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Modeling non-equilibrium electromagnetic fluctuations with scuff-neq

scuff-neq is an application code in the scuff-em suite for studying non-equilibrium (NEQ) electromagnetic-fluctuation-induced phenomena--specifically, for computing radiative heat-transfer rates and non-equilibrium Casimir forces and torques for bodies of arbitrary shapes and arbitrary (linear, isotropic, piecewise homogeneous) frequency-dependent permittivity and permeability.

scuff-neq implements the fluctuating-surface current (FSC) approach approach to numerical modeling of non-equilibrium fluctuation phenomena. The FSC method was first developed to compute equilibrium Casimir forces (as implemented by scuff-cas3d) [1] and was subsequently extended to handle non-equilibrium radiative heat-transfer problems [2] and later non-equilibrium forces and torques [3].

As discussed in more detail below, what scuff-neq actually computes are temperature-independent quantities known as generalized fluxes that describe frequency-resolved contributions to various quantities of interest in fluctuational electromagnetism, including both (a) spatially-integrated (SI) quantities such as the total rate of heat absorbed or radiated by, or the total force or torque on, a material body, and (b) spatially-resolved (SR) quantities such as the Poynting flux or Maxwell stress tensor at individual points in space. scuff-neq outputs data files giving values of generalized fluxes at various frequencies you specify; these data may then be integrated by the separate utility code scuff-integrate to compute full thermally-averaged values of SI and SR quantities for various body temperatures.

When requesting spatially-integrated quantities, you will specify which of the various computational techniques you want the code to use to compute total energies and forces/torques on bodies; the main options are the displaced surface-integral (DSI) method, which integrates the Poynting vector or Maxwell stress tensor over a bounding surface surrounding the body, or the energy-momentum transfer (EMT) method, which integrates the local power absorption and force over the volume of the body. (Algorithms for spatially-integrated PFT calculation are discussed in Refs. [ [3] and [4] ](scuff-neq.md#bibliography) below.

When requesting spatially-resolved quantities, you will specify a list of evaluation points, and you will get back a data file giving (generalized fluxes for) the Poynting vector and Maxwell stress tensor at each point.

scuff-neq also offers options for generating visualization files showing the spatial distribution of energy and momentum absorption by bodies.

As in scuff-cas3d, you can specify an optional list of geometrical transformations describing various displacements and rotations of the bodies in your geometry; in this case you will get back multiple copies of all output quantities, one for each transformation in your list.

For Casimir forces and torques, the quantities computed by scuff-neq are only the non-equilibrium contributions to the total force and torque---that is, the contributions arising from the temperature differences between individual bodies and the surrounding environment. To get the total force, these must be added to the equilibrium contributions, which are the Casimir forces and torques for the case in which all bodies are at the temperature of the environment. These contributions must be computed by doing a separate scuff-cas3d calculation, on the same geometry, at the temperature of the external medium. (For heat-transfer rates there is of course no equilibrium contribution, as there is no net power transfer between bodies at thermal equilibrium.)

One difference between scuff-cas3d and scuff-neq is that, whereas in scuff-cas3d you use command-line options like --zForce or --xTorque to request computation of specific SI quantities, in scuff-neq you automatically get all 8 SI quantities (heat radiated, heat absorbed, plus all 3 components of the force and torque).

Another difference between scuff-cas3d and scuff-neq is that, whereas scuff-cas3d reports only the Casimir force on one body in a geometry (namely, the first body listed in the .scuffgeo file), scuff-neq reports forces and heat-transfer rates for all bodies in the geometry. [The extra information would typically be redundant in an equilibrium Casimir calculation, since the equilibrium Casimir force on the second body (in a two-body geometry) is just equal and opposite to the force on the first body; but in general no such relation holds in the non-equilibrium case.]

In fact, the output from scuff-neq is even more detailed than that: in addition to the total power/force/torque on each body, you also get the contributions of each individual source body to those quantities. All of this means that the output from scuff-neq requires some effort to interpret, as discussed in more detail below.

Table of Contents

1. What scuff-neq actually computes
2. scuff-neq command-line options
3. scuff-neq output files
4. Using scuff-integrate to perform frequency integrals
4. Examples of calculations using scuff-neq
- Bibliography

1. What scuff-neq actually computes

Consider a collection of one or more homogeneous material bodies, each maintained at a given temperature, embedded in an finite- (or zero-) temperature environment. The electromagnetic fields radiated by thermally-induced fluctuating sources in each body carry energy and momentum, which we may characterize either in terms of (a) spatially-resolved (SR) quantities like the time-average Poynting vector or Maxwell stress tensor at given points in space, or (b) spatially-integrated (SI) quantities like the total time-average heat absorbed or radiated by, or force or torque on, entire bodies. In the FSC approach to non-equilibrium fluctuation phenomena implemented by scuff-neq, the total thermally-averaged value of any SR or SI quantity $Q$ is computed by summing the contributions of fluctuations from all bodies at all frequencies:

$\big\langle Q\big\rangle = \int_0^\infty \, \sum_b \, \Theta(T_b,\omega) \Phi_b(\omega)\,d\omega$ where $T_b$ is the temperature of body $b$ , $\Phi_b(\omega)$ is a temperature-independent generalized flux describing the contribution of frequency- $\omega$ source fluctuations in body $b$ , and $\Theta(T_b,\omega) = \frac{\hbar\omega}{e^{\hbar \omega/kT_b} - 1}$ is the Bose-Einstein factor.

The sum over bodies $b$ in this equation includes the contributions of the external environment. To isolate these contributions it is convenient to decompose $\langle Q \rangle$ into a sum of two terms: $\begin{array}{rcl} \big\langle Q\big\rangle &=& \big\langle Q\big\rangle^{\small EQ} + \big\langle Q\big\rangle^{\small NEQ} \\[8pt] \big\langle Q\big\rangle^{\small EQ} &\equiv& \displaystyle{\int_0^\infty} \Theta(T_{\small env},\omega) \sum_b \, \Phi_b(\omega)\,d\omega \\[8pt] \qquad \big\langle Q\big\rangle^{\small NEQ} &\equiv& \displaystyle{\int_0^\infty \sum_b } \Big[ \Theta(T_b, \omega) - \Theta(T_{\small env},\omega)\Big] \Phi_b(\omega)\,d\omega \\[4pt] &=& \displaystyle{\int_0^\infty \sum_b} \Delta \Theta(T_b, \omega) \Phi_b(\omega)\,d\omega \end{array}$

where

$\Delta \Theta(T_b, \omega) \equiv \Theta(T_b, \omega) - \Theta(T_{\small env},\omega).$

The quantity $\langle Q\rangle^{\small EQ}$ is the average value of $Q$ that would obtain if the temperature in all material regions were equal to the environment temperature $T_{\small env}$ ---that is, it is the equilibrium value of $\langle Q\rangle$ at temperature $T_{\small env}$ . The equilibrium value of PFT quantities may be computed by methods that are less costly than scuff-neq. (For example, if $Q$ is a spatially-integrated force or torque, then $\langle Q\rangle^{\small EQ}$ is just the equilibrium Casimir force, which is computed efficiently by scuff-cas3d. On the other hand, if $Q$ is a spatially-integrated power transfer quantity, then $\langle Q\rangle^{\small EQ}=0$ identically.) Thus this contribution is not computed by scuff-neq.

The quantity $\langle Q\rangle^{\small NEQ}$ is the extent to which $\langle Q\rangle$ deviates from its equilibrium value, and the sum in its definition ranges only over the source bodies in the geometry, not including the environment contribution.

The job of scuff-neq is to compute the quantity $\Phi_b(\omega)$ ---known as a generalized flux--- that enters the integral defining $\langle Q\rangle^{\small NEQ}$ ; scuff-neq will do this computation at each of multiple frequencies that you specify in advance. These data are then used by the separate scuff-integrate utility to evaluate the actual $\omega$ integrals and compute thermally-averaged power, force, and torque quantities at various temperatures.

2. scuff-neq command-line options

Options specifying the geometry

````--geometry MyGeometry.scuffgeo --TransFile MyTransFile

{.toc}

The mandatory `--geometry` option specifies the
[<span class=SC>scuff-em</span> geometry file][scuffEMGeometries]
describing your geometry.

The optional `--TransFile` option specifies a
[list of geometrical transformations][scuffEMTransformations]
to be applied to your geometry. Each output quantity you
request will be separately computed and reported for 
each transform you request.

#### Options specifying frequencies

````--omega 1.23
--OmegaFile MyOmegaFile

The first option here requests computation at just a single angular frequency (interpreted in standard scuff-em units of $\omega_0=3\cdot 10^{14}$ rad/sec).

The second option specifies a file containing a list of frequencies $\omega$ (one per line) at which to compute generalized fluxes $\Phi(\omega)$ .

Options requesting spatially-integrated output quantities

--DSIPFT
--OPFT

Request calculation of spatially-integrated quantities via one of the available methods: the energy/momentum transfer (EMT) method, the displaced-surface-integral (DSI) method, or the overlap (O) method. You may specify none, any combination, or all of these options.

Note that all of these methods are computing the same quantities, so in principle their results should be equivalent, and specifying more than one should be redundant. However, in practice it is often helpful to compute SI quantities via multiple methods as a confirmation and accuracy check.

For the particular case of --DSIPFT, there are several options controlling the particular implementation of the algorithm; see below.

Option requesting spatially-resolved output quantities

--EPFile MyEPFile

Specifies a file containing a list of evaluation points, one per line (three Cartesian coordinates), at which values of the Poynting vector and Maxwell stress tensor are to be computed.

Options controlling DSI calculations

The displaced-surface-integral (DSI) method total absorbed/radiated power, force, and torque on each body in a geometry by integrating the Poynting flux and Maxwell stress over a bounding surface surrounding the body. You have two options for what bounding surface is used: (a) a gmsh mesh file you provide giving a discretized version of your bounding surface, or (b) a sphere centered at the body. (In both cases, the surface is displaced and re-centered at the origin of each body to compute DSIPFT for that body.)

To use option (a), say

 --DSIMesh    MyDSIMesh.msh

If the --DSIMesh option is not specified, scuff-neq defaults to a spherical bounding surface. In this case you may use the command-line options

 --DSIRadius  2.0
 --DSIPoints  302

to set the radius of this sphere and the number of cubature points used to approximate the surface integral over it. (Use scuff-neq --help to see a list of the available values of --DSIPoints). If you don't specify values for these options, they will be set automatically.

 --DSIPoints2 770

For the case of a spherical bounding surface, you may specify -DSIPoints2 to request a "second opinion" DSI calculation with a different number of cubature points to check the accuracy of the numerical cubature. The second-opinion DSI data will be written to a file with file extension .SIFlux.DSI2 (This is optional; if you don't specify --DSIPoints2 no second-opinion calculation is done.)

 --DSIOmegaFile DSIOmegaFile

Because the DSI calculation can be slow, in some cases you may wish to request that it only be performed at a subset of the frequencies in your --OmegaFile. You specify that subset using the -DSIOmegaFile option. For example, you might say --EMTPFT to request EMT calculations at all frequencies, but use --DSIPFT with a --DSIOmegaFile containing only every 5th or 10th frequency in your full OmegaFile to obtain a sanity check on the EMTPFT results.

Option requesting visualization output

--PlotFlux

This option directs scuff-neq to produce visualization files (in addition to its usual output files) which may be opened in gmsh to visualize, for each spatially-integrated PFT quantity you requested, the spatial distribution of the Poynting flux or Maxwell stress on the surfaces of objects or the displaced bounding surfaces over which those quantities are integrated to compute the total PFT quantity.

Other options

--FileBase MyFileBaSe

Specifies the base filename for all output files. If this option is not specified, --FileBase is set to the base filename of the .scuffgeo file.

--SourceObject MySourceObject
--DestObject   MyDestObject

For a geometry containing $N$ bodies, by default computes the full $N\times N$ matrix of generalized flux quantities $Q_{s\to d}$ describing the energy/momentum transfer to destination body $d$ due to fluctuations in source body $s$ . You may use either or both of --SourceObject and --DestObject to compute only quantities sourced by a given source body and/or only energy/momentum transfer to a given destination body; the arguments to these options are the labels you specified for the objects in question in the .scuffgeo file.

Note: Because of the way the EMTPFT method works, this algorithm always computes results for all destination bodies, so --DestObject has no effect on EMTPFT calculations. (But --SourceObject can still be used to reduce the number of EMTPFT calculations performed.)

--OmitSelfTerms

This flag may be set to bypass the calculations of self-contributions (i.e. fluxes of the form $\Phi_{s\to s}$ ).

3. scuff-neq output files

The `.log` file

Like all command-line codes in the scuff-em suite, scuff-cas3d writes a .log file that you can monitor to keep track of your calculation's progress.

Output files for spatially-integrated PFTs: The `.SIFlux.METHOD` file

If you used an option like --EMTPFT or --DSIPFT to request computation of spatially-integrated PFT data, you will get back a file with extension .SIFlux.METHOD (where SIFlux stands for "spatially-integrated flux" and where METHOD is replaced by the method used, i.e., EMTPFT, DSIPFT, etc.) reporting values of the generalized fluxes $\Phi(\omega)$ at each frequency you requested.

Generalized fluxes represent temperature-independent frequency-resolved contributions to thermally-averaged rates of energy and momentum transfer; as noted, above, they are multiplied by temperature-dependent Bose-Einstein factors and integrated over frequency to yield full thermally and temporally-averaged data. This integration operation is carried out by the utility code scuff-integrate, as described below.

Output files for spatially-resolved PFTs: The `.SRFlux` file

If you requested the computation of spatially-resolved power and momentum flux (by specifying the --EPFile command-line option), you will get back a file with extension .SRFlux reporting temperature-dependent frequency-resolving contributions to the Poynting flux and Maxwell stress at each point in your EPFile. (See below for more information on what these values mean.)

4. Using scuff-integrate to perform frequency integrals

After running scuff-neq to get temperature-independent generalized flux data at various frequencies, we run scuff-integrate to evaluate the frequency integrals that compute total thermally-averaged rates of energy and momentum transfer.

Specifying temperatures

To convert frequency-resolved data into thermally-averaged data, we need to know the temperature of all bodies in the geometry, as well as that of the environment; in general, we will want results for multiple different combinations of body and environment temperatures This information is specified to scuff-integrate in the form of a simple text file (I usually call it simply TemperatureFile) in which each line describes a separate assignment of temperatures to the environment and to the various bodies in the geometry, with format

 TEnv   T1   T2   ...   TN

where TEnv is the environment temperature and T1, T2, ..., TN are the temperatures of the first, second, ..., Nth bodies in your geometry (indexed in order of their appearance in the .scuffgeo file).

For example, here's a TemperatureFile for a two-body geometry in which we request data for the first body held at temperatures of $T_1=\{50,100,150\}$ K with the second body held fixed at $T_2=100$ K and the environment kept at $T_{\text{env}}=0$ K (temperatures are always specified in units of Kelvin):

0 50  100
0 100 100 
0 150 100

Spatially-integrated quantities

For spatially-integrated quantities (total heat radiation/absorption, force, and torque on bodies), scuff-integrate inputs a .SIFlux produced by scuff-neq and produces two further output files with extensions .NEQPFT and .SIIntegrand, which report respectively the total (frequency-integrated) thermally-averaged quantities and the temperature-weighted integrand of the frequency integral that produced them.

To understand the various quantities written to these 3 files, let $Q_d$ be the spatially-integrated PFT on a destination body $d$ , and write the FSC decomposition of the thermal average of $Q_d$ in the form

$\big\langle Q_d\big\rangle = \underbrace{ \Bigg[ \int_0^\infty \underbrace{ \bigg\{ \hbar\omega_0^2 \sum_b \, \Delta \widehat \Theta_b(u) \underbrace{ \Phi_{s\to d}(u)}_{\texttt{.SIFlux.XXXPFT}} \bigg\} }_{\texttt{.SIIntegrand}} \,\,du \Bigg] }_{\texttt{.NEQPFT}}$

In this equation,

$u$ is a dimensionless frequency variable: $u=\omega/\omega_0$ , where $\omega_0=3\cdot 10^{14}$ rad/sec. (Thus $u$ agrees numerically with the arguments to the --omega option.)
$\widehat\Theta(u)$ is a dimensionless version of the usual Bose-Einstein factor, defined by $\Theta(\omega)=\hbar \omega_0 \widehat\Theta(u)$ .
$\Delta \widehat \Theta_b(u)=\widehat \Theta_b(u) - \widehat \Theta_{\small env}(u)$ is the difference between the dimensionless Bose-Einstein factors of source body $s$ and the environment.
$\Phi_{s\to d}$ is the temperature-independent generalized flux describing the contributions of fluctuations in source body $s$ to the power, force, or torque on destination body $d$ . Values of this quantity are written to the .SIFlux.METHOD output file (where METHOD is a string like EMTPFT identifying the spatial-integration method used).
$\hbar\omega_0^2 \Delta \widehat \Theta_b(u) \Phi_{s\to d}(u)$ is the spectral density of temperature-weighted contributions from fluctuations in source body $s$ to the PFT on destination body $d$ . Values of this quantity are written to the .SIIntegrand output file.
Finally, $\langle Q_d \rangle$ is the total thermally-averaged PFT on body $d$ . Values of this quantity are written to the .NEQPFT output file.

In all of these files, each single line corresponds to a single frequency, a single geometric transformation, and a single pair of (source,destination) objects.

At the top of each output file you will find a file header explaining the significance of each of the various columns in the file. One of the columns will be described in the header as # (source object, dest object), and will take values like 12, 22, or 02. The first case (12) indicates that the data on that line correspond to the contributions of object 1 to the PFT on object 2. (The ordering of objects corresponds with the order of their appearance in the .scuffgeo file). The second case (22) indicate that the data on that line correspond to the self-contributions of object 2 to its own PFT. The third case (02) indicates that the data on that line correspond to the total PFT on object 2---that is, the sum of contributions from all source objects.

Spatially-resolved quantities

For spatially-resolved data, scuff-integrate inputs a .SRFlux produced by scuff-neq and produces two further output files with extensions .PVMST and .SRIntegrand, which report respectively values of the total (frequency-integrated) thermally-averaged Poynting Vector and Maxwell Stress Tensor and the temperature-weighted integrand of the frequency integral that produced them.

The breakdown here is similar to that described above for spatially-integrated quantities. To understand this, let $Q(\mathbf{x})$ be a spatially-resolved PFT quantity (a component of the Poynting vector or Maxwell stress tensor) at a point $\mathbf{x}$ . Then the thermal average of $Q$ may be written in the form

$\big\langle Q(\mathbf{x})\big\rangle = \underbrace{ \Bigg[ \int_0^\infty \underbrace{ \bigg\{ \hbar\omega_0^2 \sum_b \, \Delta \widehat \Theta_b(u) \underbrace{ \Phi_{s\to\mathbf x}(u)}_{\texttt{.SRFlux}} \bigg\} }_{\texttt{.SRIntegrand}} \,\,du \Bigg] }_{\texttt{.PVMST}}$

In this equation,

$\Phi_{s\to \mathbf{x}}$ is the temperature-independent generalized flux describing the contributions of fluctuations in source body $s$ to the Poynting flux or Maxwell stress at $\mathbf{x}$ . Values of this quantity are written to the .SRFlux output file.
$\hbar\omega_0^2 \Delta \widehat \Theta_b(u) \Phi_{s\to \mathbf{x}}(u)$ is the spectral density of temperature-weighted contributions from fluctuations in source body $s$ to the Poynting flux or Maxwell stress at $\mathbf{x}$ . Values of this quantity are written to the .SRIntegrand output file.
Finally, $\langle Q(\mathbf{x})\rangle$ is the total thermally-averaged Poynting vector or Maxwell stress tensor at $\mathbf{x}$ . Values of this quantity are written to the .PVMST output file.

In all of these files, each line corresponds to a single frequency, a single geometric transformation, and a single source object. At the top of each output file you will find a file header explaining how to interpret the various data columns on each line.

Units of output quantities

The units of the total (frequency-integrated) spatially-integrated output quantities reported in the .NEQPFT file are watts for power, nanoNewtons for force, and nanoNewtons $\times$ microns for torque.
The quantities in the .SIIntegrand output file are the PFT quantities per unit dimensionless frequency, so have the same units as the corresponding quantities in the .NEQPFT file.
The quantities in the .SIFlux output file are the quantities per unit dimensionless frequency per watt of thermal energy, so these quantities have the same units as the quantities in the .NEQPFT and .SIIntegrand file, but divided by watts: thus the power flux is dimensionless, the force flux has units of nanoNewtons / watts, and the torque flux has units of nanoNewtons microns/watts.

4. Examples of calculations using scuff-neq

Bibliography

Here are the original papers cited above describing the FSC approach to fluctuational electromagnetism:

The various algorithms for computing spatially-integrated data (DSI, EMT, etc.) are described in Ref. 3 here; see also this paper.