# Integrating frequency-dependent data with scuff-integrate

Many application codes in the scuff-em suite compute physical quantities defined by definite integrals over real or imaginary frequencies, with the numerical value of the integrand at each point obtained by solving individual scuff-em scattering problems at that frequency. For example,

• scuff-cas3d and scuff-caspol compute zero-temperature Casimir quantities by integrating contributions from imaginary frequencies $\xi$: Here $Q$ is a zero-temperature Casimir energy/force/torque (scuff-cas3d) or Casimir-Polder potential (scuff-caspol) and $I(\xi)$ is the spectral density of contributions to $Q$ from fluctuations at imaginary frequency $\xi$, which may be obtained by solving scuff-em scattering calculations at imaginary frequency $\xi$.

• scuff-neq computes the total rate of energy or momentum transfer from a source body $s$ to a destination body $d$ by integrating contributions from real frequencies $\omega$: Here $Q_{s\rightarrow d}$ is the contribution of body $s$ to quantity $Q$ (a heat-transfer rate, force, or torque) for body $d$, $\Theta(T,\omega)=\frac{\hbar\omega}{e^{\hbar\omega/kT}-1}$ is the Bose-Einstein statistical factor, $T_s$ and $T_\text{env}$ are the temperatures of the source body and the environment, and $\Phi(\omega)$ is a "generalized flux" quantity that may be computed by solving scuff-em scattering calculations at frequency $\omega$.

The integrals over $\xi$ and $\omega$ are evaluated by numerical cubature---that is, as weighted sums of integrand samples. In a perfect world, it would be possible for scuff-em application codes to choose appropriate integration strategies automatically, hiding these details from the user and reporting just the frequency-integrated quantities $Q$. This is in fact the strategy that was adopted in early incarnations of the scuff-em codes.

In the real world, however, the behavior of integrand functions like $I(\xi)$ and $\Phi(\omega)$ varies widely from problem to problem, depending on factors such as the shapes and materials of bodies in the scattering geometry and the quantity being computed. For this reason, it's hard for scuff-em to make intelligent automatic choices of integration strategies, and attempts to do so without user input may result in misleading or even flat-out incorrect data.

For this reason, the modern approach to frequency integration in scuff-em is to ask users to define a list of frequencies at which to sample the integrand; this list is passed to scuff-em application codes using the --XiFile or --OmegaFile command-line options, and in response the code produces an output file reporting values of the integrand at the specified points. The frequency integral may then be calculated as as post-processing step using the information reported in the frequency-resolved data files, and this is the task for which scuff-integrate exists.

scuff-integrate is a general-purpose tool, not limited to scuff-em applications

Actually, scuff-integrate is designed to be a general-purpose numerical-integration tool, not particularly tied to scuff-em; given a data file containing samples of some function $f(x)$ at sample points $\{x_n\}$ distributed over an interval $[x_a, x_b]$, scuff-integrate approximates the integral $\int_{x_a}^{x_b} f(x) \, dx$. (It does this by constructing a third-order spline interpolant through the data, then integrating the interpolant via numerical quadrature.) The code offers command-line options to specify how the data file is to be interpreted (for example, which columns are the $x$ samples and which are the $f$ samples). There is no restriction on the number or spacing of the sample points; the variable x need not correspond to a real or imaginary frequency, and the function $f$ need not be a generalized flux, Casimir integrand, or any other quantity reported by a scuff-em calculation.

But for data files that do come from scuff-em calculations, scuff-integrate knows what to do automatically with minimal user input

In addition to the fully general-purpose mode described above, scuff-integrate also has built-in automatic support for certain specialized types of scuff-em data files---such as the .SIFlux files produced by scuff-neq---designed to make it easy to run the tool to get the frequency-integrated data you want with minimal user input required.

## 1. Using scuff-integrate as a general-purpose numerical integrator

### Integrating a single function of a single variable

The simplest usage of scuff-integrate is to integrate a single function $f$ of a single variable $x$; as noted above, this is a general-purpose use of the code, independent of scuff-em, in which $x$ and $f$ need not be a frequency and a generalized flux, but could have any arbitrary significance. Suppose we have a file called fData in which are tabulated numerical pairs $(x_n, f_n)$, $n=1,2,\cdots,N$, where $f_n=f(x_n)$:

x1 f1
x2 f2
...
xN fN


Then to compute $\int_{x_1}^{x_N} f(x) \, dx$ we can just go like this:

% scuff-integrate --datafile fData --freqColumn 1 --dataColumn 2


This will produce a file named fData.Integrated containing a single line of data: the integrated value of $f$.

If the frequency and/or integrand values are printed in different columns of the data file, just adjust the --freqColumn and --dataColumn options accordingly. For example, if the data file looks like this:

<stuff> <stuff> x1 <stuff> <stuff> f1 <stuff> ...
<stuff> <stuff> x2 <stuff> <stuff> f2 <stuff> ...
...
<stuff> <stuff> xN <stuff> <stuff> fN <stuff> ...


you would use --freqcolumn 3 --datacolumn 6. In this case, the content of the other columns (the <stuff> in the above snippet) is ignored.

### Integrating multiple functions of frequency

More generally, you may have multiple integrand functions $f_1, \cdots, f_N$, each sampled at the same set of $x$ values, You can integrate all of these at once simply by specifying multiple --dataColumn options. For example, if you have functions $f$ and $g$ and you have a file named fgData with the format

x1 f1 g1
x2 f2 g2
...
xN fN gN


then you can say

% scuff-integrate --datafile fgData --freqColumn 1 --dataColumn 2 --datacolumn 3


and the resulting output file fgData.Integrated will report values for both $\int f(x)\,dx$ and $\int g(x)\,dx$.

### Giving names to data columns

In the legend at the top of the .Integrated output file, the values of the various integrated functions will by default be labeled data 0, data 1, etc. If you want to give more descriptive names, just follow each --dataColumn option with a --dataName option.

For example, if columns 8 and 11 of your data file respectively report values of force and torque integrands, you might say , say --dataColumn 8 --dataName Force --dataColumn 11 --dataName Torque.

### Integrating functions of frequency and other parameters

In many cases we will have functions that depend on various parameters beside the integration variable, i.e. functions of the form $f(x;p_1, \cdots, p_N)$ where the data file includes integrand samples for multiple sets of values of the $(p_1,\cdots,p_n)$ parameters. (In scuff-cas3d, for example, we might compute Casimir forces between particles separated by various distances $d$, so the integrand function may be thought of as a function $f(x,d)$ of both distance and frequency.) In these cases you will generally want to evaluate separate integrals over $x$ for each set of parameter values represented in your data file.

For example, suppose your data file is called pxfgData and looks something like

p1 x1 f11 g11
p1 x2 f12 g12
....
p1 xN f1N g1N
p2 x1 f21 g21
p2 x2 f22 g22
....
p2 xN f2N g2N
....
pM xN fMN gMN


where p1, p2, ..., pM denote $M$ distinct values of some parameter $p$ and fmn,gmn are the numerical integrand values $f(p_m, x_n), g(p_m,x_n)$. In this case you can't simply say --freqColumn2 --dataColumn 3 --dataColumn 4, because then data for all parameter values will be mashed all together and integrated as a single function of frequency, yielding nonsense.

Instead, you handle this situation by specifying the additional command-line parameter --tagColumn 1 to tell scuff-integrate to interpret data lines with different values in column 1 as samples of different functions:

% scuff-integrate --datafile pxfgData --tagcolumn 1 --freqColumn 2 --dataColumn 3 --dataColumn 4


In this case, the output file pxfgData.Integrated will report $x$-integrated values of $f$ and $g$ separately for each value of $p$.

If your integrands depend on multiple parameters $(p,q,\cdots)$, you may specify multiple --tagColumn options to specify the columns in which values of the various parameters live. Then each line of the .Integrated output file will report $x$-integrated values of all functions for a single tuple of parameter values $(p,q,\cdots).$

## 2. Using scuff-integrate as a specialized integrator for scuff-em data

As noted above, scuff-integrate has built-in knowledge of the structure of various data files produced by scuff-em calculations. This streamlines calculations by allowing you you to feed those data files directly into scuff-integrate without needing to tell the code which column of the data file is which.

### 2A. Integrating scuff-neq data

As discussed at the top of the page, the frequency-resolved data reported by scuff-neq are generalized fluxes describing temperature-independent rates of energy and momentum transfer from each body to each other body in your geometry. To turn these into total heat-transfer rates, forces, and torques for a given set of object and environment temperatures, we must (a) integrate over frequency with Bose-Einstein thermal weight factors appropriate for the given temperatures, and (b) sum the contributions of multiple source bodies to yield the total rates of power and momentum transfer to each destination body. scuff-integrate already knows the file format of the .SIFlux and .SRFlux frequency-resolved data files produced by scuff-neq, so there is no need to specify --FreqColumn and --DataColumn options.

#### Specifiying temperatures

Instead, the only input you need to supply (besides the .SIFlux or .SRFlux data file) is a specification of the temperatures of the bodies in your geometry, and of the surrounding environment. By default, all bodies and the environment are at absolute zero, $T=0$ K; if you do not modify this situation by setting a nonzero temperature for at least one body (or the environment), all heat-transfer rates and forces/torques will be zero. The command-line option for setting

• --Temperature N T

If $N\ge 1$, this sets the temperature of the $N$th object/surface in the geometry to T Kelvin. (Here objects/surfaces are indexed using a 1-based convention; to set the temperature of the first object/surface specified in the .scuffgeo file to room temperature you would say --Temperature 1 300.)

If $N=0$, this instead sets the temperature of the environment to T.

#### Specifiying multiple temperature sets

A bonus feature of the separation between scuff-neq and scuff-integrate is that a single set of frequency-resolved flux data (in an .SIFlux or .SRFlux file) can be used to obtain data on heat-transfer rates and forces and torques at multiple sets of temperatures for the bodies and the environment, simply by evaluating integral (2) multiple times with different temperatures but the same flux data.

#### Spatially-integrated flux data (.SIFlux files)

knows how to do this automatically

In this case, for a geometry containing $N$ bodies, each line of the .SIFlux output file is tagged with a data field of the form $sd$ (where $s$ and $d$ are integers between 1 and $N$) to label the contributions of sources in body $s$ to the power, force, and/or torque (PFT) on body $d$. (For example, lines for which this field reads 13 give contributions of body 1 to the PFT for body 3). The actual data quantities reported in the .SIFlux file are the generalized fluxes $\Phi_{s\rightarrow d}$ in equation (2) above, and to evaluate the $\omega$ integral here we need to know the temperatures of the environment and of all bodies in the geometry, which enter through the Bose-Einstein factors in (2).

To handle these complications, scuff-integrate supports the following additional command-line options:

• --sdColumn xx

Specifies that the $sd$ indicator field appears on column xx of the data file. (The default is --sdColumn 3, matching the default file format of the .SIFlux files produced by scuff-neq, so for those files this option may be omitted.)

• --TemperatureFile TFile

Specifies a file containing multiple temperature configurations at which to compute total PFTs. For an $N$-body geometry, each line of TFile should contain $N+1$ space-separated numbers in the same format as the arguments to the --Temperature option, i.e. TEnv T1 ... TN.

For example, to compute PFTs in a two-body geometry with the temperature of body 1 scanned from 10 to 300 Kelvin, the temperature of body 2 held fixed at room temperature, and the environment temperature fixed at 0, TFile would look like

0 10  300
0 20  300
...
0 300 300


## 4. Miscellaneous notes

### Only numerical data columns are counted as columns

There is one potentially confusing aspect of the way scuff-integrate interprets column indices as specified by command-line arguments such as --FreqColumn or DataColumn. This is that scuff-integrate treats non-numerical data columns as white space, and in particular does not include data columns containing text strings when counting column indices.

Thus, for example, if your data file contains frequency and integrand data in the second and third columns, with the first column containing a character string, like this:

DEFAULT 0.1 3.45e-5
DEFAULT 0.2 7.82e-5
DEFAULT 0.3 1.10e-4
...


then scuff-integrate ignores the DEFAULT column and considers the first column with numerical data to be column 1, so here you would say --freqColumn 1 --dataColumn 2.

4.00000 0.1 3.45e-5

you would want to say --freqColumn 2 --dataColumn 3.