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 to numerical modeling of non-equilibrium fluctuation phenomena.
As in scuff-cas3d, you can request either (a) frequency-resolved information on heat-transfer rates and NEQ Casimir forces (in which case you will specify a list of frequencies and will get back a list of frequency-specific values of energy and momentum fluxes) or (b) frequency-integrated information, in which case you will assign temperatures to each body in your geometry and numerically integrate the fluxes, weighted by appropriate Bose-Einstein factors, to obtain the total heat-transfer rate or NEQ Casimir force. (scuff-neq doesn't actually do the frequency integration---that's handled by the separate utility code scuff-integrate, as discussed below in the section what scuff-neq actually computes.)
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 values of the frequency-resolved or frequency-integrated quantities for each transformation you specify.
A bonus feature of scuff-neq that is not present in scuff-cas3d is the ability to obtain spatially-resolved information on energy and momentum fluxes. More specifically, you can specify to scuff-neq a list of evaluation points and you will get back values of the frequency-resolved Poynting flux and Maxwell stress tensor at each spatial point you requested, which scuff-integrate will integrate to yield thermally- and temporally-averaged quantities.
In addition to numerical output on heat-transfer rates and Casimir quantities, you can also request visualization outputs that plot the spatial distribution of the heat or momentum flux.
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 scuff-cas3d reports only the Casimir force on one body in
a geometry (namely, the first body listed in the
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.
- 1. What scuff-neq actually computes
- 2. scuff-neq command-line options
- 3. scuff-neq output files
- 4. Examples of calculations using scuff-neq
1. What scuff-neq actually computes
scuff-neq implements the FSC approach to non-equilibrium fluctuation phenomena, an algorithm for computing the thermal averages of power, force, and torque (PFT) quantities in geometries consisting of homogeneous material bodies at various temperatures embedded in an finite-temperature or zero-temperature environment. scuff-neq can compute both spatially-resolved and spatially-integrated PFT quantities. (Examples of spatially-resolved quantities include components of the average Poynting flux or Maxwell stress tensor at individual points in space. Examples of spatially-integrated quantities include the total power absorbed by, or the total force or torque on, a compact homogeneous body. Spatially-integrated quantities are generally obtained by integrating spatially-resolved quantities over closed bounding surfaces, although this is not necessarily the way they are computed by scuff-neq.)
In general, for a geometry consisting of multiple homogeneous material regions, PFT quantities receive contributions from source fluctuations in all regions and at all frequencies, and the the thermal average of a PFT quantity Q may be written in the form
where is the temperature of region , is a temperature-independent generalized flux describing the contribution of frequency- source fluctuations in region , and is the Bose-Einstein factor.
The sum over regions in this equation includes the contributions of the external environment. To isolate these contributions it is convenient to decompose into a sum of two terms:
The quantity is the average value of that would obtain if the temperature in all material regions were equal to the environment temperature ---that is, it is the equilibrium value of at temperature . The equilibrium value of PFT quantities may be computed by methods that are less costly than scuff-neq. (For example, if is a spatially-integrated force or torque, then is just the equilibrium Casimir force, which is computed efficiently by scuff-cas3d. On the other hand, if is a spatially-integrated power transfer quantity, then identically.) Thus this contribution is not computed by scuff-neq.
The quantity is the extent to which 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 ---known as a generalized flux--- that enters the integral defining ; 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 integrals and compute thermally-averaged power, force, and torque quantities at various temperatures.
2. scuff-neq command-line options
scuff-neq recognizes the following subset of the list of commonly accepted options to scuff-em command-line codes.
Options requesting output quantities
Specifies the quantities in which you are interested:
absorbed power (
--PAbs), radiated power (
Cartesian force components, or Cartesian torque components.
You may specify none, all, or any subset of these options,
but each option you specify will generally increase
the computation time (you can scrutinize the
.log file to see how much additional time each
extra output quantity takes to compute).
Option requesting visualization output
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.
Options controlling the computation of power, force, and torque
As detailed in this paper, there are several ways to compute PFTs in surface-integral formulations, including the "displaced-surface-integral" (DSIPFT), "equivalence principle" (EPPFT), and "overlap" (OPFT) methods.
By default, scuff-neq uses different algorithms for different cases of the PFT computation:
- Power computation (self term): EPPFT
- Power computation (non-self terms): EPPFT
- Force/Torque computation (self term): DSIPFT
- Force/Torque computation (non-self term): OPFT
However, you can override this default behavior
--ForceDSI, in which case DSIPFT
will be used in all cases.
The other options here set specific parameters that are only used for DSIPFT calculations.
--DSIMesh specifies the name of a mesh file
(such as a GMSH-produced
.msh file) that
defines the bounding surface over which the
surface integrals are computed. (This surface is
automatically displaced and rotated appropriately
for each object in accordance with any
geometrical transformations that may be
specified in the
.scuffgeo file and/or the
.trans file.) The surface integral is evaluated
via a one-point cubature over the surface of
each panel in the bounding mesh; thus, the
finer the bounding mesh, the more accurate and
the more expensive the computation).
If you do not specify a
--DSIMesh, then the
surface integral for each object is computed
using Lebedev cubature over a bounding sphere
centered at the origin of coordinates of the
object mesh (appropriately displaced if the
object has been displaced via statements in
.trans files.) In this case,
you may use
set the radius of this sphere (in microns) and
the number of Lebedev cubature points (the more
points, the more accurate and expensive the calculation).
To see the possible values that may be specified
Finally, you may use
--DSIFarField to request
that the Poynting vector and Maxwell tensor
on the bounding surface be computed using only
the far-field (radiation-zone) contributions
of the surface currents to the fields.
Omit the contributions of sources in individual bodies to the total PFTs on those bodies themselves.
3. scuff-neq output files
Output files for spatially-integrated PFTs: The
If you requested the computation of any spatially-integrated
PFTs (by setting command-line options such as
you will get back a file with extension
"fpatially-integrated flux") reporting values
of the generalized fluxes at each frequency you requested.
Generalized fluxes represent frequency-resolved contributions to
thermally-averaged rates of energy and momentum transfer;
as noted, above, they are multiplied by Bose-Einstein
factors and integrated over frequency to yield full
thermally and temporally-averaged data.
To understand what is written to these files, let be the spatially-integrated PFT on a destination body , and write the FSC decomposition of the thermal average of in the form
In this equation,
is a dimensionless frequency variable: , where rad/sec. (Thus agrees numerically with the arguments to the
is a dimensionless version of the usual Bose-Einstein factor, defined by .
is the difference between the dimensionless Bose-Einstein factors of source body and the environment.
is the temperature-independent generalized flux describing the contributions of fluctuations in source body to the power, force, or torque on destination body . Values of this quantity are written to the
is the spectral density of temperature-weighted contributions from fluctuations in source body to the PFT on destination body . Values of this quantity are written to the
Finally, is the total thermally-averaged PFT on body . Values of this quantity are written to the
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
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
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 (
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.
Output files for spatially-resolved PFTs: The
If you requested the computation of spatially-resolved
power and momentum flux (by specifying the
command-line option), you will get back files reporting
various contributions to these quantities. The breakdown
here is similar to that described above for spatially-integrated
quantities. To understand this, let be
a spatially-resolved PFT quantity (a component
of the Poynting vector or Maxwell stress tensor)
at a point . Then the thermal average of
may be written in the form
In this equation,
is the temperature-independent generalized flux describing the contributions of fluctuations in source body to the Poynting flux or Maxwell stress at . Values of this quantity are written to the
is the spectral density of temperature-weighted contributions from fluctuations in source body to the Poynting flux or Maxwell stress at . Values of this quantity are written to the
Finally, is the total thermally-averaged Poynting vector or Maxwell stress tensor at . Values of this quantity are written to the
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
.NEQPFTfile are watts for power, nanoNewtons for force, and nanoNewtons microns for torque.
The quantities in the
.SIIntegrandoutput file are the PFT quantities per unit dimensionless frequency, so have the same units as the corresponding quantities in the
The quantities in the
.SIFluxoutput 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
.SIIntegrandfile, 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.