# Modeling change in public sentiment with nonlocal reaction-diffusion
equations

Joseph L. Shomberg∗ ∗Department of Mathematics and Computer Science,
Providence College, Providence, Rhode Island 02918, USA,
<EMAIL_ADDRESS>
###### Abstract.

This is a brief “proof of concept” article that shows a nonlocal variant of a
reaction-diffusion equation, which is already well suited to the study of
pattern formation, is a plausible tool for the particular application of
modeling change in public sentiment. Public sentiment is ubiquitous in modern
society; from the marketing of consumer goods to the choice of political
rhetoric, we are all affected by it. Modeling change in public sentiment has
an important role when analyzing and predicting the course society is on. Of
course, change is permanent. Perhaps the best-known feature in public
sentiment is polarization. By this, we mean the development or establishment
of a certain subset of a population whose views on a given subject are fixed
to one end of the spectrum. It is when public sentiment undergoes a change do
we witness the emergence of polarization. In large part we see polarization
arise through the use of persuasion and rhetoric. Moreover, some marketing or
arguments are more effective on certain individuals, thus leading these people
into a polarized regime. Our model captures this emergence of the polarized
regime. In addition, in some cases, we see the development of true polar
opposites (which we call mixed polarity). Our method features:

  * •

We use a nonlocal Chafee–Infante reaction-diffusion equation to model the
evolution of public sentiment in a population that interacts with other
individuals.

  * •

We employ a pseudo-random convolution kernel as a symmetric matrix of
lognormally distributed values. This kernel models the influence of
individuals when interacting with others.

  * •

Change in sentiment emerges and may converge to a polarized state expressed by
a double-well potential. Other more complicated states occur whereby a mixed
polarization emerges.

###### Key words and phrases:

Nonlocal diffusion; pattern formation; public sentiment.

##  1\. Introduction and motivation

In this article, we introduce the notion of nonlocal diffusion as a means to
describe the change, and possible polarization, that occurs in some public
sentiment. Public sentiment can be collected by a variety of sources ranging
from websites that perform online surveys to Bayesian classifiers digesting
our social media behavior. Undoubtably, such data is important to advertisers,
to name just one group, and the amount of data that is accessible today is
increasing at an ever faster rate (cf. e.g. [6, 7, 8]). According to a recent
story at blacklinko.com [1], “The social media growth rate since 2015 is an
average of 12.5% year-over-year.” A quick Google search will bring up
thousands of pages pertaining to ‘modeling public sentiment’, but here we are
interested in its evolution brought forward through nonlocal diffusion.

Differential equations have long been used to predict the evolution of an
initial state. The size of a population with a limited amount of resources can
be modeled with a logistic equation, for example. It is sometimes important to
incorporate mixing, or diffusion, into a population. This is evident whenever
a substance of high concentration prefers a region of lower concentration. We
find [2] analyzes a problem involving competing species where each “consumes
resources in some area around their average position”. This later class of
diffusion mechanism is now referred to as nonlocal diffusion. One important
feature in nonlocal diffusion is the interaction kernel. Such kernels are
functions where one can assign relevant interaction values, according to a
certain probability distribution perhaps. Besides population dynamics,
nonlocal diffusion has already earned its place in modeling several phenomena
important to the social sciences, especially economics. Notably, recent
applications to wealth distribution appear in [3] and [4].

Whereas (local) diffusion is championed for its success in modeling classical
phenomena, such as heat flow (see J. Fourier’s “The Analytical Theory of
Heat”), nonlocal diffusion is far-reaching. Rather than one particle
interacting with only strictly adjacent particles, nonlocal diffusion provides
a mechanism in which one particle can interact with all others. The underlying
rule is that one particle is allowed to interact with all others according to
the probability/interaction kernel. We will use (positive) numbers to
represent the number of interactions between two individuals. We can assume
such an interaction also represents an exchange of information. Of course, the
form of that exchange could be persuasive. Thus, nonlocal diffusion is an
ideal mathematical tool for modeling the type of interaction that often occurs
when studying sentiment.

##  2\. The model problem

We will suppose that sixteen people are given a questionnaire containing
sixteen questions asking them for their opinion on different topics (the
number of people does not have to be the same as the number of questions, but
it makes for a nicer presentation when they are). Each question is indicated
by one of the sixteen (vertical) columns of Figure 1. The sixteen different
people are represented as one of the sixteen (horizontal) rows. So, the fifth
row indicates the answer (or ‘sentiment’) to each of the questions
corresponding to the fifth person. Responses can range in the interval of
numbers $[-1,1]$, where the value $-1$ corresponds to “strongly disagree” (in
this case the color white) to the value $+1$ which corresponds to “strongly
agree” (color black). Figure 1 contains sample responses. Here, the responses
are uniformly randomly chosen.

Figure 1. A sample of sixteen individuals, represented as rows, and their
‘sentiment’ to sixteen topics, in columns.

People are allowed to discuss the topics with others who participated in the
questionnaire, as well as with people outside the group of sixteen. In this
experiment, we do limit the total number of people to 31 (the sixteen who
answered the same questions and fifteen who did not–the number 15 could be
increased or decreased). We assume that each individual may interact with each
of the others on a certain fixed timescale (e.g. daily) and that any
interaction produces a degree certain influence. We assume interaction occurs
with an exchange of communication during physical or virtual meetings. All
interactions are possible and given a nonnegative value representing the
(average) number of interactions over a specified time period. Here, influence
is a certain persuasive intent. When it comes to interacting with different
people, we may barely see someone in a month, or that person could make poor
arguments about certain topics and is unable to sway many others to their
point of view. On the other hand, some are easily persuaded and some seek out
others in an attempt to change someone’s point of view. For the purposes of
modeling change in sentiment, we do not assume one interacts with oneself. So
here, one does not change their mind on their own (however, one could easily
assign a nonzero probability for this occurrence to any of the individuals).
Finally, interactions are assumed to be symmetric, which means the influence
of Person A on Person B is not different from the influence of Person B on
Person A. (This assumption could also be dropped.)

We need to record the influence of all the possible interactions. For this we
will use a symmetric matrix, or when displayed graphically, a $31\times 31$
array where each entry contains a number represented by some small grayscale
square. Each interaction score is a random number determined by a lognormal
distribution with mean $\mu=1.0$ and standard variation $\sigma=1.7$. This
means all interactions have a nonnegative value, but increasing interaction
scores occur with less frequency. We take these interaction values to be
lognormally distributed. (It is interesting to note that the lognormal
distribution is also used in pricing stock options.) Anyway, the group of
sixteen who answered the survey make up a $16\times 16$ square inside the
$31\times 31$ interaction array shown in Figure 2.

Figure 2. Thirty-one individuals in rows and their interaction values with the
sixteen individuals participating in the survey (contained in the red square).
The white diagonal is the zeros for self interactions.

To capture polarization in public opinion, we will rely on a function that has
two attractive fixed states. The function is called a double-well potential
and is given by $F(x)=\frac{1}{4}(x^{2}-1)^{2}$.

Figure 3. A graph of the double-well potential
$F(x)=\frac{1}{4}(x^{2}-1)^{2}$.

Observe, $F(0)=0$ and $F(\pm 1)=0.$ The numbers $x=\pm 1$ refer to the stable
steady states whereas $x=0$ is said to be unstable. The graph of $F(x)$ is
like a ‘W’ and one can imagine the middle hump, $x=0$, separates sentiment
into one of the two lower ‘wells’, $x=-1$ or $x=1$. Hence, this function works
to promote positive and negative polarity. The derivation used below actually
requires the derivative of the double-well potential, $F^{\prime}$ which we
will denote by $f$.

In order to communicate the evolution, or change, of any given initial
sentiment, such as that given in Figure 1 for example, we rely on a
differential equation made from balancing the rate of change in time to the
interaction of mixing influence in the presence of two stable steady states.
We put these concepts together by borrowing a model known to applied
mathematicians as a nonlocal Chafee–Infante reaction-diffusion equation. This
equation has a celebrated history because of its archetypical impact on the
development of dynamical systems (indeed, see [5]). That is,

$\partial_{t}p_{i}+\sum_{m=1}^{16}K_{m}*p_{i}+f(p_{i})=0,\quad\text{for}\
i=1,2,\dots,16,$

where $p_{i}=p_{i}(t,x)$ is the $i$-th individual’s unknown sentiment to
question/topic $x$ at time $t$. The term $K_{m}$ represents the $m$-th column
of the extended interaction kernel $K$. The chosen convolution expresses a
Neumann-type no flux boundary condition,

$(K_{m}*p_{i})(t,x)=\int_{\Omega}K_{m}(x-y)\left(p_{i}(t,y)-p_{i}(t,x)\right)dy.$

OK, now we answer the question, “why the extra 15 people and why use the
extended interaction kernel?” Inside the kernels $K_{m}$, we see the argument
contains the difference $x-y$. The way a convolution works is like one
function, say the kernel, slides from left to right over another function, in
this case let us simply say $p_{i}(t,y)-p_{i}(t,x)$, and we sum the area of
the overlap. So to compute the integral at $x=1$, we need to account for all
$y$, i.e., $y=1,2,\dots,16.$ Hence, we need to know the value of
$K_{m}(1-16)=K_{m}(-15)$ and other terms like this. When we are at a point $x$
inside the red box in Figure 2, we need to look up 15 points to find
$K_{m}(-15)$. Note that the boundary is “wrapped” in the sense that when we go
out the top, we reappear at the bottom. Moving on, in this setting the
derivative of the double-well potential is

$f(p_{i})=p_{i}^{3}-p_{i}$

and the stationary polar sentiments $p_{i}=\pm 1$ are apparent.

We implement a simple forward-Euler numerical scheme in Python to produce two
numerical simulations that illustrate the change of individual initial
sentiment after interacting with other members of a population in the presence
of two attractive steady states. These simulations appear in Figures 4 and 5.
The simulation was run until the numerical solution approached an equilibrium.
In this case, each simulation ran until the maximum difference taken over each
of the $16\times 16$ squares between the last two iterations was less than
$0.001$. For comparison, we also provide an image of the steady-state as if no
interaction (mixing) were present. Polarization in public opinion is apparent
in the development of pure white and pure black pools of color. Interestingly,
we see the appearance of entire columns of black or white also develop. This
can be explained by the presence of sufficient influence in the interaction
kernel.

Finally, we also report another method for detecting various degrees of
sentiment change. A so-called difference map records the difference between
the initial condition and the equilibrium state. When a white pixel in an
initial condition (which $=-1$ in our scheme) remains white in the
equilibrium, the difference maps presents this pixel with the value
$-1-(-1)=0$. Similarly, when a black initial condition pixel remains black
until the end, the difference map indicates $0$ for that pixel. So, any pixel
that does not change color will appear with a $0$ in the difference map. Now
we consider the case when change does occur. This happens when a white pixel
changes to black, or visa versa. The difference maps will show a white pixel
that changed to black with the value of $1-(-1)=2$, and a black pixel that
changes to white is valued with $-2$. So far, subtle changes are difficult to
detect with this method. To help alleviate this, we define the difference map
as follows,

${\rm diff}(x):={\rm sgn}\\{\phi_{N}(x)\\}-{\rm sgn}\\{\phi_{0}(x)\\}.$ (2.1)

Equation (2.1) says the difference map value at pixel $x$ is the difference of
the sign function applied to the equilibrium state (this gives values of only
$-1$ or $+1$) and the sign function of the initial condition (again, only
possible value here are $-1$ or $+1$). The following difference maps
illustrate a variety of effects produced by interactions modeled by nonlocal
diffusion with random kernels. These appear in Figure 6. It should be noted
that to better highlight the different emergent structures, we rely on a
larger $32\times 32$ system.

(a) Extended interaction kernel

(b) Initial sentiment

(c) Sentiment at 475 iterations

(d) Sentiment at 950 iterations

(e) Final sentiment

(f) End state with no interaction

Figure 4. Sum of initial sentiment is $4.017$ vs. sum of final sentiment after
1425 iterations is $-121.965$. This indicates a dominantly negative (white)
change in public sentiment.

(a) Extended interaction kernel

(b) Initial sentiment

(c) Sentiment at 475 iterations

(d) Sentiment at 950 iterations

(e) Final sentiment

(f) End state with no interaction

Figure 5. Sum of initial sentiment is $-4.031$ vs. sum of final sentiment
after 1944 iterations is $106.869$. This indicates a dominantly positive
(black) change in public sentiment.

(a) Emergent negative polarity: White pixels indicate black pixels changing to
white.

(b) Emergent positive polarity: Black pixels indicate white pixels changing to
black.

(c) Emergent mixed polarity: Grouped/separated changes in polarity.

(d) Emergent negative polarity: Strong changes with white bands in
polarization.

(e) Emergent positive polarity: Strong changes with black bands in
polarization.

(f) Emergent mixed polarity: Black and white bands in polarization.

Figure 6. Several $32\times 32$ simulations produce these different difference
maps.

##  3\. Conclusions

We have presented a nonlocal differential equation model to simulate the
change and subsequent polarization in public sentiment. The following
observations and questions arise.

  * •

Dependency: The columns of the sentiment data are implicitly assumed to
represent independent questions. This should not always the case. The model
could be fitted to express dependence between certain questions. This means we
will also expect to see change along the (horizontal) rows during the
sentiment evolution.

  * •

Larger systems: Thanks to modern computing architecture (GPUs) and modern
programming libraries (Python’s NumPy), the present algorithm is scalable to
larger systems and even subject to more complicated interaction kernels.

  * •

A sensitivity problem: Which one sentiment change produces the greatest
deviation from the current final sentiment? Solving this problem could help us
understand and detect the propagation of misinformation.

  * •

Big data: Beyond relying on people to fill out surveys, we can find other
sources for our model’s initial sentiment data. There is data collected by our
smart phones, cellular servicer, computers, internet service provider, etc.

Conflicts of Interest. The author declares no conflict of interest.

Funding. This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors.

## References

  * [1] Brian Dean, “Social network usage & growth statistics: How many people use social media in 2021?”, blacklinko.com, March 9, 2021. Date accessed: April 1, 2021.
  * [2] V. Volpert, Nonlocal Reaction-Diffusion Equations. In: Elliptic Partial Differential Equations. Monographs in Mathematics, vol. 104 (2014) Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0813-2_9.
  * [3] S. Anita, V. Capasso, H. Kunze, and D. La Torre, Dynamics and optimal control in a spatially structured economy growth model with pollution diffusion and environmental taxation. _Appl. Math. Lett._ 42 (2015), 36–40.
  * [4] M. Banerjee, S.V. Petrovskii and V. Volpert, Nonlocal reaction-diffusion models of heterogeneous wealth distribution. _Mathematics_ 2021, 9(4), 351; https://doi.org/10.3390/math9040351.
  * [5] Nathaniel Chafee and E.F. Infante, Bifurcation and stability for a nonlinear parabolic partial differential equation. _Bull. Amer. Math. Soc._ 80 (1974), 49–52.
  * [6] Johan Bollen and Alberto Pepe and Huina Mao, Modeling public mood and emotion: Twitter sentiment and socio-economic phenomena. arXiv:0911.1583.
  * [7] M.Y. Chen and M.H. Fan and T.H. Chen and R.P. Hsieh, “Modeling Public Mood and Emotion: Blog and News Sentiment and Politico-economic Phenomena”, A. Visvizi and M.D. Lytras, Politics and Technology in the Post-Truth Era (Emerald Studies in Politics and Technology), Emerald Publishing Limited (2019), Bingley, pp. 57–71.
  * [8] Mu-Yen Chen and Chien-Hsiang Liao and Ren-Pao Hsieh, Modeling public mood and emotion: Stock market trend prediction with anticipatory computing approach. _Computers in Human Behavior_ 101 (2019), 402–408.


# Supplementary Material for
“Multi-Messenger Measurements of the Static Structure of Shock-Compressed
Liquid Silicon at 100 GPa”

H. Poole<EMAIL_ADDRESS>Department of Physics, University of
Oxford, Oxford OX1 3PU, UK  M. K. Ginnane  Laboratory for Laser Energetics,
University of Rochester, Rochester, New York 14623, USA  Department of
Mechanical Engineering, University of Rochester, Rochester, New York 14611,
USA  M. Millot  Lawrence Livermore National Laboratory, Livermore, CA 94550,
USA  G. W. Collins  Laboratory for Laser Energetics, University of Rochester,
Rochester, New York 14623, USA  S. X. Hu  Laboratory for Laser Energetics,
University of Rochester, Rochester, New York 14623, USA  D. Polsin  Laboratory
for Laser Energetics, University of Rochester, Rochester, New York 14623, USA
R. Saha  Department of Physics, University of Nevada, Reno, Nevada 89557, USA
J. Topp-Mugglestone  Department of Physics, University of Oxford, Oxford OX1
3PU, UK  T. G. White  Department of Physics, University of Nevada, Reno,
Nevada 89557, USA  D. A. Chapman  First Light Fusion Ltd, Unit 9/10 Mead Road,
Oxford Pioneer Park, Yarnton, Oxford OX5 1QU, UK  J. R. Rygg  Laboratory for
Laser Energetics, University of Rochester, Rochester, New York 14623, USA
Department of Mechanical Engineering, University of Rochester, Rochester, New
York 14611, USA  S. P. Regan  Laboratory for Laser Energetics, University of
Rochester, Rochester, New York 14623, USA  G. Gregori  Department of Physics,
University of Oxford, Oxford OX1 3PU, UK

††preprint: APS/123-QED

##  I Spectrally resolved X-ray scattering

Table S1:  Experimental parameters for all shots including the total incident energy of the shock-compression drive, $E_{\mathrm{drive}}$, and X-ray, $E_{\mathrm{xray}}$, lasers. Shot | $2\theta_{\mathrm{XRTS}}\,\,(${}^{\circ}$)$ | Pinhole | $h_{\mathrm{CH}}\,$\mathrm{(}\mu\mathrm{m}\mathrm{)}$$ | $h_{\mathrm{Si}}\,$\mathrm{(}\mu\mathrm{m}\mathrm{)}$$ | $E_{\mathrm{drive}}\,($\mathrm{J}$)$ | $E_{\mathrm{xray}}\,($\mathrm{J}$)$ | $t_{\mathrm{drive}}\,($\mathrm{n}\mathrm{s}$)$ | $t_{\mathrm{xray}}\,($\mathrm{n}\mathrm{s}$)$ | $t_{\mathrm{SBO}}\,($\mathrm{n}\mathrm{s}$)$ | $L_{\mathrm{coh}}$
---|---|---|---|---|---|---|---|---|---|---
Background target - No Silicon
$30970$ | $70$ | Ta | $11\pm 2$ | - | - | $3760.7$ | - | - | - | $260\pm 9$
Ambient conditions
$30968$ | $70$ | Ta | $11\pm 2$ | $51\pm 1$ | - | $3618.6$ | - | - | - | $430\pm 10$
Shock-compressed conditions
$30964$ | $95$ | Ta | $11\pm 2$ | $51\pm 1$ | $441.3$ | $3707.5$ | $-0.049\pm 0.025$ | $5$ | $5.35\pm 0.04$ | $615\pm 10$
$30967$ | $70$ | Ta | $11\pm 2$ | $51\pm 1$ | $429.0$ | $3908.4$ | $-0.15\pm 0.025$ | $4.8$ | $6.22\pm 0.06$ | $580\pm 10$
$33538$ | $70$ | Ag | $11\pm 2$ | $51\pm 1$ | $437.9$ | $3778.3$ | $-0.002\pm 0.025$ | $5$ | $5.20\pm 0.04$ | $875\pm 20$
$33541$ | $98$ | Ag | $11\pm 2$ | $51\pm 1$ | $422.5$ | $3835.0$ | $0.014\pm 0.025$ | $5$ | $4.91\pm 0.03$ | $534\pm 8$
Figure S1:  (a) Raw intensity of calibration s33544 where the Cu foil was
placed in the target holder of the PXRDIP box. (b) In the top plot the XRTS
spectrum extracted after integrating along each energy arc is shown. A further
polynomial background fit (red dash-dotted line) is subtracted from the
overall signal to produce the spectrum in the lower plot. The energy axis is
calibrated to the Cu He-alpha peak. The expected positions of the remaining Cu
transitions are shown as vertical dashed lines. (c) Comparison of the
spectrally resolved XRTS signal for the post-SBO shots where
$V_{\mathrm{l}}/V_{\mathrm{s}}>0.9$. The source function from (b) is scaled to
the $\theta_{\mathrm{XRTS}}=70${}^{\circ}$$ scattering signal and is shown as
a red dashed line.

The zinc-spetrometer (ZSPEC) diagnostic used to record the X-ray Thomson
scattering consisted of a highly-oriented pyrolitic graphite (HOPG) crystal
placed $12.8\,\unit{cm}$ after the sample. The central Bragg angle was
$13.2^{\circ}$ with an angular spread of $2.5^{\circ}$. Its image plate (IP)
was protected by a $5\,\unit{mm}$ Be filter and mounted $13.1\,\unit{cm}$ from
the crystal meaning the ZSPEC was fielded out of perfect von-Hamos focusing.
This resulted in the spectrally resolved X-rays being scattered on to a cone
which can allow for some spatial information to be resolved. In the raw
intensity image shown in Figure S1(a), the X-ray scattering cone is
highlighted in black and the scattered photon energy increases to its point.
The dispersion of the ZSPEC is [1] ${\Delta E}/{\Delta x}\sim
7\,\unit{eV/pixel}$ where the pixels are $50\,\unit{\mu m}$ wide. To extract
the spectrally resolved spectrum, the total scattering signal along each
energy arc is determined. The pixels falling along an arc, such as those
highlighted in red, have an average pixel background value (determined from
the orange arcs which fall outside of the X-ray scattering cone) removed, and
are then summed. This gives an integrated scattering signal, $I(x)$, for a
given spatial position $x$ on the IP. The spectrum is converted from spatial
to energy space using,

$I(E)=\mbox{\Large$\frac{hc}{2d\,\,\mathrm{sin}\left(\mathrm{tan}^{-1}\left(\frac{I(x)+D_{c}\mathrm{sin}\theta_{0}+D_{ip}\mathrm{sin}\theta_{0}}{D_{ip}\mathrm{cos}\theta_{0}+D_{c}\mathrm{cos}\theta_{0}}\right)\right)}$}\,,$
(1)

where $2d=0.67\,$\mathrm{n}\mathrm{m}$$ is the HOPG lattic spacing,
$D_{c}=12.8\,$\mathrm{c}\mathrm{m}$$ is the distance from the source to the
HOPG crystal, $D_{ip}=13.1\,$\mathrm{c}\mathrm{m}$$ is the distance from the
HOPG crystal to the IP, and $\theta_{0}=13.2${}^{\circ}$$ is the central Bragg
angle on the crystal. In the upper plot of Figure S1(b), the XRTS spectrum
from the Cu X-ray source is shown. After a further polynomial fit to the
background is removed, the total background removed spectrally resolved XRTS
spectrum is given in the lower plot of Figure S1(b). The energy calibration is
performed by aligning the peak signal with the Cu Helium-alpha X-rays at
$8.358\,$\mathrm{k}\mathrm{e}\mathrm{V}$$. It can be seen in the lower plot
that the remaining expected Cu transitions align with the lower intensity
peaks on the spectrum.

A comparison of the XRTS spectra produced by the data shots taken after SBO
(i.e. where $V_{\mathrm{l}}/V_{\mathrm{s}}>0.9$) are shown in Figure S1(c)
against the source function. As expected with low silicon ionization, there is
no resolvable inelastic scattering signal above the source function. The XRTS
signals can therefore not be used independently to extract the plasma
parameters, but can be combined with the X-ray diffraction.

To compare the XRTS to the angularly resolved X-ray diffraction signal, the
XRTS must be spectrally integrated to give $I(k)=\sum_{\omega}I(k,\omega)$.
However, as only the coherent scattering signal is clearly observed over the
background noise, an integration over all $\omega$ cannot be performed. Taking
the peak intensity of the spectrally resolved signals shown in Figure S1(c)
increases the measurement uncertainty due to the compounded error of isolating
the X-ray scattering cone and from the integration methods performed over each
energy arc. Instead, as demonstrated in Figure S2, the relative coherent
scattering signals, $L_{\mathrm{coh}}$, are determined by removing fitted
background Gaussian distributions from the overall signal intensities -
isolating the target scattering. The signals are corrected for filtering,
polarization, backlighter distance and relative thickness of the silicon seen
by the X-ray source. The subsequent Gaussian fits to these scattering
histograms yields, $L_{\mathrm{coh}}=(\mu_{s}+2\,\sigma_{s})-\mu_{b}$, where
$s$ and $b$ denote the Gaussian distributions to the scattering and background
signals, respectively.

Figure S2:  The XRTS signal intensities focused around the coherent scattering
arc for ambient s30968 (a), pre-SBO s30967 (b) and post-SBO s33538 (c) at
$\theta_{\mathrm{XRTS}}=70${}^{\circ}$$. The raw scattering image is shown on
the right of each plot and their corresponding scattering signal histograms
are shown on the left. The red dashed lines are the mean coherent signal
intensities, $L_{\mathrm{coh}}$. (d) Relative intensity of elastic XRTS signal
($\propto S_{ii}$) against the fraction of liquid silicon,
$V_{\mathrm{l}}/V_{\mathrm{s}}$, for all $2\theta_{\mathrm{XRTS}}$.
Highlighted in red are the shots taken after shock breakout. For these shots
the liquid fraction is greater than $\sim 0.9$ and the scattering is assumed
to be only from liquid silicon. The liquid fraction of the unfilled diamond
points at $95${}^{\circ}$$ and $98${}^{\circ}$$ are determined from HELIOS
simulations due to insufficient ambient data at these scattering locations.

For the ambient s30968, where $2\theta_{\mathrm{XRTS}}=70${}^{\circ}$$, the Ta
Bragg peak from the (112) lattice plane is resolved by the XRTS. This pinhole
scattering signal is isolated from the silicon scattering by subtracting the
XRTS coherent signal from the reference s30970, where only the Ta pinhole was
loaded in the target holder. The coherent scattering intensities are listed in
Table S1.

As discussed in the main paper, comparing the coherent scattering intensities
of ambient and shocked silicon provided information on the fraction of shocked
(fluid) silicon within the probe volume. Using a simple scattering model as
described by Pelka et al. [2], which is based on the approach of Chihara [3],
the time-averaged volume fraction of liquid ($l$) to solid ($s$) silicon
present during the scattering event is calculated as,

$\frac{V_{l}}{V_{\mathrm{s}}}=\frac{L_{\mathrm{coh}}^{l}}{L_{\mathrm{coh}}^{s}}\,\frac{S_{\mathrm{tot}}^{s}}{S_{\mathrm{tot}}^{l}}=\frac{L_{\mathrm{coh}}^{l}}{L_{\mathrm{coh}}^{s}}\,\frac{Z_{Si}\left[1-I_{\mathrm{coh}}(k)/Z_{Si}^{2}\right]}{I_{\mathrm{coh}}(k)S_{ii}(k)}\,,$
(2)

where $S_{\mathrm{tot}}$ are the static structure factors and
$Z_{\mathrm{Si}}=14$ is the nuclear charge. As shown in Figure S2, the volume
fraction for the shock-compressed silicon states was found to be
$V_{l}/V_{s}>0.6$.

As shown in Figure 3(a) of the main paper, the post-shock breakout XRTS data
shots (s30964, s33538 and s33541) – where the fraction of liquid silicon is
greater than $\sim 0.9$ – are compared to the diffuse angularly resolved
scattering recorded on s30967. These $L_{\mathrm{coh}}$ signals are scaled by
fitting the value at $2\theta_{\mathrm{XRTS}}=70^{\circ}$ to the range of
accepted MCMC fits. As only the coherent contribution is resolved using ZSPEC,
this fitting procedure uses the MCMC fits, $I_{\mathrm{fit}}$, prior to adding
the incoherent scattering, $I_{\mathrm{incoh}}$. The higher
$2\theta_{\mathrm{XRTS}}$ signals are then determined by their scattering
intensity relative to the $70^{\circ}$ data. The total signal errors are
compounded by their respective coherent scattering uncertainties and the
scaling error of $L_{\mathrm{coh}}$ at $70^{\circ}$.

##  II Angularly resolved X-ray scattering

Figure S3:  (a) and (b): Raw images plates for background s30970 and shocked
silicon s30967, respectively. The lines shown are the Bragg diffraction peaks
of the Ta pinhole which are used to calibrate the geometry of the PXRDIP box.
Their corresponding warped $2\theta$-$\phi$ signals at the pinhole position
are shown in (c) and (d), respectively. The scattering distributions for the
pinhole used a SNIP background removal process [4] which enhances sharp peaks,
improving signal-to-background ratios.  Figure S4:  (a) and (b): The warped,
intensity corrected signals at the sample position for background s30970 and
shocked silicon s30967, respectively. The superimposed red and black dashed
horizontal lines are the calibrated $2\theta$ Bragg diffraction peaks of the
Ta pinhole and the expected ambient silicon peaks, respectively. (c) Shocked
Si scattering after background removal. Artifacts from this removal process
are seen at the edges of the image plates. The region selected for
$\phi$-integration is highlighted in gray. (d) Shown in blue, the
partial-$\phi$ integration of (c) to obtain the liquid scattering signal in
$2\theta$. The purple and green lines show the effect filter thickness
uncertainties have on the inferred liquid shape. The overall signal
uncertainty is taken over the gray shaded region.

Using Lawrence Livermore National Laboratory’s AnalyzePXRDIP procedure [4],
the raw X-ray diffraction IP’s, shown in Figures S3(a) and (b), are warped
into $2\theta$-$\phi$ by using the Ta pinhole Bragg peaks as calibrants. The
pinhole calibration for each shot is shown in (c) and (d). To isolate the
liquid silicon scattering the background signal must be removed from the
shocked data. Whilst a variant of the Statistics-sensitive Non-linear
Iterative Peak-clipping (SNIP) algorithm is often used to isolate signal from
the background in PXRDIP scattering data [4], this process is only appropriate
when dealing with sharp Bragg peaks, such as shown in Figure S3. As discussed
in the main paper, to quantify the background signal we instead performed a
series of shots to isolate each contributor. In s30966, we investigated the
signal intensity recorded on the IP’s when only the drive laser was present
(i.e. no X-ray probe). At the investigated pressure, this was found to be
negligible in comparison to the data collected with X-ray scattering events.
Comparison of the scattering recorded on the PXRDIP using an ambient silicon
sample (e.g. s30968) versus for a reference s30970, where the silicon sample
was removed, showed the dominant background scattering contributor to be the
pinhole. Subsequent analysis therefore uses s30970 as the background
reference.

The scattering for the background s30970 and shocked s30967 shown in Figure S4
(a) and (b), respectively, are obtained by accounting for the filtering
($12.5\,\unit{\mu m}$ Cu and $25\,\unit{\mu m}$ Kapton), incident solid angle,
polarization ($(1+\cos^{2}2\theta)/2$) and the attenuation of the scattered
X-rays. The background subtracted shocked signal is shown in Figure S4(c).
Contamination at the edge of the IPs means a full integration in $\phi$-space
cannot be performed. The partial-$\phi$ integration region, highlighted in
gray, therefore focuses on the IP center, and excludes the $2\theta$ regions
around the Ta Bragg peaks. The resultant liquid scattering signal is shown in
blue in Figure S4(d). The $12.5\,\unit{\mu m}$ Cu and $25\,\unit{\mu m}$
Kapton filters used for the PXRDIP IPs have a $20\%$ thickness uncertainty.
Propagating this error through the background removal process results in the
purple and olive plots shown in (d). The intensity error of each signal is
given by the standard deviation from the mean signal along the $\phi$
integration. As we did not record to high-$k$, their absolute signal intensity
is not applicable and they are normalized to their broad liquid scattering
peak around $45\unit{{}^{\circ}}$. This process demonstrates the effect filter
uncertainties have on the overall shape of the liquid scattering feature. The
total liquid scattering signal uncertainty, as shown in Figure 3(a) of the
main paper, therefore covers the region highlighted in gray.

##  III VISAR analysis

Figure S5:  (a) Raw VISAR data from s30967. The fringes are reflected from the
rear of the $51\,$\mu\mathrm{m}$$ thick Si sample. Superimposed in red is the
laser drive power. The integrated VISAR intensity is projected underneath. The
time of shock-breakout, $t_{\mathrm{SBO}}$, is determined as the point at
which there is a sharp gradient decline in reflected VISAR signal. (b)
Impedance matching CH and Si shock conditions, using equations 4, 5 and 6, to
find (shown in the upper right quadrant) the linear relationship between
$D_{\mathrm{CH}}$ and $D_{\mathrm{Si}}$. The statistical errors arising from
the CH and Si model uncertainties are shown throughout as fainter green and
blue lines, respectively. The appropriate shock speed for each material
(highlighted as a magenta diamond) is found by substituting the
$D_{\mathrm{CH}}$-$D_{\mathrm{Si}}$ relation into equation 3. The
corresponding $1\sigma$ error due to both model and experimental uncertainties
are shown as magenta circles. This information is carried through the
remaining plots to find the corresponding $P$-$\rho$ space for CH and Si. (c)
Eulerian HELIOS 1D simulation, produced by scaling the drive laser profile to
match the measured $t_{\mathrm{SBO}}$, for s30967. The silicon shock
trajectory is shown as a thick red line and the timing of the X-ray laser
pulse is highlighted in orange.

Impedance matching techniques are used to infer the average shock speed
through the silicon sample, $D_{\mathrm{Si}}$. The total shock time through
both the CH ablator and Si sample is $t=t_{\mathrm{SBO}}-t_{\mathrm{drive}}$,
where $t_{\mathrm{SBO}}$ and $t_{\mathrm{drive}}$ are the measured shock-
breakout (SBO) and laser driver timings. These timings are listed in Table S1
and an example of the VISAR diagnostic for s30967 is shown in Figure S5(a).
This total time can be related to the shock speeds in the Si and CH via,

$t=\frac{h_{\mathrm{Si}}}{D_{\mathrm{Si}}}+\frac{h_{\mathrm{CH}}}{D_{\mathrm{CH}}}\,,$
(3)

where $h$ is the thickness of each material.

As described in the main paper, silicon is opaque to the VISAR laser, meaning
a direct measurement of particle velocity, $U$, cannot be obtained on each
shot. The particle velocity in the shocked silicon is therefore inferred as
[5],

$D_{\mathrm{Si}}=10.3(\pm 0.1)+1.8(\pm 0.1)\left[U-4.95\right]\,,$ (4)

which is valid for
$4<U\,\,$\mathrm{(}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{)}$<6.5$
and is based on the explosively driven data collected by Pavlovski in Ref.
[6]. The corresponding linear relationship used for the CH ablator is [7],

$D_{\mathrm{CH}}=21.029(\pm 0.057)+1.305(\pm 0.015)\left[U-14.038\right]\,.$
(5)

As detailed in Figure S5(b) these equations are used alongside the Rankine-
Hugoniot relations, which are derived from the conservation of mass, momentum
and energy across a shock front, to infer the post-shock $P$-$\rho$ state,

$P=P_{0}+\rho_{0}U_{s}U_{p}\,\,\,\,,\,\,\,\,\rho=\frac{\rho_{0}U_{s}}{U_{s}-U_{p}}\,,$
(6)

where $P_{0}=0\,$\mathrm{M}\mathrm{b}\mathrm{a}\mathrm{r}$$ and
$\rho_{0}=2.329\,$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$$ are the pre-
shock conditions. The silicon shock speed, pressure and density for each
shock-compressed experiment are listed in Table S2.

Table S2:  Comparison of the inferred plasma conditions for each shot using Rankine-Hugoniot relations, on the left, and the Si mass-averaged conditions from HELIOS simulations during the scattering event on the right. Shot | $\mathbf{D_{\mathbf{Si}}}\,\unit{(km/s)}$ | $\mathbf{P}\,\unit{(GPa)}$ | $\bm{\rho}\,\unit{(g/cm^{3})}$ | $\mathbf{D_{\mathbf{Si}}}\,\unit{(km/s)}$ | $\mathbf{P}\,\unit{(GPa)}$ | $\bm{\rho}\,\unit{(g/cm^{3})}$
---|---|---|---|---|---|---
VISAR Transit Time | HELIOS Simulations
$30964$ | $11.4\pm 0.3$ | $146\pm 9$ | $4.54\pm 0.07$ | $10.9\pm 0.2$ | $70\pm 7$ | $3.3\pm 0.2$
$30967$ | $9.5\pm 0.2$ | $101\pm 6$ | $4.43\pm 0.08$ | $9.4\pm 0.1$ | $48\pm 4$ | $3.2\pm 0.1$
$33538$ | $11.7\pm 0.3$ | $155\pm 10$ | $4.56\pm 0.07$ | $11.6\pm 0.2$ | $81\pm 10$ | $3.3\pm 0.2$
$33541$ | $12.4\pm 0.2$ | $178\pm 10$ | $4.60\pm 0.07$ | $12.0\pm 0.2$ | $82\pm 10$ | $3.2\pm 0.2$
Figure S6: Normalized probability distribution functions (PDFs) of silicon
conditions during the X-ray scattering event as predicted by the HELIOS
simulations. The vertical red dashed lines are their respective mass-averaged
values, $\left<X\right>_{m}$. (a), (b) and (c) show the density, pressure and
$T_{e}$, respectively, for s30967, taken before shock-breakout. (d), (e) and
(f) as above for s33538, taken after shock-breakout.

##  IV HELIOS Simulations

Using the $t_{\mathrm{SBO}}$ measured with the VISAR diagnostic and by scaling
the drive laser profile, HELIOS 1D simulations were produced for each shot
using SESAME-EOS [8]. An example of the mass density through the target is
shown in Figure S5(c). The simulations emphasize the non-uniformity of the
conditions within the silicon during the X-ray scattering event, which is
highlighted in orange. The simulated shock speed through the silicon was
determined by tracking the mass density gradient. An example of this shock
tracking is shown as a thick red line in Figure S5(c).

The simulated mass-averaged plasma conditions, $\left<X\right>_{m}$ (where $X$
is a plasma parameter), within the silicon during the X-ray probe were
calculated as the mass-weighted average in space and averaged over the probe
duration in time [9]. This is determined as,

$\left<X\right>_{m}(t)=\frac{\sum_{i}V_{i}(t)\rho_{i}(t)X_{i}(t)}{\sum_{i}V_{i}(t)\rho_{i}(t)}\,,$
(7)
$\left<X\right>_{m}\equiv\left<\left<X\right>_{m}(t)\right>_{t}=\frac{\sum_{j_{\mathrm{min}}}^{j_{\mathrm{max}}}\left<X\right>_{m}(t_{j})t_{j}}{\Delta
t_{\mathrm{xray}}}\,,$ (8)

where $V_{i}(t)$ is the volume in the $i$-th cell at time $t$. The inferred
$P$-$\rho$ states for each shot are listed in Table S2. These mass-averaged
values are compared to the overall parameter distributions during the X-ray
scattering event in Figure S6. In the pre-SBO data the sharp peaks
corresponding to the low density region indicate the presence of ambient
silicon.

##  V Markov-Chain Monte Carlo Analysis

Figure S7:  (a) Ion-ion structure factors, $S_{ii}(k)$, for each $V_{ii}(k)$
using the mean parameters $\left<\Psi\right>_{\mathrm{NH}}$ as shown in Figure
3(a) of the main paper,
($\rho=4.5\,$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$$, $Z=1.3$,
$T=6800\,$\mathrm{K}$$). (b) Comparison of the tabulated incoherent scattering
signal $I_{\mathrm{incoh}}$ [10], the bound electron form factor, $f(k)$, and
the screening cloud, $q(k)$, produced with DH and FW models using
$\left<\Psi\right>_{\mathrm{NH}}$. (c) The synthetic diffraction signals
produced with each $V_{ii}(k)$, $q(k)$, $f(k)$ and $I_{\mathrm{incoh}}$, as
shown in (a) and (b), compared to the scaled experimental scattering signal.
For this representative plot, the scaling parameter, $\Gamma$, from equation 9
was chosen using the NH $I_{\mathrm{fit}}$ (solid red curve).

MCMC is a robust method for exploring complex multi-parameter spaces to
overcome the challenge of inverse problem instability, which implies that the
same measured experimental data can be fitted equally well by very different
conditions. Given a specific set of parameters, $\Psi(\rho,T,Z)$, the multi-
component scattering spectra (MCSS) code [11] is used to produce a theoretical
diffraction signal, $I_{\mathrm{fit}}(k)$. As discussed in the main paper, the
scaling parameter $\gamma$ for the experimentally measured liquid diffraction
signal, $I_{\mathrm{liq}}(k)$, cannot be obtained experimentally. Instead we
scale the experimental signal to the simulated fit using,

$\frac{I_{\mathrm{liq}}(k)}{\gamma}\equiv
I_{\mathrm{scal}}(k)=I_{\mathrm{liq}}(k)\,\times\,\Gamma\,\,\frac{I_{\mathrm{fit}}^{\mathrm{max}}}{I_{\mathrm{liq}}^{\mathrm{max}}}\,,$
(9)

where $\Gamma$ is a free random Gaussian scalar with a standard deviation
equal to the noise of the raw data, $I_{\mathrm{fit}}^{\mathrm{max}}$ and
$I_{\mathrm{liq}}^{\mathrm{max}}$ are the peak values in the MCSS fit and raw
X-ray scattering data, respectively.

The MCMC process then calculates the likelihood of these parameters producing
the given scaled X-ray scattering spectrum, $I_{\mathrm{scal}}(k)$, as [12],

$P(\Psi|I_{\mathrm{scal}}(k))=\frac{P\left(I_{\mathrm{scal}}(k)|\Psi\right)P(\Psi)}{P(I_{\mathrm{scal}}(k))}\,,$
(10)

where $P(\Psi)$ is the prior distribution of possible parameters,
$P(I_{\mathrm{scal}}(k))$ is the marginal likelihood of the observed data over
all possible parameters and the forward model likelihood,
$P(I_{\mathrm{scal}}(k)|\Psi)$, is as described in the main paper.

Figure S8:  Probability density functions for the liquid silicon density,
pressure and temperature state using the Debye-Hückel (a), effective Coulomb
(b) and non-linear Hulthén (c) $V_{ii}(k)$ models. As in Figure 3 of the main
paper, the lower quadrant plots compare the $1$, $2$ and $3\sigma$ parameter
correlations for the MCMC converged X-ray scattering analysis (gray heat maps)
and the combined density functions (red heat maps). The diagonal histograms
show the probability densities for each parameter. The VISAR distributions are
added in blue for density and pressure.

Model sensitivities for producing synthetic X-ray scattering signals using
MCSS are shown in Figure S7. This demonstrates how the dominant contributor to
the synthetic scattering signal is the chosen ion-ion interaction potential,
$V_{ii}(k)$. The screening models, $q(k)$, cannot be differentiated within the
measured experimental error. The potentials for each $V_{ii}(k)$ are given by,

$V_{ii}^{ec}(k)=\frac{\mathcal{K}}{k^{2}}\,,$ (11)

where $\mathcal{K}=-4\pi Z^{2}e^{2}/3k_{B}T_{e}$,

$V_{ii}^{dh}(k)=\frac{\mathcal{K}}{\kappa_{e}^{2}+k^{2}}\,,$ (12)

where $\kappa_{e}$ is the inverse screening length, and

$V_{ii}^{nh}(k)=\mathcal{K}\frac{i}{4k\kappa_{e}}\left[{\psi^{(1)}\left(1+i\frac{k}{2\kappa_{e}}\right)-\psi^{(1)}\left(1-i\frac{k}{2\kappa_{e}}\right)}\,,\right]$
(13)

where $\psi^{(1)}(x)$ is a trigamma function. The inverse screening length is
taken as,

$\kappa_{e}=\sqrt{\frac{n_{e}e^{2}}{k_{B}T_{e}\epsilon_{0}}\frac{\mathcal{F}_{-1/2}(\eta_{e})}{\mathcal{F}_{1/2}(\eta_{e})}}\,,$
(14)

where $\mathcal{F}_{j}$ denotes the complete Fermi-Dirac integral of order $j$
and $\eta_{e}$ is the dimensionless chemical potential of the free electrons,

$\eta_{e}=\mathcal{F}_{1/2}^{-1}\left[\frac{n_{e}\hbar^{3}}{2}\left(\frac{2\pi}{m_{e}k_{B}T_{e}}\right)^{3/2}\right]\,.$
(15)

The converged shock-compressed silicon conditions using the effective Coulomb,
Debye-Hückel and non-linear Hulthén models are shown in gray in Figure S8.

From the convergence parameters, the total plasma pressure is determined using
the ‘two-fluid’ model described by Vorberger et al. [13, 14],

$\displaystyle P=$
$\displaystyle\,\left(1+\langle{Z_{i}}\rangle\dfrac{\mathcal{F}_{3/2}(\eta_{e})}{n_{e}\Lambda_{e}^{3}/2}\right)n_{i}k_{\text{B}}T-n_{i}^{2}\left.\dfrac{\partial
f_{e}^{\text{XC}}}{\partial n_{i}}\right|_{T}$
$\displaystyle-\dfrac{n_{i}^{2}}{4\pi^{2}}\dfrac{\partial}{\partial
n_{i}}\left.\\!\\!\left(\int_{0}^{\infty}\text{d}k\,k^{2}\,\chi_{ee}(k,0)\\!\left[V_{ei}^{\text{eff}}(k)\right]^{2}\right)\right|_{T}$
$\displaystyle-2\pi
n_{i}^{2}\int_{0}^{\infty}\text{d}r\,r^{2}\left[g_{ii}(r;V_{ii}^{\text{eff}}(r))-1\right]\mathcal{V}_{ii}^{\text{eff},P}(r)\,,$
(16)

in which the effective ion-ion interaction potential for the evaluation of the
excess ion pressure is defined as

$\displaystyle\mathcal{V}_{ii}^{\text{eff},P}(r)=$
$\displaystyle\,\left(\dfrac{r}{3}\dfrac{\partial}{\partial
r}-n_{i}\dfrac{\partial}{\partial n_{i}}\right)V_{ii}^{\text{eff}}(r)\,.$ (17)

In Eq. (V), $\Lambda_{e}$ is the thermal de Broglie wavelength of the free
electrons. The first term is the sum of the ideal gas contributions from the
ions and free electrons. In the second term, $f_{e}^{\text{XC}}$ is the
exchange-correlation contribution to the free energy (per particle) of the
interacting electron gas [15]. The third term gives the contribution arising
from electron-ion interactions in linear response, where the static electron
density response function, $\chi_{ee}(k)$, is taken in the random phase
approximation and the local field correction, $G_{ee}(k)$, is determined using
the effective static approximation [16]. The fourth term gives the excess
pressure due to ionic correlations, where $g_{ii}(r;V_{ii}^{\text{eff}}(r))$
is the pair distribution function of the ions [17] evaluated with a consistent
effective ion-ion potential.

Table S3:  Comparison of the liquid silicon conditions within $1\sigma$ for the X-ray scattering MCMC convergence and the combined VISAR state using each ion-ion interaction potential. $\mathbf{V_{ii}(k)}$ | $\bm{\rho}\,\unit{(g/cm^{3})}$ | $\mathbf{P}\,\unit{(GPa)}$ | $\mathbf{T}\,\unit{(K)}$ | $\mathbf{\bar{Z}}$
---|---|---|---|---
MCMC Convergence
EC | $5.5\pm 0.2$ | $200\pm 60$ | $6600\pm 2900$ | $1.3\pm 0.3$
DH | $4.9\pm 0.2$ | $250\pm 90$ | $6900\pm 3000$ | $3\pm 1$
NH | $5.2\pm 0.2$ | $140\pm 40$ | $6500\pm 3000$ | $1.5\pm 0.4$
MCMC and VISAR Combined
EC | $4.72\pm 0.05$ | $110\pm 5$ | $4500\pm 500$ | $1.01\pm 0.03$
DH | $4.51\pm 0.09$ | $104\pm 6$ | $3400\pm 1000$ | $1.5\pm 0.1$
NH | $4.57\pm 0.07$ | $106\pm 6$ | $6900\pm 1500$ | $1.46\pm 0.06$

##  VI Combining VISAR and MCMC analysis

As shown in Figure 3 of the main paper, given the VISAR and MCMC $P$-$\rho$
probability density functions, their subsequent joint probability is defined
as,

$\mathrm{Pr}_{j}(\rho,P)=\frac{\mathrm{Pr}_{m}(\rho,P)\times\mathrm{Pr}_{v}(\rho,P)}{\sum_{\rho,P}\left[\mathrm{Pr}_{m}(\rho,P)\times\mathrm{Pr}_{v}(\rho,P)\right]}\,.$
(18)

These 2D combined density functions are shown as red heat maps in the lower
left plots in Figure S8. This formalism can be extended into multi-parameter
dimensions as the X-ray scattering analysis has dependencies beyond density
and pressure. However, as VISAR provides no direct measurement of temperature,
we must define its 3D density function such that
$\mathrm{Pr}_{v}(\rho,P,T_{i})\equiv\mathrm{Pr}_{v}(\rho,P,T_{j})$, where
$i(\neq j)$ describes a position along the temperature axis. By using equation
18, the 2D phase space for pressure and temperature can subsequently be found
as,

$\mathrm{Pr}_{j}(P,T)=\frac{\sum_{\rho}\mathrm{Pr}_{j}(\rho,P,T)}{\sum_{P,T}\left[\sum_{\rho}\mathrm{Pr}_{j}(\rho,P,T)\right]}\,.$
(19)

This process can be repeated in the ionization space.

The $1\sigma$ errors of each parameter’s joint probability density function,
i.e. $\mathrm{Pr}_{j}(\rho)$, are given in Table S3. The phase diagrams in
Figure 4 of the main paper detail the $1\sigma$ errors from
$\mathrm{Pr}_{j}(\rho,P)$ and $\mathrm{Pr}_{j}(P,T)$, shown as the dark red
contours in the lower left and central density maps in Figure S8.

## References

  * Pak et al. [2004] A. Pak, G. Gregori, J. Knight, K. Campbell, D. Price, B. Hammel, O. L. Landen, , and S. H. Glenzer, Review of Scientific Instruments 75, 3747 (2004).
  * Pelka et al. [2010] A. Pelka, G. Gregori, D. O. Gericke, J. Vorberger, S. H. Glenzer, M. M. Gunther, K. Harres, R. Heathcote, A. L. Kritcher, N. L. Kugland, et al., Physical review letters 105, 265701 (2010).
  * Chihara [2000] J. Chihara, Journal of Physics: Condensed Matter 12, 231 (2000).
  * Rygg et al. [2020] J. R. Rygg, R. F. Smith, A. E. Lazicki, D. G. Braun, D. E. Fratanduono, R. G. Kraus, J. M. McNaney, D. C. Swift, C. E. Wehrenberg, F. Coppari, et al., Review of Scientific Instruments 91, 043902 (2020).
  * Henderson et al. [2021] B. J. Henderson, M. C. Marshall, T. R. Boehly, R. Paul, C. A. McCoy, S. X. Hu, D. N. Polsin, L. E. Crandall, M. F. Huff, D. A. Chin, et al., Physical Review B 103, 094115 (2021).
  * Pavlovskii [1968] M. N. Pavlovskii, Soviet Physics Solid State 9, 2514 (1968).
  * Barrios et al. [2010] M. A. Barrios, D. G. Hicks, T. R. Boehly, D. E. Fratanduono, J. H. Eggert, P. M. Celliers, G. W. Collins, and D. D. Meyerhofer, Physics of Plasmas 17, 056307 (2010).
  * Bennett et al. [1978] B. I. Bennett, J. D. Johnson, G. I. Kerley, and G. T. Rood, Los Alamos National Laboratory, Los Alamos, NM pp. LA–7130 (1978).
  * Chapman et al. [2014] D. A. Chapman, D. Kraus, A. L. Kritcher, B. Bachmann, G. W. Collins, R. W. Falcone, J. A. Gaffney, D. O. Gericke, S. H. Glenzer, T. M. Guymer, et al., Physics of Plasmas 21, 082709 (2014).
  * Hubbell et al. [1975] J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton, Journal of physical and chemical reference data 4, 471 (1975).
  * Chapman [2017] D. A. Chapman, _User Guide and Theoretical Basis for the Multi-Component Scattering Spectra (MCSS) Thomson Scattering Analysis Code_ (AWE, 2017).
  * Kasim et al. [2019] M. Kasim, T. P. Galligan, J. Topp-Mugglestone, G. Gregori, and S. M. Vinko, Physics of Plasmas 26, 112706 (2019).
  * Vorberger et al. [2013] J. Vorberger, D. O. Gericke, and W.-D. Kraeft, High Energy Density Physics 9, 448 (2013).
  * Ebeling et al. [2020] W. Ebeling, H. Reinholz, and G. Röpke, The European Physical Journal Special Topics 229, 3403 (2020).
  * Groth et al. [2017] S. Groth, T. Dornheim, T. Sjostrom, F. D. Malone, W. M. C. Foulkes, and M. Bonitz, Physical review letters 119, 135001 (2017).
  * Dornheim et al. [2021] T. Dornheim, Z. A. Moldabekov, and P. Tolias, Physical Review B 103, 165102 (2021).
  * Hansen and McDonald [2013] J.-P. Hansen and I. R. McDonald, _Theory of simple liquids: with applications to soft matter_ (Academic press, 2013).