act on it, using the $\varepsilon$-\TeX{} protected mechanism is the
recommended course of action to prevent expansion of macros in
table cells. (As is shown in the table, \TeX's expansion of
\LaTeXe{} robust commands can lead to unexpected results.)
\bigskip

\begin{tabular}{E{2.0}E{3.0}}
63 & 126 \cr
49 &  49 \cr
\end{tabular}
\end{document}


text:
# §8.13 Zeros

## §8.13(i) $x$-Zeros of $\gamma^{*}\left(a,x\right)$

The function $\gamma^{*}\left(a,x\right)$ has no real zeros for $a\geq 0$. For $a<0$ and $n=1,2,3,\dots$, there exist:

1. (a)

one negative zero $x_{-}(a)$ and no positive zeros when $1-2n;

2. (b)

one negative zero $x_{-}(a)$ and one positive zero $x_{+}(a)$ when $-2n.

The negative zero $x_{-}(a)$ decreases monotonically in the interval $-1, and satisfies

 8.13.1 $1+a^{-1} $-1. ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $a$: parameter Permalink: http://dlmf.nist.gov/8.13.E1 Encodings: TeX, pMML, png See also: Annotations for §8.13(i), §8.13 and Ch.8

When $-5\leq a\leq 4$ the behavior of the $x$-zeros as functions of $a$ can be seen by taking the slice $\gamma^{*}\left(a,x\right)=0$ of the surface depicted in Figure 8.3.6. Note that from (8.4.12) $\gamma^{*}\left(-n,0\right)=0$, $n=1,2,3,\dots$.

For asymptotic approximations for $x_{+}(a)$ and $x_{-}(a)$ as $a\to-\infty$ see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012).

## §8.13(ii) $\lambda$-Zeros of $\gamma\left(a,\lambda a\right)$ and $\Gamma\left(a,\lambda a\right)$

For information on the distribution and computation of zeros of $\gamma\left(a,\lambda a\right)$ and $\Gamma\left(a,\lambda a\right)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a).

## §8.13(iii) $a$-Zeros of $\gamma^{*}\left(a,x\right)$

For fixed $x$ and $n=1,2,3,\dots$, $\gamma^{*}\left(a,x\right)$ has:

1. (a)

two zeros in each of the intervals $-2n when $x<0$;

2. (b)

two zeros in each of the intervals $-2n when $0;

3. (c)

zeros at $a=-n$ when $x=0$.

As $x$ increases the positive zeros coalesce to form a double zero at ($a_{n}^{*},x_{n}^{*}$). The values of the first six double zeros are given to 5D in Table 8.13.1. For values up to $n=10$ see Kölbig (1972b). Approximations to $a_{n}^{*}$, $x_{n}^{*}$ for large $n$ can be found in Kölbig (1970). When $x>x_{n}^{*}$ a pair of conjugate trajectories emanate from the point $a=a_{n}^{*}$ in the complex $a$-plane. See Kölbig (1970, 1972b) for further information.


text:
1. ## Laplace transform help

Hey guys, I need some help with this problem:
[tex]\int e^-^s^t cost dt lower limit is 0 and the upper limit is \infty [\math]

The problem I am running into is when I use integration by parts. I let u = cost and dv = [tex] e^-^s^[\math]

I am somehow getting stuck with an infinite integration by parts problem. Any help is greatly appreciated!!! Thank you!

Sorry, for some reason the latex is not working the way I want it to.......

2. Let

$I=\int_{0}^{\infty}e^{-st}\cos(t)\,dt.$

Let

$u=e^{-st},\quad du=-se^{-st}\,dt,\quad dv=\cos(t)\,dt,\quad v=\sin(t).$

We get

$I=e^{-st}\sin(t)\Big|_{0}^{\infty}+\int_{0}^{\infty}se^{-st}\sin(t)\,dt=s\int_{0}^{\infty}e^{-st}\sin(t)\,dt.$

Next, we let

$u=e^{-st},\quad du=-se^{-st}\,dt,\quad dv=\sin(t)\,dt,\quad v=-\cos(t).$

We get

$I=s\left[-e^{-st}\cos(t)\Big|_{0}^{\infty}-s\int_{0}^{\infty}e^{-st}\cos(t)\,dt\right]=s\left[1-sI\right].$

It is that last step there, recognizing that we've arrived back at I, that is the crucial one. You now have an equation for I. Solve for it.

3. I think we're back to [TEX ] ... [/TE X]

4. Originally Posted by Jeonsah
Hey guys, I need some help with this problem:
[tex]\int e^-^s^t cost dt lower limit is 0 and the upper limit is \infty [\math]

The problem I am running into is when I use integration by parts. I let u = cost and dv = [tex] e^-^s^[\math]

I am somehow getting stuck with an infinite integration by parts problem. Any help is greatly appreciated!!! Thank you!

Sorry, for some reason the latex is not working the way I want it to.......
Don't use integration by parts! Either use the complex exponential form for $\cos (t)$:

$\cos(t) = \frac{e^{it}+e^{-it}}{2}$

$\int_0^{\infty}e^{-st}\cos(t)\; dt= \text{re}\left(\int_0^{\infty}e^{-st}e^{it}\; dt \right)$


text:
## Introduction

Patient-level data build the foundation for a plethora of healthcare research endeavors such as drug discovery, clinical trials, biomarker discovery, and precision medicine1. Collecting such data is extremely time-consuming and cost-intensive, and additionally access-restricted by ethical and legal regulations in most countries. Individual organizations, such as hospitals, pharmaceutical companies, and health insurance providers are currently limited in their ability to collect data that are fully representative of a disease population. This issue is especially pronounced in clinical studies, where patients are usually recruited based on predefined inclusion and exclusion criteria that introduce cohort-specific statistical biases2. These biases, in turn, can negatively impact the generalization ability of machine learning models, since the usual i.i.d. assumption is violated3. A naive idea to counteract this issue might be to build up large data repositories pooling diverse clinical studies from several organizations. However, here, a major obstacle is that sharing patient-level data across different organizations is exceedingly difficult due to legal restrictions, as formulated, for example, in the General Data Protection Rule of the European Union.

The idea we propagate in this paper is to learn a continuous-time generative machine learning model from clinical study data. Given the distribution of the real training data was appropriately learned by such a model, the generated synthetic datasets maintain the real data signals, such as variable interdependencies and time-dependent trajectories. Furthermore, these synthetic datasets can overcome crucial limitations of their real counterparts like missing values or irregular assessment intervals, hence opening the opportunity to make at least subsets of variables from different studies statistically comparable. A further strong motivation for generating synthetic datasets is the aim to use the generated data as an anonymized version of its real-world counterpart and thereby mitigate the increased restrictions for sharing human data4,5,6. However, synthetic patient-level datasets open opportunities that reach far beyond data sharing. For example, trained generative models could be used for synthesizing control arms for clinical trials based on data from previously conducted trials, or from real-world clinical routine data7. This helps addressing major ethical concerns in disease areas, such as cancer, where it is impossible to leave patients untreated. Both, the American Food and Drug Administration and the European Medicines Agency have recognized this issue and taken initiatives to allow for synthetic control arms7.

Over the last years, generative models (mostly generative adversarial networks [GANs]) have found notable success, mostly in the medical imaging domain8,9,10,11,12,13. However, GANs are often found to show a collapse in the statistical mode of a distribution, which raises concerns regarding coverage of the real patient distribution by synthetic data. Moreover, these methods are not necessarily suited to cope with the complex nature of clinical data collected in observational, longitudinal cohort studies, which is the main focus of our work: In addition to the previously mentioned issue of irregular measurement frequencies and missing values not at random (e.g., due to participant drop-out), clinical studies often comprise several modalities combining time-dependent variables (e.g., measures of disease severity) and static information (e.g., biological sex). One approach specifically designed for the joint modeling and generation of multimodal, time-dependent, and static patient-level data containing missing values is the recently introduced Variational Autoencoder Modular Bayesian Networks (VAMBN)4. However, VAMBN only operates on a discrete time scale while relevant clinical indicators such as, for example, disease progression expressed through a cognitive decline or rising inflammatory markers, are intrinsically time continuous. Recently, Neural Ordinary Differential Equations (NODEs) have been introduced as a hybrid approach fusing neural networks and Ordinary Differential Equations (ODEs)14. While NODEs are time continuous and thus enable smooth interpolation between observed data points and extrapolation beyond the observations in the data, they are not able to integrate static variables.

In this work, we present the Multimodal Neural Ordinary Differential Equations (MultiNODEs) as an extension of the NODEs. MultiNODEs allow learning a generative model from multimodal longitudinal and static data that may contain missing values not at random. To demonstrate MultiNODEs’ generative capabilities, we applied the model to clinical, patient-level data from an observational Parkinson’s disease (PD) cohort study (the Parkinson’s Progression Markers Initiative [PPMI]15) and, additionally, a longitudinal Alzheimer’s disease (AD) data collection (National Alzheimer’s Coordination Center [NACC]16). We compared the generated trajectories and correlation structure with the real counterpart. In this context, we additionally evaluated MultiNODEs’ performance against the previously published VAMBN approach. Furthermore, we assessed MultiNODEs’ interpolation and extrapolation performance. Finally, we investigated the influence of sample size, noisiness of the data, and longitudinal assessment density on the training of MultiNODEs in a systematic benchmark on data simulated from a mathematical model well-known in the epidemiology field.

## Results

### Conceptual introduction of the MultiNODEs

MultiNODEs represent an extension of the original NODEs framework14 that overcomes the limitations of its predecessor such that an application to incomplete datasets consisting of both static and time-dependent variables becomes feasible. Conceptually, MultiNODEs build on three key components (Fig. 1): (1) latent NODEs, (2) a variational autoencoder (more specifically a Heterogenous Incomplete Variational Autoencoder [HI-VAE], designed to handle multimodal data with missing values17), and (3) an implicit imputation layer18. The latent NODEs enable the learning and subsequent generation of continuous longitudinal variable trajectories. The longitudinal properties of the initial condition (i.e., the starting point for the ODE system solver of the latent NODEs) are defined by the output of a recurrent variational encoder that embeds the longitudinal input data into a latent space (Fig. 1, orange box). To allow for an additional influence of static variables on the estimation of the longitudinal variable trajectories, the second component, a HI-VAE, is introduced (Fig. 1, blue box). This component transforms the static information into a distinct latent space and the resulting embedding is used to augment the latent starting condition of the NODEs by concatenating the static variable embedding and the latent representation of the longitudinal variables (Fig. 1, “augmentation”). The HI-VAE component itself holds generative properties and conducts the synthesis of the static variables when MultiNODEs are applied in a generative setting. Conclusively, MultiNODEs integrate static variables (e.g., biological sex or genotype information) both to inform the learning of longitudinal trajectories, and in the generative process. Finally, to mitigate the original NODEs’ incapability of dealing with missing values, we introduced the imputation layer which implicitly replaces missing values during model training with learned estimates (Fig. 1, green box). For further details on the model architecture, training, and hyperparameter optimization, we refer to the Method section and Supplementary material, respectively.

### Synthetic data generation using MultiNODEs

Generating synthetic data using MultiNODEs starts by randomly sampling a latent representation for both the static and longitudinal variables, respectively. The longitudinal variables in data space are then generated by first constructing the initial conditions of the latent ODE system (i.e., concatenating the static latent representation to the longitudinal one), followed by solving the ODE system given these initial conditions, and finally by decoding the result into data space. The static variables are generated by directly transforming their sampled latent representation into data space using the HI-VAE decoder.

MultiNODEs support two different approaches for the initial sampling of the latent representations, namely sampling from the prior distribution employed during model training and sampling from the learned posterior distribution of the input data.

During the posterior sampling procedure, the reparameterization trick19 is applied to draw a latent representation from the posterior distribution learned from the training data. The amount of noise added in this process can be tuned, whereas greater noise will lead to a wider spread of the generated marginal distributions of the synthetic data. Alternatively, the latent representations can be sampled from the prior distributions imposed on the latent space during variational model training. We ensure statistical dependence between static and longitudinal variables by drawing their values from a Bayesian network that connects both latent representations such that the longitudinal variables are conditionally dependent on the static variables. More detailed descriptions of both generation procedures are provided in the Method section.

### Application cases: Parkinson’s disease and Alzheimer’s disease

We applied MultiNODEs to longitudinal, multimodal data from two independent clinical datasets with the goal of generating realistic synthetic datasets that maintain the real data properties. Details about the data preprocessing steps are described in the Supplementary material.

The first dataset was the PPMI, an observational clinical study containing 354 de-novo PD patients who participated in a range of clinical, neurological, and demographic assessments which form the variables of the dataset. In total, a set of 25 longitudinal and 43 static variables was investigated.

Furthermore, as a second example, we applied MultiNODEs to longitudinal, multimodal data from the NACC. NACC is a database storing patient-level AD data collected across multiple memory clinics. After preprocessing, the dataset used in this study contained 2284 patients, and a set of three longitudinal and four static variables was investigated.

In the following sections, we will focus on the results achieved on the PPMI data and refer to the equivalent experiments based on the NACC data that are presented in the Supplementary material.

### MultiNODEs generate realistic synthetic patient-level datasets

We applied prior as well as posterior sampling for comparison purposes. With each method, we generated the same number of synthetic patients as encountered in the real dataset to allow for a fair comparison. To assess whether the generated data followed the real data characteristics, we conducted thorough comparisons of the marginal distributions using qualitative and visual assessments and further, quantitatively compared the Jensen–Shannon divergence (JS-divergence) between the generated data and real distributions. The JS-divergence is bound between 0 and 1 with 0 indicating equal distributions. In addition, we investigated the underlying correlation structure of the measured variables. Finally, we trained a machine learning classifier (Random Forest) that evaluated whether real and synthetic patients showed similar clinical characteristics when compared to real healthy control individuals from their respective studies. Across all these aspects, we evaluated MultiNODEs’ performance in comparison to the previously published VAMBN approach4.

The synthetic data generated using MultiNODE generally exhibited marginal distributions that bore high similarity to their corresponding real counterparts (Fig. 2, Supplementary Table 1, and Supplementary Fig. 1; equivalent figures for the NACC data are presented in Supplementary Fig. 4). The average JS-divergences between the real and synthetic distributions calculated across all variables and timepoints amounted to 0.018 ± 0.015 and 0.011 ± 0.009 for the PPMI data generated from the prior and posterior, respectively. For NACC the average JS-divergence was 0.071 ± 0.055 and 0.029 ± 0.031 for prior and posterior sampling, respectively. With respect to PPMI, data generation from the posterior distribution resulted in synthetic data that resembled the real data significantly closer than those generated from the prior distribution (Mann–Whitney U test, p < 0.02).

Compared to VAMBN, the prior sampling method seemed to be inferior with respect to the average JS-divergence when using NACC (U test, p = 0.038). However, no statistically significant difference in the performance of VAMBN compared to MultiNODE’s posterior sampling could be observed (U test, p = 0.80). For PPMI, no significant differences were found between VAMBN and any of MultiNODEs’ generation approaches (U test, p = 0.31 for the prior approach; U test, p = 0.24 for the posterior).

In order to evaluate whether MultiNODEs learned not only to reproduce univariate distributions but actually captured their interdependencies accurately, we compared the correlation structure of the generated data to that of the real variables. Visualizations of the Spearman rank correlation coefficients showed that both the prior and posterior sampling generated synthetic data which successfully reproduced the real variables’ interdependencies (Fig. 3). Comparing the results against VAMBN-generated data revealed that both generation procedures of MultiNODEs were significantly better at reproducing the real data characteristics: the Frobenius norm of real data correlation matrix resulted in 45.3, and with a Frobenius norm of 25.66 the VAMBN-generated data placed substantially further from the real data than the MultiNODEs approaches with 62.63 and 56.47 for the prior and posterior sampling, respectively. This shows that MultiNODEs slightly overestimated the present correlations, while VAMBN underestimated them. Concordantly, the relative error (i.e., the deviation of the respective synthetic dataset’s correlation matrix from the real one normalized by the norm of the real correlation matrix), was 0.81, 0.62, and 0.46, respectively, for VAMBN and MultiNODEs’ prior and posterior sampling, leaving MultiNODEs with a substantially lower error than the VAMBN approach.

### Assessment of the utility of generated synthetic patients for machine learning

To evaluate whether the generated synthetic patients could be reliably used in a machine learning context, we built a Random Forest classifier that aimed to distinguish between healthy individuals and diseased patients. The classifier was trained within a five-fold cross-validation scheme once using real and once using synthetic diseased patients. In addition, we trained a classifier on each respective synthetic dataset (comprising synthetically generated diseased and healthy subjects) and evaluated their performance on the real data (Table 1). As predictors, we used clinical symptoms and genetic markers that are characteristic of the disease in question. For PD (PPMI), these were the UPDRS scores that describe a series of motor and non-motor symptoms commonly encountered in PD patients, for AD (NACC), we predominantly used cognitive assessments and a genetic risk factor. Technical details about the classifiers can be found in the Supplementary material. Distinguishing real patients from healthy control subjects was possible with a 10 times repeated five-fold cross-validated performance of 0.97 ± 0.02 area under the receiver operator curve (AUC) and 0.90 ± 0.01 AUC for PPMI and NACC, respectively. On PPMI, all evaluated generative methods achieved almost equal performance, indicating that clinical characteristics of synthetic patients followed the same patterns as in real patients. In addition, the most relevant features were the same across the real and all synthetic data-trained classifiers (Supplementary Fig. 13).

For NACC, some deviations were found between a classifier’s cross-validated performance on real data and the synthetic-data-based performances. Here, MultiNODEs’ posterior and VAMBN showed similar deviations in opposite directions, with the posterior slightly overperforming and VAMBN slightly underperforming. The performance on the data generated via MultiNODE’s prior sampling method deviated the most (Table 1). When trained on synthetic data and evaluated on real data, all trained classifiers underperformed compared to classifiers trained on real data. The feature importances of predictors were highly similar between the real data-trained and the respective synthetic data-trained classifiers.

### Generating data in continuous time through smooth interpolation and extrapolation

One particular strength of MultiNODEs, that sets it apart from alternative approaches such as VAMBN, is its ability to model variable trajectories in continuous time. The latent ODE system allows for the estimation of variable trajectories at any arbitrary timepoint and thereby opens possibilities for (1) the generation of smooth trajectories, (2) overcoming panel-data limitations through interpolation, and finally, (3) extrapolation beyond the time span covered in training data themselves. Again, we evaluated these capabilities based on the PPMI and NACC datasets (for brevity, NACC results are presented in the Supplementary material). For the following, we only focused on the MultiNODE posterior sampling approach to generate synthetic subjects.

Comparing the median trajectories of variables from the real data to those generated using MultiNODEs revealed that MultiNODEs accurately learned and reproduced the longitudinal dynamics exhibited in the real data (Fig. 4). Generation from both the prior and posterior distribution led to synthesized median trajectories that closely resembled the real median trajectories. Equivalently, also the 97.5% and 2.5% quantiles of the synthetic data approximated the corresponding real quantiles closely, indicating a realistic distribution of the synthetic data across the observed timepoints. This observation held true for most of the time-dependent variables (plots for all variables are linked in the Supplementary material).

We further assessed the interpolation and extrapolation capabilities of MultiNODEs. For interpolation, one timepoint was excluded from model training and subsequently, data were generated for all timepoints including the one left-out. Contrasting the interpolated/imputed values against the corresponding real values showed that MultiNODEs accurately reproduced the longitudinal dynamics of a variable, even for unobserved timepoints (Fig. 5a, c). In this context, we further compared the interpolated values against synthetic data that was generated based on the complete real data trajectory. We observed that the mean JS-divergence calculated across all variables between the interpolated data and the real data was slightly higher (0.025 ± 0.011) than that of the real data and the synthetic data generated after training MultiNODEs on the complete trajectory (0.016 ± 0.011). Similarly, the relative error between the interpolated correlation matrix and the real data was again only marginally higher than between the complete data and the real data (0.48 and 0.46, respectively; Supplementary Fig. 4).

In order to test MultiNODEs’ extrapolation capabilities, only the first 24 months of assessment follow-up and the static variables were used during model training. The trained model was then applied to generate data for the remaining, left-out timepoints of the longitudinal variables. In this course, 77 values were extrapolated while not every variable had the same number of follow-up assessments after month 24. Comparing the extrapolated synthetic data to the left-out real data demonstrated reliable extrapolation beyond the training data (Fig. 5b, d). As in the interpolation setting, we also compared the average JS-divergence between the extrapolated data and the real data with that between the real data and synthetic data that were generated after training MultiNODEs on the complete trajectory. As expected, we could see a larger difference between the JS-divergences compared to the interpolation setting with 0.037 ± 0.024 for the extrapolated data and 0.016 ± 0.009 for the synthetic data based on the complete trajectory. The correlation structure in the extrapolation culminated in a relative error of 0.64 compared to 0.46 when using the complete trajectory for training MultiNODEs (Supplementary Fig. 4).

In addition, the marginal distributions at both the interpolated and extrapolated timepoints also followed those of the real data (Fig. 5c, d).

### Systematic model benchmarking on simulated data

To explore the learning properties of MultiNODEs more systematically, we investigated how alternating training conditions with respect to measurement frequency, sample size, and noisiness of the data influence MultiNODEs’ generative performance.

The benchmarking data was simulated via the well-established Susceptible-Infected-Removed (SIR) model that is often used to describe the spread of infectious diseases and follows a highly nonlinear structure: Let S(t) be the number of susceptible individuals at a timepoint t, I(t) be the number of infectious individuals at a timepoint t and R(t) be the number of removed or recovered individuals at a timepoint t. With β as transmission rate, γ as mean recovery/death rate, and N = S(t) + I(t) + R(t) as fixed population size the SIR model can be defined by the ODE system presented in Eq. (1):

$$\begin{array}{*{20}{c}} {\frac{{dS}}{{dt}} = \frac{{ - \beta SI}}{N}} \\ {\frac{{dI}}{{dt}} = \frac{{\beta SI}}{N} - \gamma I} \\ {\frac{{dR}}{{dt}} = \gamma I} \end{array}$$
(1)

Details about the SIR parameter settings are described in the Supplementary material.

As baseline settings for each investigation, we simulated 1000 data points with 10 equidistant assessment timepoints each, distributed over a span of 40 time intervals, and added 5% Gaussian noise to each measurement. That means we added a normally distributed variable with the standard deviation set to 5% of the theoretical range of each of the variables S(t), I(t), and R(t). During the benchmarking, we individually alternated the sample size, timepoints, and noise level. For the timepoint investigation, we compared MultiNODEs trained on 5, 10, and 100 equidistant assessments; for the sample size we considered 100, 1000, and 5000 samples; and for the noise level, we tested 50%, 75%, and 100% of the maximum encountered value added as noise.

Alternating the amount of equidistant, longitudinal timepoints exposed a strong dependency of MultiNODEs on the longitudinal coverage of the time-dependent process (Fig. 6a). While the general trends in the data were appropriately learned for all explored assessment frequencies, the position of the observations in time influenced how close the learned function approximated the true data-underlying process. Especially the peak of the “Infected”-function represented a challenge for MultiNODEs if no data point was located close to it (Fig. 6a, “Infected”). Similarly, the start of the decline in the “Susceptible”-function and the incline in the “Removed”-function were shifted, depending on the positioning of measurements. In conclusion, and as expected, a higher observation frequency of the data-underlying the time-dependent process significantly increased the fit of MultiNODEs to the process, although, general trends could already be approximated for lower assessment frequencies.

Investigating the effect of the sample size on training MultiNODEs, we observed that an increase of the sample size led to an expected improvement of the model fit to the SIR dynamics (Fig. 6b). While the general trends could again be learned from limited data (n = 100), sample sizes of 1000 or 5000 substantially reduced the model’s deviation from the true SIR model. With 1000 samples, the learned dynamic is less stable than when trained on 5000 samples, where a smooth dynamic was learned that closely resembled the true underlying process. In conclusion, MultiNODEs can already learn longitudinal dynamics based on only a few data points, however, they tend to underfit under these circumstances and benefit from larger sample sizes.

Adding an increasing noise level to the SIR training data revealed that MultiNODEs remain very robust (Fig. 6c). Only when introducing 100% of the maximal encountered value as additional noise, a clear deviation from the underlying true model could be observed.

## Discussion

In this work, we presented MultiNODEs, a hybrid AI approach to generate synthetic patient-level datasets. MultiNODEs are specifically designed to consider the characteristics of clinical studies, extend its predecessor, the Neural ODEs, and enable the application of the latent ODE system to multimodal datasets comprising both time-dependent and static variables with values missing not at random. MultiNODEs learn a latent, continuous time trajectory from observed data. This concept fits well with processes like disease progression, where relevant observations (e.g., biomarkers and disease symptoms) only indirectly mimic the true, underlying disease mechanism. Consequently, MultiNODEs are well suited for an application to heterogeneous datasets holding complex signals as encountered, for example, in biomedical research.

Our evaluations showed that MultiNODEs successfully generated complex, synthetic medical datasets that accurately reproduced the characteristics of their real-world counterparts. In a direct comparison MultiNODEs’ outperformed the state-of-the-art VAMBN approach, most notably with respect to the integrity of the correlation structure. This finding implies that the single data instances generated using MultiNODEs exhibit more realistic properties and that the real data characteristics are not only reproduced at the population level. Out of MultiNODEs two generative methods, the posterior sampling expectedly led to more realistic synthetic patients; however, generating from the prior distribution comes with the benefit that the model itself can be shared and used for data generation without needing any real data points in the process.

Machine learning classifiers that discriminated between real healthy controls and diseased subjects showed almost equal performance when trained on data from synthetic and real diseased subjects, respectively. Here, we only observed small deviations from the performance on real data for the NACC dataset, where classifiers trained and tested on synthetic patients and real healthy controls within a cross-validation setting showed a slightly increased performance to those trained on real data. Interestingly, at the same time, we found a lower prediction performance compared to real data when we trained on synthetic subjects and evaluated on the real data. A possible explanation is that synthetic data can contain noise that is introduced during the generation of synthetic data points (e.g., through overestimated correlations between variables). Therefore, synthetically generated diseased patients are better discriminated against real healthy controls than real diseased patients. At the same time, this situation leads to the fact that a classifier trained on synthetic data (synthetic patients as well as healthy controls) shows a slightly lower prediction performance on real data compared to a classifier trained entirely on real data. Altogether our results demonstrate that synthetically generated subjects share patterns of real patients, but they are not completely identical.

Besides the reproduction of marginal distributions and synthesis of realistic data instances, MultiNODEs most prominent strength lies in the generation of smooth longitudinal data. The latent ODE system allows MultiNODEs to learn dynamics that are continuous in time and cover the unobserved time intervals of real-world data. Here, both the prior and posterior sampling approach resulted in synthetic trajectories that obey real variables’ dynamics.

Furthermore, the time-continuous generative capabilities of MultiNODEs create opportunities to fill gaps in the real data through interpolation and go beyond the observation time by extrapolating the longitudinal dynamics. Hence, MultiNODEs could be used to support the design of longitudinal clinical studies, in which the maximum observation period, as well as visit frequency, is always a crucial decision to make. Here, the question of how patients might develop between two visits or after the last one determines the optimal follow-up time, to demonstrate, for example, the most significant treatment effect. Furthermore, synthetic disease trajectories generated based on data from one clinical study can be compared to those generated based on other studies, even if the visit intervals employed in the real studies were not identical.

Our benchmark experiments on the simulated SIR model data demonstrated that MultiNODEs are applicable under a variety of different data settings. While the general trends of a data-underlying process could already be learned from a relatively limited dataset, similar to any machine learning task, the accuracy and trustworthiness of the model critically depends on the available data. Especially for complex, nonlinear processes, a sufficiently high observation frequency should be considered. Here, the position of the observation timepoints relative to the true underlying process is crucial for MultiNODEs to accurately learn nonlinear dynamics. The sample size of the training data mainly impacts how well MultiNODEs fitted the data dynamics and we observed that lower sample sizes can lead to underfitting and rather rigid ODE systems. On the other hand, only severe noise levels led to a model deviation from the true data-underlying process, and thus, with respect to noise, MultiNODEs proved to be highly robust. In conclusion, MultiNODEs’ requirements toward the training data ultimately depend on the complexity of the data-underlying process, whereas the learning of more complex processes requires more frequent observations and larger sample size, while more linear systems can already be learned from rather limited datasets.

One limitation of MultiNODEs in their current form only allows static categorical variables. This is because the variational encoder for longitudinal data maps trajectories to a latent Gaussian distribution. Sampling from this distribution (even, if conditioned on the distribution of the static data) and decoding will result in real valued features rather than categorical ones. In future work, we will thus explore whether a recurrent version of the HI-VAE encoder can be used instead of a recurrent variational long-short term memory (LSTM) encoder.

In addition, MultiNODEs are sensitive to several hyperparameters that should be optimized for optimal performance. The training process and all relevant hyperparameters are explained in the Method section.

Synthetic data generated using models trained on sensitive personal information can bear a risk of information disclosure (e.g., attribute disclosure or dataset membership disclosure), if an attacker has information about properties of real patients that are similar to a synthetic subject. Therefore, before synthetic data are distributed, it must be assured that the probability of private information disclosure remains within task-appropriate boundaries20. Disclosure risk often stands in a direct trade-off with data utility and a sensible compromise should be taken balancing the two according to the application in question. Several approaches are described in the literature that can reduce the risk of information disclosure21, one of which is based on the concept of differential privacy4. MultiNODEs themselves provide a way to tune the deviation from the real data when sampling from the posterior distribution by changing the amount of noise injected in the latent space. We would like to mention that a rigorous quantification of the re-identification risk is a non-trivial and challenging task for its own requiring several assumptions and is thus beyond the scope of this paper.

## Methods

### Application case datasets

Both datasets, namely PPMI and NACC, are well-known staples in their respective fields and can be accessed after successful data access applications. For PPMI see https://www.ppmi-info.org/. For NACC we refer to https://naccdata.org/. More details on the investigated variables are presented in the Supplementary material.

Both studies retrieved informed consent from their participants for data collection and sharing and followed the declaration of Helsinki to ensure ethical data collection. Both studies got ethical approval from their respective review boards. We followed their employed regulations and thus did not seek further ethical approval, as we did not work with human participants ourselves.

### Neural ODEs (NODEs)

NODEs are a hybrid of neural networks and ODEs14. They can be seen as an extension of a ResNet22, which does not rely on a discrete sequence of hidden layers, but on a continuous hidden dynamical system defined by an ODE.

For 0 < t < M and $$z_0 \in R^D$$ the dynamics of the hidden layer of a NODE are given as Eq. (2).

$$\begin{array}{*{20}{c}} {\frac{{dz\left( t \right)}}{{dt}} = f\left( {z\left( t \right),t,\theta } \right)} \\ {z\left( 0 \right) = z_0} \end{array}$$
(2)

where z(0) may be interpreted as the first hidden layer and z(T) as the solution to the initial value problem at timepoint T. Importantly, f is a feed-forward neural network parameterized by θ.

#### NODEs as generative latent time series models

As demonstrated by the authors in their publication, NODEs can be trained as a continuous time Variational Autoencoder. The basic idea is to learn the initial conditions z0 of the dynamical system in Eq. (2) from observed time series data using a variational LSTM recurrent encoder23. Hence, Eq. (2) now describes the dynamics of a latent system, resulting in a classical state-observation model. Accordingly, a feed-forward neural network decoder is required to project the solution of Eq. (2) back to observed data at defined timepoints (Supplementary Fig. 10).

Overall NODEs are trained at once by maximizing the evidence lower variational bound (ELBO): let the training data be $$D = \{ ( {x_{t_i}^n,t_i}) | n = 1, \ldots ,N,i = 1, \ldots ,M\}$$, where N is the number of patients and $$t_{i_1}, \ldots ,t_{i_M}$$ the observed timepoints / patient visits. That means $$x_{t_i}^n \in R^p$$ is the p-dimensional vector of measurements taken for the nth patient at visit ti. The ELBO for NODEs is then given as Eq. (3).

$$\begin{array}{l}ELBO^{NODE} = \frac{1}{N}\mathop {\sum}\limits_{n = 1}^N \mathop {\sum}\limits_{i = 1}^M - D_{KL}\left( {q\left( {z_{t_0}^n | \left\{ {x_{t_i}^n,t_i} \right\}_i} \right)\parallel p\left( {z_{t_0}^n} \right)} \right)\\ \qquad\qquad\qquad\quad+\, E_{q\left( {z_{t_0}^n | \left\{ {x_{t_i}^n,t_i} \right\}_i} \right)}\left( {log\left( {p\left( {x_{t_i}^n | z_{t_i}^n} \right)} \right)} \right)\end{array}$$
(3)

where $$p\left( {z_{t_0}^n} \right) = N\left( {0,I} \right)$$, as usual. For details, we refer to Chen et al.14.

### Multimodal Neural ODEs (MultiNODEs)

#### Handling missing values

To handle missing values (potentially not at random) in longitudinal clinical data we build on our previously published work, in which we introduced an imputation layer to implicitly estimate missing values during neural network training18: let $$A: = \left\{ {x_{t_i,j}^n{{{\mathrm{|}}}}x_{t_i,j}^n\ {\rm{is\ not\ missing}}} \right\}$$, 1A be the indicator function on set A with cardinality $$\left| A \right|$$. The imputation layer can be defined as a data transformation $$\tilde x_{t_i,j}^n = x_{t_i,j}^n \times 1_A\left( {x_{t_i,j}^n} \right) + b_{t_i,j} \times \left( {1 - 1_A\left( {x_{t_i,j}^n} \right)} \right)$$, where parameters $$b_{t_i,j}$$ are trainable weights. That means missing values in a patient’s data vector $$x_{t_i,j}^n$$ are replaced by $$b_{t_i,j}$$. The accordingly completed data is subsequently mapped through a recurrent neural network encoder to a static, lower dimensional vector, which is interpreted as the initial condition of the latent ODE system (Supplementary Fig. 11).

To learn parameters $$b_{t_i,j}$$ the NODEs’ loss function needs to be adapted. More specifically, we use the modified ELBO criterion presented in Eq. (4).

$$\begin{array}{l}ELBO_{IMP}^{NODE} = \frac{1}{N}\mathop {\sum}\limits_{n = 1}^N \mathop {\sum}\limits_{i = 1}^M - D_{KL}\left( {q\left( {z_{t_0}^n{{{\mathrm{|}}}}\left\{ {x_{t_i}^n,t_i} \right\}_i} \right)\parallel p\left( {z_{t_0}^n} \right)} \right) \\\qquad\qquad\quad\quad\,\,\,\,\,\,+\, \frac{{DM}}{A}\mathop {\sum}\limits_{n = 1}^N \mathop {\sum}\limits_{i = 1}^M {\mathop {\sum}\limits_{j = 1}^D {1_A\left( {x_{t_i,j}^n} \right)\left( {x_{t_i,j}^n - \hat x_{t_i,j}^n} \right)^2} }\end{array}$$
(4)

where $$\hat x_{t_i,j}^n$$ denotes the reconstructed data. Note that we only aim for reconstructing the observed data, but not the imputed one. Due to the layer-wise architecture of a neural network $$\hat x_{t_i,j}^n$$ implicitly depends on $$b_{t_i,j}$$.

In practice, we initialize $$b_{t_i,j}$$ for neural network training as $$\frac{1}{N}\mathop {\sum }\nolimits_{n = 1}^N x_{t_i,j}^n$$.

#### Dealing with multimodal data

In addition to implicit missing value imputation, the second main idea of MultiNODEs is to complement NODEs with a HI-VAE encoder17 for static variables (Supplementary Fig. 11). A HI-VAE is an extension of a Variational Autoencoder that can implicitly impute missing values via an input drop-out model and handle heterogeneous multimodal data, including categorical data and count data, via an accordingly factorized generative model. In addition, a HI-VAE uses a Gaussian Mixture Model (GMM) as a prior distribution rather than a single Gaussian. We refer to Nazabal et al.17 for details.

The HI-VAE results in a lower dimensional latent representation zstat of static variables, which can be used to augment the initial conditions zt0 learned from time series data. Consequently, we arrive at the following formulation of the latent ODE system given in Eq. (5).

$$\begin{array}{*{20}{c}} {\frac{d}{{dt}}z^{aug}\left( t \right) = \frac{d}{{dt}}\left[ {\begin{array}{*{20}{c}} {z\left( t \right)} \\ {\tilde z\left( t \right)} \end{array}} \right] = f\left( {\left[ {\begin{array}{*{20}{c}} {z\left( t \right)} \\ {\tilde z\left( t \right)} \end{array}} \right],t,\theta _f^{aug}} \right)} \\ {z_{t_0}^{aug} = \left[ {\begin{array}{*{20}{c}} {z_{t_0}} \\ {z_{stat}} \end{array}} \right]} \end{array}$$
(5)

This approach resembles the Augmented Neural ODEs by Dupont et al.24. In contrast to our work, in their work no additional features were added during the augmentation step, i.e., zstat = 0. According to Dupont et al. the purpose of Augmented Neural ODEs is to smoothen f, whereas we focus here on multimodal data integration.

For training MultiNODEs, we have to jointly consider $$ELBO_{IMP}^{NODE}$$ as well as $$ELBO^{HI - VAE}$$. After bringing both quantities on a comparable numerical scale, we use a weighted sum as our final training objective (see Eqs. (6) and (7)):

$$\begin{array}{*{20}{c}} {\widetilde {ELBO}_{IMP}^{NODE} = \frac{{ELBO^{HI - VAE}}}{{ELBO^{HI - VAE} + ELBO_{IMP}^{NODE}}}ELBO_{IMP}^{NODE}} \\ {\widetilde {ELBO}^{HI - VAE} = \frac{{ELBO_{IMP}^{NODE}}}{{ELBO^{HI - VAE} + ELBO_{IMP}^{NODE}}}ELBO^{HI - VAE}} \end{array}$$
(6)
$$ELBO^{MultiNODE} = \widetilde {ELBO}_{IMP}^{NODE} + \beta \widetilde {ELBO}^{HI - VAE}$$
(7)

Where β is a tunable hyperparameter. Details about hyperparameter optimization are described in the Supplementary material.

### Generating synthetic subjects

We tested two methods to generate synthetic subjects with MultiNODEs:

1. a.

The first option is drawing a sample of latent static and longitudinal representations from the respective prior distributions $$z_{t_0} \sim N\left( {0,I} \right)$$and $$z_{stat} \sim GMM\left( \pi \right)$$. To assure that interdependencies between static and longitudinal variables are conserved, we model their joint distribution $$P\left( {z_{t0},z_{stat}} \right)$$ using a Bayesian network. This network contains three nodes (random variables) representing (1) the GMM mixture coefficients π for the static data used by the HI-VAE, (2) the latent static representations $$Z_{stat} = GMM\left( \pi \right)$$, and (3) the latent longitudinal representations $$Z_{t0} = N\left( {0,I} \right)$$, respectively. The network is constrained such that directed edges can only go from si to Zstat and from there to Zt0. After randomly sampling a mixture component si from a multinomial distribution multinom(π), we can conditionally sample $$z_{stat} \sim Z_{stat} | s_i$$ and finally $$z_{t0} \sim Z_{t0} | Z_{stat}$$. Subsequently, we concatenate $$z_0 = \left[ {z_{t_0},z_{stat}} \right]$$ into a vector forming the initial conditions for the latent ODE system, solve the ODE system, and decode the solution. We call this approach “prior sampling”.

2. b.

A second option is to draw $$z_{t_0}q( {z_{t_0}^n | \{{x_{t_i}^n,t_i} \}_i} ) = N( {\lambda ( {x_{t_i}^n,t_i}),\sigma ( {x_{t_i}^n,t_i})})$$ for the longitudinal data and $$z_{stat}q\left( {z_{stat}^n{{{\mathrm{|}}}}x_{stat}^n,\pi } \right) = N\left( {\lambda \left( {x_{stat}^n,s^n} \right),\sigma \left( {x_{stat}^n,s^n} \right)} \right),s^nCategorical\left( {\pi \left( {x_{stat}^n} \right)} \right)$$ for the static data. That means we generate a blurred / noisy version of the original nth patient. We call this approach “posterior sampling” and recommend this sampling procedure for data generation. In our experiments, we doubled the posterior variance during sampling because we found the synthetic data otherwise to lie too close to the real data. Tuning the added noise can provide one option to balance identification risk versus data utility.

3. c.

Synthetic data can not only be generated for observed visits, but also for definable timepoints in between (interpolation) and after the end of the study (extrapolation). This is possible because the latent ODE system is continuous in time.

### Data preprocessing

Few steps are required to preprocess the clinical data before MultiNODEs can be applied. First, the data must be organized into a three-dimensional tensor of the shape samples × timepoints × variables for the longitudinal variables, and samples × variables for the static ones. Furthermore, the longitudinal variables are then transformed into a progression score by subtracting the baseline value and normalizing them by the standard deviation of this variable at baseline.

### Calculating the relative error for correlation matrices

The relative error between correlation matrices is calculated as the norm of the matrix describing the difference between the real correlation matrix and synthetic data correlation matrix divided by the norm of the real correlation matrix.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


text:
# Particle Beam Weapons

 Notice: Please bear with us. Your ride's still a work in progress.

Like Laser Weapons, particle beams are a popular concept in science fiction. Streams of charged or neutralized particles are accelerated to awesome velocities and projected at a remote target. Like a hail of trillions of atom-sized (or smaller) bullets, the particle beam smashes into the target, releasing thermal energy and creating radiation. The unlucky target is irradiated, cored through, or exploded into chunks!

Particle beams combine some properties of larger-projectile kinetic energy weapons and the laser directed energy weapon. Particle beams can travel at high-relativistic velocities close to the speed of light, able to reach targets only a tiny bit slower than a beam of photons. But their working medium are particles with mass, which gives them different propagation and interaction behavior. Often high-energy particles can penetrate into a target or release showers of secondary radiation that penetrate deeply. This irradiation also offers a soft, insidious means to make technology fail and kill biological beings (or anything dependent on nanometer structures and molecular systems) without having to physically melt or blow apart the target. Yet a particle beam's mass flow is generally so tiny that they are not limited by physical ammunition but the power required to drive this little mass up to high speeds and thus energy.

Depending on configuration, particle beams can be highly utilitarian systems with a wide range of not just weapon but also sensing and propulsion applications. They can deal significant radiation damage at low power or focus, or transfer enough energy for explosive vaporization at high energies. Their accuracy is exceptional and their range can be significant, providing direct-fire engagment capability out to the limits of the weapon mounts and supporting sensors.

Defending against particle beams is possible with appropiate material composition and in some cases, magnetic fields.

## The limitation of bloom

There is one fundamental problem with weaponizing particle beams. To accelerate a particle beam efficiently, we want to use electromagnetic fields - gravity is way too weak and dissipates very quickly to accelerate particle beams without completely absurd assumptions (consider: the entire mass of the Earth amounts to 1G of acceleration. The forces in a particle accelerator can be in the hundreds of millions of Gs), and neither the weak nor the strong nuclear force reach sufficiently far. Electromagnetic fields are wonderful! To accelerate particles with an electromagnetic field, they must be charged positively or negatively. But a cloud of positively or negatively charged particles will experience charge repulsion, giving rise to electrostatic bloom - the tightly bunched-together group will disperse into a weak, wide-area cloud. Too unfocused, the fearsome particle beam will not do much damage. Maybe it will irradiate the target, but the beam may disperse so widely that it becomes indistinguishable from the cosmic background radiation you find everywhere in deep space.

In older, popularized Sci-fi analysis, this issue of was generally considered the critical issue with charged particle beam weapons at ranges beyond a few hundred kilometers.

They have a disadvantage of possessing a much shorter range. The beam tends to expand the further it travels, reducing the damage density ("electrostatic bloom"). This is because all the particles in the beam have the same charge, and like charges repel, remember? Self-repulsion severely limits the density of the beam, and thus its power.

- Atomic Rockets, Particle Beams page

Particle beams disperse for a lot more reasons than laser beams, unfortunately, so it's harder to give a simple formula. It will depend on things like magnetic and electric fields in the region between the source and the target (if the particles have spin, for example, they will couple to the magnetic field gradient even if they are neutral). However, for a neutral particle beam traversing empty, field-free space, the dispersion is proportional to the temperature of the beam. Using, for the sake of a simple example, a mercury ion beam (dispersion decreases proportional to square root of atomic mass, and mercury is a convenient high-mass atom that ionizes easily), the lateral (spreading rate) velocity of the beam is:

${\displaystyle V=1.4{\sqrt {T}}\,m/sec}$ for ${\displaystyle T}$ in Kelvin

To calculate the actual angular spread of the beam, you need to know the beam velocity. For a quick calculation, you could say it's no more than the speed of light, ${\displaystyle 300,000,000m/sec}$. So the dispersion in nano-radians is ${\displaystyle 5{\sqrt {T}}}$.

So, for a beam with an effective temperature of, say, ${\displaystyle 1000K}$, dispersion for mercury is ${\displaystyle 150\,nR}$, or ${\displaystyle 0.15}$ micro-radians. Dispersion at a distance of ${\displaystyle 100,000\,km}$ would be ${\displaystyle 0.015\,km}$, or 15 meters. A hydrogen beam would disperse ${\displaystyle {\sqrt {80}}=9}$ times more. [note that if the beam is actually relativistic, you have to apply a relativistic correction, which I'll ignore here.]

- Dr. Geoffrey A. Landis, via Atomic Rockets

Particle beams come in three varieties: Charged particle beams (electrons or protons), neutral particle beams (neutrons) or neutralized atomic beams (high velocity hydrogen or helium atoms).

Each presents a number of technical problems. Electron beams are very easy to produce; indeed, anyone who’s sat in front of a CRT-based television has sat in front of one. However, electron beams are also very easy to redirect via running a modest charge through the surface of the target, and Coulomb repulsion between the electrons in the beam would cause it to disperse rapidly in roughly 40 to 80 km in a vacuum. The other variety of charged particle beam just replaces the electron with protons, which have the advantage of greater mass per particle (by a factor of a bit over 1,836). This is an advantage, in that the beams will avoid spreading quite as quickly, but also requires considerably more energy to get them up to speed in the first place.

Einsteinian time dilation from the frame of reference of the particles in the beam mean that beam spread happens more slowly from the frame of reference outside the beam, but higher speed beams have less time to interact with the atomic nuclei of their targets, making the optimum velocity of the beam somewhat hard to determine even for a theoretical model for a game. In round numbers, a proton beam would have ranges of roughly triple an electron beam for about the same energy inputs...which is still shorter than the lasers currently in the game.

The second type of particle beam is the neutralized hydrogen or helium atom beam. You impart a charge on the beam and accelerate it with a cyclotron or linear accelerator, then when the beam has enough energy stored in it, you run it through a filter that strips the electrons off, and you get a beam of neutrally charged particles. In theory, this beam of particles could penetrate armor or ignore armor within a given distance of the skin of the ship before random collisions cause it to lose energy. In practice, most filters that will de-ionize the hydrogen or helium atoms will also impart enough random momentum among the atoms in the beam that it will rapidly disperse within a few kilometers. The third type of beam is the most problematic. If you presume that there is a way to generate neutron sources, capture neutrons and accelerate them, you could, in theory, make a beam entirely out of neutrons. Neutron beams, as beams, have some very powerful advantages. They don’t have an electrical charge, so they don’t spread from mutual repulsion of like charges. This allows them to be diffraction limited, but without the inherent limitations of photons. Depending on the tolerances of the system, and the final velocity of the neutrons, they could be very long ranged indeed; a reasonable ball park figure is around 800 to 1200 km for the first range bracket. At 20 km per hex, a 40 hex ‘short range bracket’ mean they would quickly render every other weapon in the game obsolete.

Even worse, neutron beams launching particles the speeds needed to make them weaponizable wouldn’t damage the ships. They’d sweep through the ships and turn everything they crossed into a radioactive hell, killing the crew in seconds. A game of “Both ships shoot, both ships become unmanned wrecks” wouldn’t be terribly fun to play.

Now, it is also possible, that like charged particle beam, the same forces that can be used to accelerate these neutrons into a beam could also be used to shield the target; from a game play perspective, this probably results in a perfect defense that, if it fails, results in instant death for the crew, which, will somewhat better than the “two ships fire, two ships die” model above, is much less compelling than the game that’s already present.

- Designers Note: Where are the Particle beams? Ken Burnside, Attack Vector Tactical 2nd Edition Rulebook, p. 104, Ad Astra Games, Pelican Rapids, MN 56572

Expected ranges were generally poor because the charged particle beam appeared to disperse too fast. The use of neutralized ion particle beams is discussed more favorably in multiple analyses - including in a longer blog post on ToughSF - but neutralization of a charged particle beam requires the use of heavier ions as a base. At the least, single protons must be used. The neutralization of the beam removes the problem of mutual electrostatic repulsion, but the recombination of ions and electrons is not without leftover energy. This leftover energy adds random motion to what is now a relativistic gas. Once again the particle beam begins to disperse!

To get an effective weapon, we want the exact opposite! We want to fight that bloom! Only by keeping our beams focused on target, can we achieve beam intensities that do fun things like melt the target, or combust or, or explode parts of it - or just irradiate it so thoroughly that anything delicate won’t work any longer. And in deep space bloom is an even bigger problem if we want to push the range up to that of powerful lasers. Or maybe we even want to take the beams beyond the range of pesky lasers and ram a rod of particles up that smug laserstar nose to rocket nozzle!

### Solving electrostatic bloom one way

The wonderful news is that we can solve the problem of (electrostatic) bloom, and it’s surprisingly easy with the humble electron. Electrons are wonderful subatomic particles to weaponize. The electron has a mass 1840 times less that of the proton and neutron, while having the same magnitude of charge as a proton. This makes it much easier to accelerate, since the charge to mass ratio is much higher. Easier to accelerate particles can attain more energy in a shorter length of accelerator. And for this magic trick we‘ll need a lot of acceleration to happen in our high-powered hat! The fact that we can most easily accelerate electrons to very, very high energies in a short distance allows us to exploit relativity to our advantage. More specifically, relativistic time dilation.

Time dilation happens as some object moves closer to the speed of light. From the perspective of the object at relativistic speed, time slows down relative to the surrounding universe. A journey that from the perspective of an observer at rest (not at relativistic speed) takes years, can happen in seconds from the perspective of the object.

We can use the same thing in a particle beam: that the electrons will repulse from each other and disperse cannot effectively matter when from the electrons point of view, the journey to the target has happened before they had significant time to repulse each other. The beam arrives before it has had time to experience electrostatic bloom from its massively dilated point of observation. This concept has been known about for decades and has been considered for real world weapons, but accelerator technology was nowhere near up to the task to do so in a compact format. Even now our best accelerators are a bit big for current military applications and we don’t really have any need for space battleships… yet!

This is the foundation of the Ultra-relativistic electron beam (UREB). By driving electrons with energies of, minimum viable, 10 Giga-electronvolt (a measure of particle energy) at the target, the time dilation experienced by the beam checks its electrostatic repulsion. We may consider particle energies of up to 1 Tera-electronvolts and more! The equation for the doubling distance of a charged, ultra-relativistic beam is shown below just to provide an idea of what we are working with.

${\displaystyle Z=1.32\cdot 10^{2}(\gamma ^{2}-1)^{3/4}r/{\sqrt {I}}}$[1]

${\displaystyle Z}$ is the distance for the beam radius ${\displaystyle r}$ to double in size due to electrostatic bloom. The term is the Lorentz gamma ${\displaystyle \gamma }$ factor (which we will talk about later) and ${\displaystyle I}$ is the beam current in amperes. For an example of the power of this equation, if we have a ${\displaystyle 500\,GeV}$ electron beam (${\displaystyle \gamma =978474}$ and pretty high end for electron beams) and an average beam current of ${\displaystyle 2}$ milliamps (pretty normal and even low for particle accelerators) and a beam radius of ${\displaystyle 1\,cm}$, we get a doubling distance of nearly ${\displaystyle 30}$ million km! Although this is an elementary first approximation, the potential is clear!

A UREB will still eventually disperse. However, significant defocusing at the discussed particle energy will only occur well after other limitations to weapons range. There is the constant spectre of space plasmas and electromagnetic fields complicating this basic analysis, but that is multiple PhD thesis levels of work and most likely only an issue at really extreme ranges.

In theory, we could consider using ultra-relativistic speeds to fight bloom on particle beams of heavier particle beams as well. Practically, this gets prohibitively complicated due to the much higher mass of ions: with masses thousands to tens of thousands or more times heavier than a single electron, the speed gained per invested energy is much lower, and getting up to ultra-relativistic speeds requires enormous amounts of energy and gigantic accelerators. However, arguably this is also not necessarily required. On ion beams, we can fight bloom in other ways.

### Solving electrostatic bloom the other way

In the introduction, we mentioned using neutral ion beams. This is actually a quite powerful technique, though the devil is in the details! Instead of using relativity to suppress the problems of charge repulsion, we can remove the problem of charge repulsion from the equation by using neutral particles.

The catch is, we need electric charge for all of our acceleration tricks to work. (Since gravity has the gall to lack in easily applicable fields and is a horrendously weak force anyway - look at how much mass we need for a paltry 1G of acceleration compared to the trillions of G’s in a typical particle accelerator!) So during the acceleration process, we require ions - atoms with fewer or more electrons in their orbits than protons in their cores.

We also need the aforementioned much heavier particles! Our mass per particle rises by at least three orders of magnitude in using protons. Using light ions, we are now working with particles four orders of magnitude heavier, and for heavy atoms, we are at five orders of magnitude and more! This causes us a bunch of problems in accelerating these heavy beasts quickly, and pretty much leaves neutralization as the only way to work with them at significant ranges.

For comparison, if we were to use the ion beam without neutralization: let us take a hydrogen beam fired at 100 MeV. This beam doubles its radius in just 1.9 km! The quotes from the introduction come to mind. A heavy ion beam can end up much, much worse due to the higher charge per particle!

With neutralization, our particle beam is accelerated and handled internally as ions and then neutralized upon firing. There are two primary ways to do this, either strip an extra electron that was added to make a negative ion or add the needed electrons to a positive ion. Each method has its own advantages and disadvantages to consider. For the extra electron version you get some accelerator physics advantages for hydrogen beams, but beams of any heavier ions are just harder to accelerate properly due to the poor charge to mass ratio (charge of one electron vs the mass of some sort of large ion). Stripping that extra electron also can be very difficult with the most popular method being gas or foil strippers that make the beam spot bigger and can take huge (up to 50%) cuts out of the beam power. This won’t do for a weapon beam at! There are also optical electron stripping methods available, although those can potentially require large amounts of laser power to pull off. The real advantage of this method though, is that you end up with a truly neutral beam of relativistic atoms that will laugh at anything but a solid chunk of matter.

The second method of ionization involves accelerating a beam of electrons up to the same speed as the ions and then injecting them into the final beam optics with the ions. Luckily this is very easy with a 100 MeV proton beam only needing electrons at 51 keV to keep up with it. That can be generated with an old CRT TV tube! These electrons will naturally mix with the ions and form a cold plasma that travels to the target. Yes, you heard me right A PLASMA! Now you would think that the plasma would neutralize out into a gas, and given enough time you would be right! However, the time scale for this to happen at any large scale is typically too long to matter to a beam traveling a good chunk of the speed of light. If you end up shooting at other solar systems with one of these, do be sure to write and tell us how that works out! The neat thing with plasma is that it typically ends up screening out external fields as well, so attempts at electric or magnetic shields will struggle to work for a well designed weapon beam even if the needed field strengths can be reached. That is not to say it can’t be done, but it sure won’t be easy or consistent if you make your weapon right! On the other hand, this also means that a beam could be made to be easily deflected and that is wonderful for propelling spacecraft!

The catch with either of these neutralization methods is that gas or plasma has a temperature, and random brownian motion. The hotter the gas, the higher the average energy and thus velocity of the brownian motion: the quicker the beam disperses. Instead of being forced apart by electrostatic repulsion, the beam drifts apart. Due to high mass and difficulty of accelerating ions we can’t always rely on relativity to save us here. A 100 MeV proton beam is “only” doing 41% of the speed of light and carbon beam of equivalent energy is even slower! Thus relativistic time contraction is not something that will always play a role in neutral ion beams. As a consequence, controlling the temperature of the beam during generation, acceleration and neutralization is key. The stripped extra electron beams will have to be careful with how the electron is stripped (although that effort always adds “heat” to the beam) and the injected electron neutralized beams will need to be extremely careful with electron injection as well as temperature. If you know the beam temperature, the doubling distance can be found with the following equation.

${\displaystyle Z=(r/{\sqrt {T/m}})\cdot \beta \cdot c\cdot \gamma }$

Where ${\displaystyle Z}$ is the doubling distance like before, ${\displaystyle r}$ is the initial beam radius, ${\displaystyle T}$ is the beam temperature in eV, ${\displaystyle m}$ is the ion mass and ${\displaystyle \beta }$, ${\displaystyle c}$, and ${\displaystyle \gamma }$ are the typical relativistic terms for the beam and light speed respectively.

Compared to UREBs, the analysis of neutralized beam performance is more complicated, as it depends on more separate factors. While depending on analysis, neutral particle beams typically have a higher effective emittance, ions are much easier to deflect than electrons without generating synchrotron radiation. This allows for the ion beams to be re-circulated in “storage rings”, effectively creating a kind of storage device for particle beams quite akin to a weird sort of momentum wheel. This allows for the buildup of far larger particle bunches with much higher energies, which can then be projected onto targets in an instant, on demand, separated from burst power or continuous power supply capabilities of the firing vessel. One could consider a barrage of bunches built-up and stored over hours, and fired in milliseconds- each pulse significantly more powerful than what any electron beam weapon could easily provide.

### Solving electrostatic bloom the third way

If we aren’t fighting in the depths of space, but are on good ole Terra Firma (or anywhere else with an atmosphere) there is a third way to fight beam bloom that has received a LOT of R&D. In this case rather than use relativity or neutralization to fight the beam bloom, we will use the atmosphere itself! When a high current (i.e. lots of charges per second) beam is injected into the atmosphere it will turn that atmosphere into plasma (consuming some beam energy). Now a neat thing about plasma is that it always acts to neutralize any electric or magnetic fields that are around and our high current beam has a lot of both! The air will then act to neutralize and (with the right beam parameters and air pressure) further focus the beam! Sadly the physics of this phenomenon doesn’t lend itself to easy analytic equations (damn plasma physics!), but we can say that kiloamp class and MeV particle energy beam pulses are needed for this to work. Your standard CRT tube won’t cut the mustard here!

This technique has been shown to allow for low energy (single to tens of MeV) electron or ion beams to propagate much further through the atmosphere than a simple beam scattering analysis would lead one to believe. This is all thanks to a neat phenomena known as the Bennett pinch that continues to get open and classified research to this very day! During the SDI days high current, 300 MeV electron beams were investigated for killing incoming ballistic missiles and found to be one of the most promising interception methods out there. There have been a variety of experiments conducted at US national labs to show that these beams can fly kilometers through the air while still bringing sufficient energy to the target to cause an obscene amount of damage. In the SDI days the beams were expected to reach 80 km up right to the edge of space! This whole scheme works even better with heavier particles (less loss of beam to scatter in air) so if you can make a really intense proton beam instead of an electron beam it could be worth it.

## Building Weapons

That is the introduction out of the way! We know some basic things about the fundamental limitation to particle beams. The good news is: particle beams are alive, menacing and they can reach out. They are not just short-range weapons that work at hundreds of kilometers only. We can build particle beams that keep together in deep space, near and inside an atmosphere, and even pass through an atmospheric interface!

Building a weaponized particle accelerator requires understanding various bits about Particle_Accelerators in general.

Of some specific interest here are the particles we can use, the two main accelerator types of interest for us, Radiofrequency accelerators, their advanced derivative and the plasma accelerators required to implement UREBs, various |secondary facilities and the two main design shapes of LINAC and synchrotron facilities.

Weaponized particle beams in many ways follow the design considerations and general technologies of particle accelerators. The main divergence is in output parameters and support considerations. Scientific accelerators are aimed at either providing specific imaging of materials through the use of particle interactions or synchrotron radiation on one hand; or the creation of exotic conditions allowing for the probing of fundamental physics. Commercial accelerators are used to image materials, cut and solder, implant material, or other such uses.

The main goal of a weaponized particle beam is the delivery of focused energy into the target material, with at least the goal of releasing ionizing radiation that degrades or destroys sensitive parts. On top, it may deliver sufficient energy for thermal effects. If this energy is delivered rapidly, the thermal effects become explosive.

## Design It Yourself

### Example Builds

• N. Bloembergen, C. K. N. Patel, P. Avizonis, R. G. Clem, et al. "Report to The American Physical Society of the study group on science and technology of directed energy weapons", Reviews of Modern Physics, Vol 59, Issue 3, 1 July 1987, pp1-201, DOI https://doi.org/10.1103/RevModPhys.59.S1
• Andre Gsponder, "The Physics of high-intensity high-energy Particle Beam Propagation in open Air and outer-space Plasmas", Independent Scientific Research Institute, Oxford, OX4 4YS, England, 11 January 2009, http://arxiv.org/abs/physics/0409157

## Credits

Sevoris and Gerrit Bruhaug for writing

Gerrit Bruhaug for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

## References

1. Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


text:
Practice the questions of McGraw Hill Math Grade 8 Answer Key PDF Lesson 14.4 Slope to secure good marks & knowledge in the exams.

Exercises

FIND THE SLOPE

Question 1.

$$\frac{5.6}{10}$$,

Explanation:
Using two of the points on the line, we can find the slope of the line by
finding the rise and the run. The vertical change between two points is
called the rise and the horizontal change is called the run. The slope equals the rise divided by the run:
Slope =rise ÷ run = (y2 – y1) ÷ (x2 – x1),
We have given points as (-2, -5.1) and (8, 0.5),
So slope = (0.5 – (-5.1))/ (8 -(-2)) = 5.6/10.

Question 2.

–$$\frac{8.1}{14}$$,

Explanation:
Using two of the points on the line, we can find the slope of the line by
finding the rise and the run.
The vertical change between two points is called the rise and the horizontal change is called the run. The slope equals the rise divided by the run:
Slope =rise ÷ run = (y2 – y1) ÷ (x2 – x1),
We have given points as (-7, 7) and (7, -1.1),
So slope = (-1.1 – 7)/ (7 -(-7)) = -8.1/14.

Question 3.
y = 6x – 2
Slope is 6,

Explanation:
A slope of a line is the change in y coordinate with respect to
the change in x coordinate. The equation of the line is written in the
slope-intercept form, which is: y = mx + b,
where m represents the slope and b represents the y-intercept.
Therefore the given line y = 6x – 2 has slope 6.

Question 4.
y – x = 4
Slope is 1,

Explanation:
A slope of a line is the change in y coordinate with respect to
the change in x coordinate.
The equation of the line is written in the slope-intercept form,
which is: y = mx + b, where m represents the slope and b represents
the y-intercept. Therefore the given line y – x = 4, y = x + 4 has slope 1.

Question 5.
-3 = $$\frac{2}{3}$$x – y
Slope is $$\frac{2}{3}$$,

Explanation:
A slope of a line is the change in y coordinate with respect to
the change in x coordinate.
The equation of the line is written in the slope-intercept form,
which is: y = mx + b, where m represents the slope and
b represents the y-intercept.
Therefore the given line -3 = $$\frac{2}{3}$$x – y
y = $$\frac{2}{3}$$x + 3 has slope $$\frac{2}{3}$$.

Question 6.
x + y = -5