# The most general manner to injectively align true and predicted segments

Maarten Marx
Informatics Institute
University of Amsterdam
The Netherlands
<EMAIL_ADDRESS>This research was supported in part by the Netherlands
Organization for Scientific Research (NWO) through the ACCESS project grant
CISC.CC.016, and by the University of Amsterdam through Humane AI.

###### Abstract

Kirilov et al (2019) develop a metric, called _Panoptic Quality (PQ)_ , to
evaluate image segmentation methods. The metric is based on a confusion table,
and compares a predicted to a ground truth segmentation. The only non
straightforward part in this comparison is to align the segments in the two
segmentations. A metric only works well if that alignment is a partial
bijection. Kirilov et al (2019) list 3 desirable properties for a definition
of alignment: it should be simple, interpretable and effectively computable.
There are many definitions guaranteeing a partial bijection and these 3
properties. We present the weakest: one that is both sufficient and necessary
to guarantee that the alignment is a partial bijection. This new condition is
effectively computable and natural. It simply says that the number of
correctly predicted elements (in image segmentation, the pixels) should be
larger than the number of missed, and larger than the number of spurious
elements. This is strictly weaker than the proposal in Kirilov et al (2019).
In formulas, instead of |TP| > |FN| +|FP|, the weaker condition requires that
|TP| > |FN| and |TP| > |FP|. We evaluate the new alignment condition
theoretically and empirically.

_Keywords_ Panoptic Quality $\cdot$ Image segmentation $\cdot$ metric

##  1 Introduction

Kirillov et al. (2019) develop a metric, called _Panoptic Quality (PQ)_ , to
evaluate image segmentation methods. PQ is developed for images, but it can be
used for text as well, and even for any clustering problem. The only
requirement is that there is an underlying set of elements (in images the
pixels, in text segmentation typically tokens) which are _partially partioned_
(i.e., elements are combined into non overlapping segments, but not all
elements need to be assigned to a segment).

PQ is especially developed for segmentation problems in which _exact matches_
are unfeasible and not even needed for successfull applications. PQ is a very
elegant metric which can be seen as combining the _segmentation quality_ (how
well do predicted and true segments “match/overlap”?) with the _recognition
quality_ (the usual precision and recall questions).

The only non straightforward part in comparing a predicted to a true
segmentation is to match the separate segments in the two segmentations. Apart
from the fact that they are both subsets of the same underlying domain there
is no other information, and this matching can thus be done in many different
ways.

Kirilov et al (2019) require a simple, efficient and interpretable matching
process. Besides that, the alignment should be a partial bijection: for every
predicted segment $h$ there is at most one true segment $t$ and vice-verse. In
Theorem 1 they show that the natural condition stating that $h$ and $t$ match
iff $IoU(h,t)>.5$ guarantees a partial bijection.

Having this it is straightforward to define PQ. And indeed the thus defined
metric is effectively computable, interpretable and simple, as required.
Great! Except….., there are more, even many more, definitions of matching
which guarantee a partial bijection, and are also interpretable and
effectively computable. Why would we chose $IoU(h,t)>.5$?

##  2 When to match?

The condition $IoU(h,t)>.5$ is equivalent to

$|h\cap t|>|h\oplus t|,$

which says that the overlap between $h$ and $t$ is larger than their symmetric
difference.The symmetric difference consists of the two (disjoint) _error
regions_ , the _missed pixels_ and the _spurious pixels_. (Or the false
negative and false positive pixels). Stated like this, the condition is indeed
both interpretable and reasonable. We can paraphrase it as “there are more
overlapping than erroneously mapped pixels”. As the two error regions are
disjoint, the following condition is also equivalent:

$|h\cap t|>|\mathit{missed}(h,t)|+|\mathit{spurious}(h,t)|.$

Here $\mathit{missed}(h,t)=t\setminus h$ and
$\mathit{spurious}(h,t)=h\setminus t$ are the sets of false negative and false
positive pixels, respectively.

Our research question now is

> Are there other useful111We added the property _useful_ , because the
> identity matching satisfies all other properties, but obviously that is
> often too strict and thus not (very) useful., interpretable, simple and
> effective matching definitions which imply the partial bijection property?
> And if so, is there a most general one?

Indeed there is a simpler condition which is sufficient to guarantee a partial
bijection and which is also neccessary: every “reasonable” partial bijection
satisfies the condition. The condition is

$|h\cap t|>|\mathit{missed}(h,t)|\;\mathbf{\ and\ }\;|h\cap
t|>|\mathit{spurious}(h,t)|.\;\;\;\;\;\;\;\;\;(*)$

Below we will develop what “reasonable” is. But first we look deeper into the
new condition. Clearly it is simple, natural and interpretable. One may argue
that it is conceptually simpler than the $IoU>.5$ check, because adding the
spurious and missed pixels is no longer needed.

##  3 Results about the (*) alignment

Before we state and prove the characterization theorem we show that alignment
defined by (*) is effective, always a partial bijection and strictly weaker
than the $IoU>.5$ alignment.

Claim 1 Given $H$ and $T$, the condition (*) is effectively computable in
$|H|$ and $|T|$.

Proof A trivial nested for loop over $H$ and $T$ finds the alignment. This can
be optimized using the order on the elements of the underlying domain.

Claim 2 Let $H$ and $T$ be two segmentations of the same set. Then the
alignment between segments $h\in H$ and $t\in T$ defined by (*) is a partial
bijection.

In proofs it is easier to use this version of (*):

$|h\cap t|>|t\setminus h|\;\mathbf{\ and\ }\;|h\cap t|>|h\setminus
t|.\;\;\;\;\;\;\;\;\;(*)$

Proof Suppose not. We will derive a contradiction. Then there are two
possibilities. We treat one: there is a $t\in T$ and two different (and thus
dfisjoint) $h_{1},h_{2}\in H$ satisfying (*). Thus by assumption, both
$|h_{1}\cap t|>|t\setminus h_{1}|$ and $|h_{2}\cap t|>|t\setminus h_{2}|$.

Because $h_{1}$ and $h_{2}$ are disjoint, $h_{2}\cap t\subseteq t\setminus
h_{1}$ and thus

$|t\setminus h_{1}|\geq|h_{2}\cap t|.$

By assumption

$|h_{2}\cap t|>|t\setminus h_{2}|,$

and by the disjointness again

$|t\setminus h_{2}|\geq|h_{1}\cap t|,$

and thus $|t\setminus h_{1}|>|h_{1}\cap t|$, which contradicts with
$|h_{1}\cap t|>|t\setminus h_{1}|$. Note that we only used one of the
conjuncts of (*). For the other case, we need the other conjunct.

Claim 3 Let $b:H\longrightarrow T$ be the partial bijection defined by (*).
Then for each $h\in dom(b)$, $IoU(h,b(h))>\frac{1}{3}$.

Proof Denote the three sets in (*) by TP, FP and FN. All three are disjoint.
Now the $IoU$ is equal to $|TP|$ divided by $|TP|+|FP|+|FN|$. The claim is now
immediate.

Claim 4 (*) is strictly weaker than the original PQ matching condition of
$IoU(h,t)>.5$.

Proof Because the spurious and missing parts are disjoint it is weaker. For
strictness let $H=\\{\\{1\\},\\{2,3,4\\}\\}$ and
$T=\\{\\{1,2,3\\},\\{4\\}\\}$. Then with (*), $\\{2,3,4\\}$ is aligned to
$\\{1,2,3\\}$ because it has 2 elements in the overlap, and it has one missing
and one spurious element. But the $IoU$ of these two segments is equal to
$\frac{2}{4}$ and thus not strictly larger than $.5$, and thus with the
original PQ matching condition of $IoU(h,t)>.5$ the aligment is empty.

Thus the PQ of this pair $H,T$ with the original alignment definition is $0$,
while with alignment according to (*) $TP$ contains the pair
$(\\{2,3,4\\},\\{1,2,3\\})$, $FP$ contains the missed segment $\\{1\\}$ and
$FN$ the missed segment $\\{4\\}$. Thus

$PQ^{*}=\frac{IoU(\\{2,3,4\\},\\{1,2,3\\})}{|TP|+.5(|FP|+|FN|)}=\frac{\frac{2}{4}}{1+.5(1+1)}=\frac{1}{4}.$

Note that for the true segmentation $T=\\{\\{1,2,3\\},\\{4\\}\\}$, there are 5
non empty alignments defined by $IoU(h,t)>.5$, and one more, the one above,
defined by (*).

In the next claim we use SQ, RQ and PQ as in Kirillov et al. (2019). Thus SQ
is the mean $IoU$ of the True Positives, RQ is the traditional F1 value and PQ
is SQ times RQ.

Claim 5 There is an extreme case where

  * •

PQ and RQ with the $IoU>.5$ alignment are both 0, as there are no True
Positives.

  * •

RQ with the (*) alignment approaches 1.

  * •

SQ (and hence also PQ) with the (*) alignment approaches (from above)
$\frac{1}{3}$.

Proof The case is an enlarged variant of the example from Claim 4. Let $c$ and
$n$ be two integers, with $c$ even. The true segmentation consists of a
segment of length $\frac{c}{2}$ followed by $n$ segments of length $c+1$. The
predicted segmentation is simply the reverse of the true one. Now all the long
segments can be aligned to each other because for each one there is another
with $\frac{c}{2}+1$ elements overlap and thus only $\frac{c}{2}$ missing and
$\frac{c}{2}$ spurious elements. Thus $|TP|=n-2$, $|FP|=1$ and $|FN|=1$. As
$n$ gets large, the $RQ=\frac{n-2}{n-2+.5(1+1)}$ approaches 1. As $c$ gets
large, the IoU of each match approaches
$\frac{.5c+1}{.5c+1+.5c+.5c}=\frac{1}{3}$ from above, and hence also SQ.

##  4 The characterization

There are bijections between $H$ and $T$ for which (*) does not hold, but
these are in an intuitive sense not _reasonable_. Here is one over 6 elements:

$H$ | 1,2,3 | 4,5,6
---|---|---
$T$ | 1,2,3,4,5 | 6

This is a total bijection between two segmentations of a linear order. The two
first segments match even with the $IoU$ larger than .5 constraint. The two
last segments do not, but are matched just “because there is room left”.

We now develop the definition disallowing such matchings. Let $H$ and $T$ be
segmentations of $E$ and $b:H\longrightarrow T$ a partial bijection. We call
such a $H,T,b$ an _alingnment_. We will use $dom(b)$ and $ran(b)$ to denote
the domain and range of $b$ respectively.

Definition 1 Let $H,T,b$ be an alingnment.

  1. 1.

We say that $H,T,b$ is _not fair_ if there is an $h\in dom(b)$ and there is a
$t\in T$ and $t\not\in ran(b)$ such that $|h\cap t|\geq|h\cap b(h)|$.

  2. 2.

We say that $H,T,b$ is _not fair_ if $T,H,b^{-1}$ is not fair.

Definition 2 Let $H,T^{\prime},b^{\prime}$ be an alingnment.

  1. 1.

$H,T^{\prime},b^{\prime}$ is an _improvement_ of the alignment $H,T,b$ if
$dom(b)=dom(b^{\prime})$ and for all $h\in dom(b)$ it holds that $h\cap
b^{\prime}(h)\supseteq h\cap b(h)$.

  1. 2.

$H^{\prime},T,b^{\prime}$ is an _improvement_ of $H,T,b$ if
$T,H^{\prime},b^{\prime-1}$ is an improvement of $T,H,b^{-1}$.

Thus with an improvement, we may change one of the segmentations and either
the range or the domain of the alignment $b$, but only if the $IoU$ remains
the same or increases. The following new $T^{\prime}$ and now partial
bijection $b^{\prime}$ mapping $\\{1,2,3\\}$ to itself in $T^{\prime}$, and
$b^{\prime}(\\{4,5,6\\})$ remains $\\{6\\}$ is an improvement of the example
$H,T,b$ given earlier:

$H$ | 1,2,3 |  | 4,5,6
---|---|---|---
$T$ | 1,2,3,4,5 |  | 6
$T^{\prime}$ | 1,2,3 | 4,5 | 6

Definition 3 We call an alignment $H,T,b$ _fair_ if every improvement of
$H,T,b$ is fair.

The example alignment $H,T,b$ given above is not fair because the given
alignment $H,T^{\prime},b^{\prime}$ is an improvement which is not fair for
$h=\\{4,5,6\\}$. This is because $b(\\{4,5,6\\})=\\{6\\}$ and
$IoU(\\{4,5,6\\},\\{6\\})=\frac{1}{3}$, but in $T^{\prime}$ there now is the
new segment $\\{4,5\\}$ and $IoU(\\{4,5,6\\},\\{4,5\\})=\frac{2}{3}$.

Theorem 1 Let $H$ and $T$ be two segmentations of the same set and $B\subseteq
H\times T$. Then the following are equivalent:

  * •

all $(h,t)\in B$ satisfy (*);

  * •

$B$ is a partial bijection and $H,T,B$ is a fair alignment.

Proof $(\Downarrow)$ Assume that all $(h,t)\in B$ satisfy (*). By Claim 2, $B$
is a partial bijection, so we will write $B$ as the function $b$. Now suppose
to the contrary that $H,T,b$ is not a fair alignment. Then there is an
improvement of $H,T,b$ which is not fair. There are two cases. We do one, and
let the improvement be $H,T^{\prime},b^{\prime}$ with a $h\in H$ and a $t\in
T^{\prime}$ such that $|h\cap t|\geq|h\cap b^{\prime}(h)|$. Because
$H,T^{\prime},b^{\prime}$ is an improvement, it holds that $|h\cap
b^{\prime}(h)|\geq|h\cap b(h)|$. Thus we have that $|h\cap t|\geq|h\cap
b(h)|$.

Now $t$ and $b^{\prime}(h)$ are disjoint, so $h\setminus
b^{\prime}(h)\supseteq h\cap t$. Because $b^{\prime}$ is an improvement,
$h\cap b^{\prime}(h)\supseteq h\cap b(h)$ and thus $h\setminus
b^{\prime}(h)\subseteq h\setminus b(h)$, and thus $h\setminus b(h)\supseteq
h\cap t$. By chaining, we obtain $|h\setminus b(h)|\geq|h\cap b(h)|$ which
contradicts (*).

$(\Uparrow)$ Assume $H,T,b$ is a fair alignment and $b$ a partial bijection.
Suppose to the contrary that (*) does not hold. Then one of the two conjuncts
fails. Suppose the first. Thus there is a $h\in H$ such that $|h\setminus
b(h)|\geq|h\cap b(h)|$. Let $z=h\setminus b(h)$. Now create
$H,T^{\prime},b^{\prime}$ as follows.

$T^{\prime}=\\{z\\}\cup\ \\{t\in T\mid t\cap z=\emptyset\\}\cup\\{t\setminus
z\mid t\in T\mathit{\ and\ }t\cap z\neq\emptyset\\},$

and for all $h\in dom(b)$ set $b^{\prime}(h)=b(h)$ if $b(h)\in T$, and
$b(h)\setminus z$ otherwise.

We will show that $H,T^{\prime},b^{\prime}$ is an improvement which is not
fair for $h$, our required contradiction. Because $T$ and $b$ have these
properties, also $T^{\prime}$ is a partial partition, and $b^{\prime}$ a
partial bijection. To show that $H,T^{\prime},b^{\prime}$ is an improvement of
$H,T,b$, we must show that for all $\bar{h}\in H$, the overlap with its match
remained the same or increased, i.e., $\bar{h}\cap
b^{\prime}(\bar{h})\supseteq\bar{h}\cap b(\bar{h})$. This holds by definition
of $b^{\prime}$ when $b(\bar{h})$ and $z$ do not overlap. Thus in particular
for the segment $h$. If they do overlap, then as $z=h\setminus b(h)\subseteq
h$, for all $\bar{h}\neq h$, the overlap will increase because the elements in
$z$ are disjoint from $\bar{h}$ and thus taking them out of $b(\bar{h})$
reduces the number of errors.

Now we show that $H,T^{\prime},b^{\prime}$ is not fair for $h$, precisely
because of the set $z\in T^{\prime}$ which is not in $ran(b^{\prime})$. By
definition $z=h\setminus b(h)$ and thus $z=h\cap z$. By assumption on $h$,
$|h\setminus b(h)|\geq|h\cap b(h)|$, and thus $|h\cap z|\geq|h\setminus
b(h)|$. We found our desired contradiction.

##  5 Weighted P, R and F1

The idea of weighting the F1 metric with the mean similarity between the true
positives segment pairs —which is what PQ does— can be extended to Precision
and Recall as well. In order to avoid confusion or complex naming, we use the
_weighted_ prefix. For completeness, we give the definitions.

Definition 4 Let $H$ and $T$ be two segmentations of the set $D$. Thus both
$T$ and $H$ are subsets of $\mathcal{P}(D)$ in which all sets are pairwise
disjoint. Let $TP\subseteq H\times T$ be a partial bijection (thus for each
$h\in H$, there is at most one $(h,t)\in TP$, and similarly for $t\in T$).
Define

$wTP={\sum}\\{IoU(h,t)\mid(h,t)\in TP\\}.$

Then define, relative to this $T,H$ and $TP$, the weighted precision, recall
and F1 as

$P=\frac{wTP}{|H|},\ \ \ \ \ R=\frac{wTP}{|T|},\ \ \ \ \
F1=\frac{wTP}{|TP|+.5\cdot((|H|-|TP|)+(|T|-|TP|))}.$

Kirillov et al. (2019) call the mean IoU of the True Positives,
$\frac{wTP}{|TP|}$, the _segmentation quality_. When we multiply the
traditional versions of $P$, $R$ and $F1$ with $|TP|$ in the numerator with
the segmentation quality, we get the weighted versions.

##  6 Empirical evaluation

We briefly evaluate the differences between the original $IoU>.5$, and the new
$(*)$ manner of defining the aligment between predicted and true segments. We
plan to do additional experiments on image data. The new definition is weaker,
and thus it can lead to more TP’s, but those extra TP’s then have an $IoU$
value between $.33$ and $.5$, so the segmentation quality (SQ), which is the
mean IoU of the TP’s, will be lower. The recognition quality (the unweighted
metrics) will go up, because we get more TP’s. And the PQ (the weighted
metrics) can go both ways, precisely because they form the product of the
segmentation quality and the unweighted metrics.

If there is a difference in scores, this must come from those extra TP’s. So
these are segments which are hard to predict correctly, either because the
method finds them hard, or because they are hard. So we expect little or no
difference when systems score high, and we expect to see differences when the
task is hard (and even good systems score low on it).

Our results show actually very small differences, also on a hard task with a
state of the art segmenter based on a transformer model (BERT). On a synthetic
example we obtain a bit larger differences.

In the experiments below we compare the weighted and unweighted versions of
the metrics for the two ways of creating the alignment, the original $IoU>.5$
and the new $(*)$ manner. As their difference is really in the operator we
apply to the sets of missing and spurious pixels —adding them, or having two
checks—, we differentiate them by a superscript $+$ and $\&$, like in $wP^{+}$
and $SQ^{\&}$.

###  6.1 Page stream segmentation experiment with BERT

We do a page stream segmentation (PSS) experiment with a segmenter based on a
BERT transformer model developed in Guha et al. (2022). In this task one must
segment a linearly ordered stream of pages into documents. The best performing
systems learn a binary page classifier which predicts either first or internal
page, which is enough to do segmentation on this data. Systems work either on
the page represented as an image, or as text, or using both. Here we have a
system which uses the text. This is a state of the art system, which achieves
mean $.77$ weighted $F1$ score when evaluated on a dataset of 34 streams with
6.347 segments over 25K pages. The classifier was trained on 76 streams having
17.834 segments over 63.815 pages from the same distribution. These are the
findings:

  * •

of the 6.347 segments, 5.532 were TP’s with $IoU>.5$, _and only 4 more_ with
the $(*)$ matching;

  * •

mean scores over these 34 streams: $SQ^{\&}=.965$ and $SQ^{+}=.966$,
$F1^{\&}=.794$ and $F1^{+}=.793$, leading to a weighted $F1$ score of .772 for
both.

We make the task harder by testing the same classifier on a test set from
another distribution than it was trained on. The scores drop a lot, but still
hardly any difference between the two ways of alignment:

  * •

of the 1.404 segments, 590 were TP’s with the original definition and only 12
more with the $(*)$ matching;

  * •

mean scores over 108 streams: $SQ^{\&}=.861$ and $SQ^{+}=.868$; $F1^{\&}=.487$
and $F1^{+}=.479$; and $wF1^{\&}=.433$ and $wF1^{+}=.429$.

In conclusion, the scores behave as we expected, although we expected to see a
larger difference in weighted $F1$ on the hard task than an insignificant
$.004$ point.

###  6.2 Synthetic experiment

The next table shows the distributions of $SQ$, $F1$ and $wF1$ defined with
the two alignments, for all 16.384 segmentations of the first 15 digits, and
this true segmentation:

$[[1,2],[3,4,5],[6,7],[8],[9],[10,11,12],[13,14],[15]].$

Both weighted and unweighted $F1^{\&}$ values are always higher but not very
much. We can think of the table as giving the distribution of scores when
predicted with 16.384 different systems. Note that we see an increase of 229
systems finding at least one TP (1.5% of all systems; see the counts in the
columns $SQ^{+}$ and $SQ^{\&}$). The differences in unweighted $F1$ are 3
percent points and in weighted $F1$ just $1.5$ percent point.

| $SQ^{+}$ | $SQ^{\&}$ | $F1^{+}$ | $F1^{\&}$ | $wF1^{+}$ | $wF1^{\&}$
---|---|---|---|---|---|---
count | 15.556 | 15.885 | 16.384 | 16.384 | 16.384 | 16.384
mean | 0.855 | 0.819 | 0.348 | 0.379 | 0.298 | 0.314
std | 0.107 | 0.117 | 0.184 | 0.180 | 0.164 | 0.161
min | 0.600 | 0.500 | 0.000 | 0.000 | 0.000 | 0.000
25% | 0.787 | 0.750 | 0.235 | 0.250 | 0.185 | 0.196
50% | 0.867 | 0.833 | 0.333 | 0.375 | 0.292 | 0.302
75% | 0.920 | 0.889 | 0.471 | 0.500 | 0.407 | 0.419
max | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
Table 1: Descriptive statistics on the synthetic dataset from Section 6.2.

##  7 Conclusion

We found a useful, simple, interpretable and effectively computable definition
for aligning true and predicted segments which is both a necessary and
sufficient condition for the alignment being a partial bijection. If, given a
predicted and true segment, we let TP, FP and FN stand for the pixels in the
overlap, the missed and the spurious pixels, respectively, then the necessary
condition aligns the two segments if |TP| > |FN| and |TP| > |FP|. This in
contrast to the $IoU>.5$ condition which translates as |TP| > |FN| +|FP|.

Contrary to our expectations, the effect of the weaker condition was very
small, at least when measured on a text segmentation task. It would be nice to
see this tested on images as well. But even though the effect is small on
these examples, the new condition is elegant, and more (even most) general, so
we recommend and hope it will be used in future implementations of PQ.

## References

  * Kirillov et al. [2019] Alexander Kirillov, Kaiming He, Ross B. Girshick, Carsten Rother, and Piotr Dollár.  Panoptic segmentation.  In _IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA, June 16-20, 2019_ , pages 9404–9413. Computer Vision Foundation / IEEE, 2019.  doi:10.1109/CVPR.2019.00963.
  * Guha et al. [2022] Abhijit Guha, Abdulrahman Alahmadi, Debabrata Samanta, Mohammad Zubair Khan, and Ahmed H Alahmadi.  A Multi-Modal Approach to Digital Document Stream Segmentation for Title Insurance Domain.  _IEEE Access_ , 10:11341–11353, 2022.


|  Phase behavior of active and passive dumbbells†
---|---
|  Nayana Va, Shiang-Tai Linb, Prabal K Maitia
|  Using molecular dynamics simulations, we report phase separation in a 50:50
mixture of ’hot’ and ’cold’ dumbbells which interact by Lennard-Jones
potential. The ratio of the temperature difference between hot and cold
dumbbells to the temperature of cold dumbbells is a measure of the activity
$\chi$ of the system. From constant density simulations, we observe that the
’hot’ and ’cold’ dumbbells phase separate at high activity ratio
($\chi>5.80$). The critical activity of dumbbells is higher compared to that
of a mixture of hot and cold Lennard-Jones monomers $(\chi>3.44)$ . The extent
of phase separation is higher for high density. As activity increases, the
cold dumbbells cohere to form a large cluster indicating increased phase
separation which is quantified by an order parameter. The transfer of energy
from hot dumbbells to cold dumbbells due to collisions and the change in the
effective volume of hot dumbbells on phase separation affects the entropy of
hot dumbbells. The entropy of the hot dumbbells are calculated using the two-
phase thermodynamic (2PT) method and is compared with the entropy of dumbbells
in an equilibrium system where all the dumbbells have same temperature as that
of the ’hot’ dumbbells in the non-equilibrium system. We find that the entropy
of hot dumbbells is greater than the entropy of their equilibrium counterparts
once the phase separation sets in. The density of the phase separated system
varies across the interface along the length of the simulation box where the
cold dumbbells form dense clusters with density greater than the average
density of the non-equilibrium system while the density of hot dumbbells is
less than the average density of the non-equilibrium system. Also in the phase
separated non-equilibrium system, the high kinetic pressure of hot dumbbells
is balanced by the virial pressure of the cold dumbbells. We find that that
phase separation pushes the cluster of cold dumbbells to have solid like
ordering. Bond orientation order parameters reveal that the cold dumbbells
form solid-like ordering consisting of predominantly FCC and HCP packing, but
the individual dumbbells have random orientations. The simulation of the non-
equilibrium system at different ratios of number of hot dumbbells to cold
dumbbells reveals that the critical activity decreases with increase in
fraction of hot dumbbells.

##

††footnotetext: a Department of Physics, Indian Institute of Science, C. V.
Raman Ave,Bengaluru 560012, India. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS> b Department of Chemical Engineering,
National Taiwan University, No.1, Sec. 4 Roosevelt Rd. Taipei, Taiwan 10617 .
<EMAIL_ADDRESS>† Supplementary Information
available††footnotetext:

##  1 Introduction

Active matter consists of individual units which are driven out of equilibrium
as they consume energy from the surrounding medium or through internal
mechanisms. Examples of active matter include swarms of fish, flocks of birds
1, 2, 3, 4, 5, 6, 7, 8, 9, bacterial colonies, collection of cells in a
tissue, components of cellular cytoskeleton 10, 11, 12, 13, 14, 15etc. Active
matter can also be chemically synthesized and has far reaching applications in
nano-scale active transport and drug delivery16. They exhibit fascinating
properties like self-organization17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 and play very important
role in biological systems.
In many studies on active particles, the activity of the constituents is
represented by a vector model. In vector model the constituent particle(e.g.
bacteria) absorbs energy to generate a force which propels it in a particular
direction(usually along the body axis). The activity is a measure of how much
energy the constituent particle can take and convert into work (live bacteria
have high activity but dead bacteria have zero activity). Recent studies have
shown that for a system of mixture of passive and active particles(run and
tumble particles), in the limit when persistence length of active particles is
comparable to the radius of the particle, the system of active and passive
particle reduces to that of a system of ’hot’ and ’cold’ particles41. Such
representation of vectorial activity by different temperatures reduces a
vector model into scalar model. Such scalar models also help in building
analytically solvable microscopic models which gives deep insight into physics
of active matter42, 43. In scalar model, the activity differences between two
species in a system(e.g. live and dead) is represented by assigning two
different temperatures or two different diffusivities44 to two species. For
example, Ganai et al45 have proposed the use of scalar model to explain the
radial distribution of chromatin inside nucleus. Phase separation in a system
of equal mixture of hot and cold Lennard-Jones(LJ) particles at different
densities has been studied 46. Here the activity is defined as relative
temperature difference between hot and cold particles. Recently, we have done
simulation in a two temperature model of soft repulsive spherocylinders (SRS)
and have shown emergence of various liquid crystalline order at packing
fractions for which only isotropic phase is possible in equilibrium 47
In this paper, we study phase separation in system of hot and cold dumbbells48
which interact by Lennard Jones potential with aspect ratio(ratio of bond
length to diameter) 1 . Dumbbells are not only a simplest way to model
anisotropic particles but they also show interesting crystal structures. The
dumbbells with low aspect ratio form plastic crystals with no orientational
ordering49 while the dumbbells with high aspect ratio form crystals with both
translational and orientational order. In some recent works, colloidal
dumbbells have been used to build periodic crystal structures which have
partial photonic bandgap and are birefringent50. Also, there are experimental
works, which have built chiral colloidal structures51 with asymmetric
dumbbells using magnetic fields. So the dumbbells have garnered much interest
in nanotechnology of colloids.
In this paper, we study phase separation in equal mixture of passive and
active dumbbells. The active dumbbells are connected to a thermostat at high
temperature whereas passive dumbbells are connected to a thermostat at low
temperature to create difference in activity between the two components of the
system. It is observed that such a simple system shows phase separation at
sufficiently high activity.
The paper is organized as follows: in the next section we give details of the
simulation methodology. We next discuss the equilibrium properties of the
dumbbells including the entropy calculation using both thermodynamic
perturbation and 2-phase thermodynamic method. This is followed by the
discussion of the results for non-equilibrium system of equal mixture of hot
and cold dumbbells. The phase separation of hot and cold dumbbells is
quantified by order parameter $\phi$. We define another parameter $\psi$ whose
distribution P($\psi$) helps in determining the critical activity. The
critical activity is determined by observing the emergence of bimodality in
P($\psi$). The transfer of energy from hot dumbbells to cold dumbbells due to
collisions and the change in effective volume of hot dumbbells on phase
separation affect the entropy of dumbbells. So we determine entropy difference
$\Delta$S between hot dumbbells of non-equilibrium system Sneq with entropy of
corresponding equilibrium system Seq. We find that the entropy difference
$\Delta$S jumps from negative to positive value with the onset of phase
separation. Further, on performing cluster analysis of cold dumbbells, we find
that with increasing activity $\chi$ the number of cluster of cold dumbbells
decrease, while the size of largest cluster increases. See section 3.2 for
further details. The phase separation forces cold sub-system to have higher
density and hot sub-system to have lower density than the average density of
total system. The density, temperature and pressure variation along the length
of the simulation volume is given in section 3.2.4 . In section 3.2.5, bond
orientation order parameters are calculated which reveal that the cold cluster
develop solid like order with predominantly fcc and hcp arrangement. Finally,
we simulate the non-equilibrium system at different ratios of number of hot
dumbbells to the number of cold dumbbells and find that critical activity
decreases with increase in fraction of hot dumbbells.

##  2 Methodology and computational details

Fig. 1: Schematic diagram of a dumbbell. D=$\sigma$ is the diameter of each
atom. ro is the distance between centers of two atoms in a dumbbell

We start with a system of 8000 atoms where two atoms are bonded to form 4000
dumbbells. A schematic diagram of dumbbell is shown in Fig:1. Interaction
between the atoms of the dumbbells is described by Lennard-Jones potential

$V(r)=4\epsilon\bigg{[}\bigg{(}\frac{\sigma}{r}\bigg{)}^{12}-\bigg{(}\frac{\sigma}{r}\bigg{)}^{6}\bigg{]}$
(1)

where $r$ denotes the distance of separation between any two atoms, $\sigma$
and $\epsilon$ correspond to the diameter and strength of the interaction
potential of the atoms. We have used the parameters of argon ($\sigma$= 3.405
Å, $\epsilon$ = 0.238 kcal mol-1 and mass m = 39.948 g mol-1) to convert the
LJ units to real units. Two atoms within a dumbbell are held by a harmonic
bond potential of the following form

$V(r)=k(r-r\textsubscript{0})^{2}$ (2)

where r0 is the equilibrium bond length and $k$ is bond strength. In our
simulations, we have taken r0 to be equal to the diameter of the
atom($\sigma$) and $k$ equal to 3000$\epsilon/\sigma^{2}$ as per the original
LJ chain paper52.
The Nosé-Hoover thermostat with damping parameter 50$\delta$t is used to
maintain the temperature of the dumbbells. The time step of integration was
chosen to be $\delta$t = 1 fs for all the simulations reported in this work.
All the simulations are performed using the LAMMPS53 molecular dynamics
package.

##  3 Results and analysis

###  3.1 Equilibrium properties

Fig. 2: Pressure of dumbbells P* as a function of density $\rho$* at
temperature T*=3,4 and 5. Dashed lines represent pressure obtained from
perturbation theory. The solid lines represent the pressure obtained from MD
simulations.

In order to validate our system of dumbbells, we first calculate the equation
of state and entropy of dumbbells. The temperature T*(kbT/$\epsilon$) and
density $\rho*$($\rho$$\sigma^{3}$) are in reduced units. We run the MD
simulation of dumbbells at T*=3,4,5(LJ units) at different densities to obtain
the pressure of the system. The pressure obtained from simulation is compared
with pressure obtained from Wertheim’s Thermodynamic Perturbation theory52,
54. According to the Thermodynamic perturbation theory, the excess Helmholtz
free energy of Lennard-Jones chain can be obtained from excess Helmholtz free
energy of Lennard-Jones monomer fluid as below.

$\frac{A^{n}_{e}(\rho_{n})}{N}=\frac{A_{e}(\rho)}{N}+T\frac{1-n}{n}lny(\sigma)$
(3)

Here $A^{n}_{e}$ is excess free energy of Lennard-Jones(LJ) chain composed of
n monomers(for dumbbells n=2) , $A_{e}$ is the excess free energy of Lennard-
Jones monomer fluid, N is the total number of monomers and y($\sigma$) is the
pair correlation function of LJ monomer fluid at bond length $\sigma$.
Differentiating the above expression with respect to density we obtain the
expression for pressure.

$P^{n}(\rho_{n})=P(\rho)+\rho T\frac{1-n}{n}\bigg{[}1+\rho\frac{\delta
lny(\sigma)}{\delta\rho}\bigg{]}$ (4)

Here $P^{n}$ is pressure of LJ chain fluid and $P$ is the pressure of LJ
monomer fluid. The pair correlation function y($\sigma$) of LJ monomers is
obtained as an empirical fitting function where it is expressed in powers of
density and temperature. The empirical form of the pair correlation
y($\sigma$) is obtained from the work of Gubbin’s et. al.52. Pressure of the
fluid of LJ monomer $P$ is obtained from MD simulations. Pressure obtained
using equation 4 (dashed lines) is plotted as a function of density in Fig:2.
We can see that the pressure of dumbbells calculated from MD simulation (solid
lines) matches closely with the values predicted by perturbation theory(dashed
lines).

Fig. 3: Entropy of dumbbell (S) as a function of density $\rho*$ at
temperature T*=1.8. Here 2PT stands for entropy by Two phase Thermodynamic
Method , TP1 stands for entropy by Thermodynamic Perturbation with ideal
monomer gas as reference and TP2 stands for entropy by Thermodynamic
Perturbation with ideal diatomic gas(rigid rotor) as reference. .

Entropy is given by the negative derivative of free energy with respect to
temperature. So differentiating equation 3 with respect to temperature gives
the expression for entropy.

$\frac{S^{n}_{e}(\rho_{n})}{N}=\frac{S_{e}(\rho)}{N}+\frac{n-1}{n}\bigg{[}lny(\sigma)+T\frac{\delta
lny(\sigma)}{\delta T}\bigg{]}$ (5)

where $S^{n}_{e}$ is the excess entropy of LJ chain, $S_{e}$ is excess free
energy of LJ monomer fluid. The excess entropy of LJ monomer fluid $S_{e}$ is
obtained from the modified Benedict-Webb-Rubin equation of state55. To
calculate the entropy of dumbbells from excess entropy of Lennard-Jones chain
$S^{n}_{e}$, we consider two references. The excess entropy of Lennard-Jones
chain $S^{n}_{e}$ is calculated with respect to ideal monoatomic gas (TP1)
which is plotted in green dashed lines and with respect to ideal diatomic gas
(rigid rotor) (TP2) which is plotted in black dotted lines in Fig:3.
The entropy of dumbbells is also calculated from Two Phase Thermodynamic
Method(2PT)56, 57, 58. In 2PT method, first vibrational density of states of
fluid is obtained from Fourier transform of velocity autocorrelation function.
Then the vibrational density of states of the fluid is partitioned into a gas
and solid like component. The gas like component is treated as a hard sphere
gas while the solid component is treated as harmonic oscillator. This
decomposition into solid and gas like fractions is performed from the value of
vibrational density of states of the liquid at zero frequency. Once the
decomposition is done, by using the weighting functions of harmonic oscillator
and hard sphere gas, various thermodynamic properties like entropy and free
energy can be calculated. The entropy of dumbbell obtained from 2PT method as
a function of density is plotted in Fig:3 red dashed line.

Fig. 4: (a) Simulation at the end of 4ns when
$T\textsuperscript{h*}=T\textsuperscript{c*}=2$ and density is 0.8. We can see
that hot(red) and cold(blue) particles are well mixed. (b) Simulation at the
end of 1ns when $T\textsuperscript{h*}=80$ and $T\textsuperscript{c*}=2$ and
density is 0.8. We can see that hot and cold dumbbells are clearly phase
separated. (c) Simulation at the end of 1ns when $T\textsuperscript{h*}=80$
and$T\textsuperscript{c*}=2$ and density is 0.1. Here again, hot and cold
dumbbells are phase separated. The snapshots were generated using VMD59
software.

The dumbbell is approximated as a rigid rotor and the entropy of the rigid
rotor as a function of density is also plotted in Fig:3. From Fig:3 we can see
that entropy from 2PT falls between entropy calculated by Thermodynamic
perturbation with ideal monoatomic gas(TP1) and idea diatomic gas(TP2) as
reference. Entropy from 2PT matches very closely with entropy of rigid rotor
at very low density with a very small deviation. This small deviation is
present because of the contribution of the vibration of dumbbells to the
entropy. The entropy from 2PT follows the entropy of rigid rotor at low
density because the intermolecular interaction between dumbbells is negligible
at low densities but deviates from the entropy of rigid rotor as density
increases. So we believe that 2PT method can give the entropy of dumbbells
effectively.

###  3.2 Non-equilibrium Properties of Dumbbells

In the system of 4000 dumbbells, to bring in ’activity’ we introduce
temperature difference between the dumbbells. Half of the dumbbells are
labeled as ’cold’(passive) and are connected to a thermostat at low
temperature and the remaining half dumbbells are labeled as ’hot’(active) and
are connected to a thermostat at high temperature. Let Th* and Tc* denote the
temperatures of the ‘hot’ and ‘cold’ dumbbells respectively, in reduced units.
Initially to equilibrate the system, both hot and cold dumbbells are
maintained at same temperature at Th* = Tc* = 2 and at density $\rho*$=0.8.
The simulation at constant temperature and volume is run for 4 ns so that hot
and cold dumbbells mix well (Fig:4a).

Initially, the temperature of the ‘hot’ dumbbells is increased from Th* =2 to
Th*=5 and the simulation is run for 1 ns till the system reaches a steady
state. Next, the temperature of hot dumbbells is increased from Th* =5 to Th*
=10 so on up to Th* =80 in steps of five(5, 10, 15…80), and the system is
allowed to attain the steady state at each of the increased values of Th* and
the simulation is run for 1 ns at each temperature. It is observed that even
though the value of the thermostat temperature(Tc*) that is connected to the
‘cold’ dumbbells is left unchanged, there is an increase in the temperature of
the cold dumbbells due to heat exchange from the ‘hot’ dumbbells through
interactions. Hence, they reach an ‘effective’ temperature
$T_{eff}^{c*}>T\textsuperscript{c*}$ . Here, the effective temperature is
defined by the average kinetic energy of the dumbbells. In general,
$T\textsuperscript{h*}>T_{eff}^{h*}>T_{eff}^{c*}>T\textsuperscript{c*}$

As the $T\textsuperscript{h*}$ of the hot dumbbells is increased, cold
dumbbells phase separate from the hot dumbbells. In Fig:4b, for density
$\rho*=0.8$ and Th*=80, we observe that hot and cold dumbbells are phase
separated.

To determine the extent of phase separation, we divide the total volume of the
simulation box into smaller sub-cells each having equal volume. To investigate
phase separation at the sub-cell level we calculate the quantity $\phi$ given
by

$\phi=\bigg{|}\frac{(n^{h}-n^{c})}{(n^{h}+n^{c})}\bigg{|}$ (6)

where nh and nc denote the number of hot and cold dumbbells respectively in
each sub-cell . This quantity $\phi$ is denoted as the order parameter and is
then averaged over all the sub-cells and also over all the steady-state
configurations and is given by the following equation

$\phi(T\textsuperscript{h}*)=\frac{1}{N_{cell}}\bigg{<}\sum_{i=1}^{N\textsubscript{cell}}\frac{|(n^{h}(i)-n^{c}(i))|}{(n^{h}(i)+n^{c}(i))}\bigg{>}\textsubscript{SS}$
(7)

where Ncell is the total number of sub-cells and $<>\textsubscript{SS}$
denotes that average is taken over all steady-state configurations. In our
case Ncell=125, the number is chosen such that there are enough number of
dumbbells in each sub-cell.
Following the same procedure, simulations were also carried out at density
$\rho*$ of 0.5, 0.2, and 0.1.

Fig. 5: Plot of effective temperature of cold dumbbells $T^{c*}_{eff}$ versus
temperature of hot dumbbells $T^{h*}$ at different densities. It is clearly
seen that $T^{c*}_{eff}$ increases with $T^{h*}$ due to increased heat
transfer from hot to cold dumbbells. Also, $T^{c*}_{eff}$ increases with
density because of the increase in frequency of collisions between hot and
cold dumbbells.  Fig. 6: Plot of activity $\chi$ versus temperature of hot
dumbbells $T^{h*}$ at different densities. In accordance with equation 8
activity $\chi$ increases with increase in $T^{h*}$. Also $\chi$ increases
with decrease in density because $T^{c*}_{eff}$ increases as density
increases(Fig:5).  Fig. 7: Plot of order parameter $\phi$ versus activity
$\chi$ at different densities. Initially for low activity $\chi$ order
parameter $\phi$ is small but as $\chi$ increases $\phi$ rises and saturates
to a high value indicating phase separation. Also value of $\phi$ is greater
for higher density indicating greater extent of phase separation at high
densities.  Fig. 8: Distribution of $\psi$ at density 0.8 and 0.1. The
bimodality in distribution P($\psi$) of $\psi$ occurs at Th*=15 and 25 for
density $\rho*$ 0.8 and 0.1 respectively. These temperature of hot dumbbells
Th* correspond to the critical activity of 5.80 and 11.285. The distribution
of $\psi$ at density 0.5 and and 0.2 is given in Fig:S3 in supplementary
information.

####  3.2.1 Order parameter

We define another parameter called activity

$\chi=\frac{(T_{eff}^{h*}-T_{eff}^{c*})}{T_{eff}^{c*}}$ (8)

which is a measure of the difference in effective temperature of hot and cold
dumbbells. It is the activity parameter $\chi$ which is the measure of
activity difference between hot and cold dumbbells. When both hot and cold
dumbbells are at same temperature, then activity $\chi$ is 0. As the
temperature of hot dumbbells is increased, the value of $\chi$ also increases.
The plot of effective temperature of cold dumbbells $T_{eff}^{c*}$ versus
temperature of hot dumbbells Th* is given in Fig:5 and plot of activity $\chi$
versus temperature of hot particles Th* is given in Fig:6.

In Fig:5, can see that as Th* increases $T^{c*}_{eff}$ also increases because
of increased energy transfer from hot to cold dumbbells. Also as density
increases, $T^{c*}_{eff}$ increases because with increased density frequency
of collision between hot and cold dumbbells increases. So the effective
temperature of cold dumbbells $T^{c*}_{eff}$ increases with temperature of the
hot sub-system and the density of the system. In Fig:6 we can see that
activity $\chi$ increases with temperature of hot dumbbells $T^{h*}$ in
accordance with equation 8. The activity $\chi$ also decreases with density
because $T^{c*}_{eff}$ which is in denominator in equation 8 increases with
the density of the system.

When the value of activity $\chi$ is low , the cold dumbbells form small
clusters. As the activity is increased these small clusters join to form
bigger clusters and finally at sufficiently high activity the phase separation
becomes macroscopic where hot and cold dumbbells are clearly separated as
shown in Fig:4b and Fig:4c.

In Fig:7, a plot of order parameter $\phi$ versus activity $\chi$ at various
densities is given. It can be seen that, for very small initial activity
$\chi$ the order parameter $\phi$ remains small and constant. As $\chi$
increases, the value of $\phi$ also increases and finally saturates to a large
value. We can also see that value of $\phi$ increases with density. So higher
density indicates higher extent of phase separation. Also as density
decreases, the point where $\phi$ starts increasing is pushed to high
activity. So as density decreases, the initiation of phase separation requires
high activity and the extent of phase separation is also reduced.
To find out the critical activity at which macroscopic phase separation of hot
and cold dumbbells take place we plot the probability distribution of the
parameter $\psi$. The parameter $\psi$ in ith sub-cell is defined as

$\psi=\frac{(n^{h}(i)-n^{c}(i))}{(n^{h}(i)+n^{c}(i))}$ (9)

The $\psi$ takes negative values in the sub-cells where cold dumbbells
dominate and positive values where hot dumbbells dominate. The value of $\psi$
remains near zero where there is no phase separation and number of hot and
cold dumbbells in a subcell are nearly equal. So we plot the distribution of
$\psi$ at different values of Th* for each density. Fig:8 gives the
distribution $P(\psi)$ for density 0.8 and 0.1. Refer to Fig:S3 in
Supplementary Information for distribution $P(\psi)$ for density 0.5 and 0.2.
As we increase the activity, the distribution of $\psi$ becomes bimodal in
shape when phase separation becomes macroscopic. The activity at which the
distribution of $\psi$ becomes bimodal is taken as critical activity. The
bimodality in distribution of $\psi$ occurs at Th*=15,20,20 and 25 for density
$\rho*$ 0.8, 0.5, 0.2 and 0.1 respectively. These temperatures of hot
dumbbells Th* correspond to the critical activity of 5.80, 8.41, 8.74 and
11.28. For a system of Lennard-Jones Monomer46 at density $\rho*$= 0.8, the
critical activity is 3.44 but at the same density for a system of dumbbells,
the critical activity is 5.80. So, the dumbbells start phase separating at a
higher activity than the system of LJ monomers.

Fig. 9: Plot of number of clusters of cold dumbbells Ncl versus activity
$\chi$(left) and plot of number of cold dumbbells in the largest cluster N0
versus activity $\chi$(right) . We can see that number of clusters of cold
dumbbells Ncl decreases with activity $\chi$ as small clusters join to form
large clusters. Complementary to Ncl the number of cold dumbbells in the
largest cluster N0 increases with activity $\chi$. Fig. 10: Plot of entropy
difference $\Delta$S= Sneq-Seq versus activity $\chi$ at different densities.
We can see that $\Delta$S is negative before the system phase separates but
jumps to a positive value once the phase separation sets in.

####  3.2.2 Cluster analysis

As stated earlier, initially for low activity, small clusters of cold
dumbbells are formed. As activity is increased, these small clusters join to
form larger clusters ultimately leading to phase separation. To study the
number and size of clusters, we perform cluster analysis. In this work, we
define a cluster using the following criterion: if the distance between two
atoms of different dumbbells is within a specified cut off distance rc , then
the dumbbells are considered to be part of the same cluster. The value of the
cut off distance rc is determined from the position of the first peak in the
radial distribution function calculated between the passive–passive dumbbells.
Fig:14 shows the radial distribution function between cold dumbbells at
$\rho$*=0.8 and Th*=80. The value of rc=1.069$\sigma$. Plot of number of
clusters of cold dumbbells Ncl versus activity $\chi$ is given in Fig:9(left).
We can see that initially for small activity, when there is no phase
separation number of clusters of cold dumbbells Ncl is large and constant. As
activity $\chi$ is increased , phase separation sets in and the number of
clusters of cold dumbbells decreases and saturates to a low value.
Complementary to the number of clusters of cold dumbbells Ncl, the size of the
largest cluster increase with activity $\chi$ as phase separation sets in. The
number of cold dumbbells N0 in the largest cluster is a measure of the size of
the cluster. Plot of number of cold dumbbells in the largest cluster N0 versus
activity $\chi$ is given in Fig:9(right). Initially for low activity, when
there is no phase separation, the size of largest cluster given by N0 is small
and constant. As the activity $\chi$ increases, the the size of largest
cluster grows and reaches a constant value. We can see that as density
increases, number of dumbbells in the largest cluster N0 and hence the size of
the cluster also increases indicating higher extent phase separation.

####  3.2.3 Entropy Calculation

As mentioned earlier, there is a transfer of energy from hot dumbbells to cold
dumbbells due to collisions between them. Also once the phase separation sets
in, since the cold dumbbells form closely packed clusters as shown in section
3.2.2, the hot dumbbells have larger effective volume. These factors affect
the entropy of hot dumbbells. Hence we compare the entropy of hot dumbbells
with their equilibrium counterparts. Here we have calculated the entropy per
dumbbell for the hot dumbbells in the non-equilibrium system of hot and cold
dumbbells which we call Sneq. It is compared with the entropy per dumbbell of
an equilibrium system of dumbbells where the dumbbells have the same
temperature as that of hot dumbbells Th* of the non-equilibrium system and the
density of equilibrium system is taken to be equal to the average density of
non-equilibrium system, which is represented by Seq. We use the 2-Phase
Thermodynamic(2PT) 56 method to calculate both entropy of hot dumbbells Sneq
in non equilibrium system and entropy of dumbbells in corresponding
equilibrium system Seq. We calculate the difference $\Delta S$ defined as

$\Delta S=S\textsuperscript{neq}-S\textsuperscript{eq}$ (10)

Plot of $\Delta$S versus $\chi$ at various densities is given in Fig:10. We
find that, initially for small activity $\Delta$S is negative, but as activity
increases $\Delta$S jumps to a positive value. There are two factors which
determine value of $\Delta$S.
1\. In non-equilibrium system, there is energy transfer from hot dumbbells to
cold dumbbells. It decreases the entropy of hot dumbbells Sneq compared to
that of equilibrium system Seq.

2\. When there is phase separation, cold dumbbells form dense clusters and
hence effective volume for hot dumbbells increases. This increase in effective
volume increases Sneq compared to Seq. So we can see that initially for small
activity where phase separation has not yet started, according to reason 1,
$\Delta$S is negative. As the activity is increased, phase separation takes
place and reason 2 dominates reason 1 and $\Delta$S is positive.
In Fig:10, we can also observe that the value of $\Delta$S increases with
density.

####  3.2.4 Density and temperature variation

Fig. 11: Plot of density variation $\rho$*(x) across the length of simulation
box when total density $\rho$*=0.8 at Th*=2 and 80 . We can see that density
of cold dumbbells for Th*=80 exceeds the average density 0.8 and the density
of hot dumbbells decreases below average density. We can see that the
interface is positioned near $x/L\sim 0.3$ Fig. 12: Plot of temperature
variation T*(x) across the length of simulation box for $\rho$*=0.8 at
different Th*. The temperature of the hot dumbbells (x/L>0.4) varies and
decreases as we move towards the interface due to transfer of energy from hot
to cold dumbbells.  Fig. 13: Plot of pressure variation P*(x) across the
length of simulation box at density $\rho$*=0.8 at Th*=80. The kinetic Pkin
and virial Pvir contribution to the total pressure is plotted separately. The
kinetic pressure of hot dumbbells is balanced by virial pressure of cold
dumbbells.

In order to study how density and temperature varies along the length of the
simulation box, the simulation box is divided in 20 sub-volumes along the
direction phase separation takes place(say x axis). Then density in ith sub-
volume is given by

$\rho(i)=\frac{n(i)}{V(i)}$ (11)

where n(i) is number of atoms of the dumbbells and V(i) is the volume of ith
sub-volume in reduced units. Fig:11 gives plot of density $\rho$*(x) versus
x/L for $\rho$=0.8 at Th*=2 and Th*=80, where L is the length of simulation
box along x-axis. The density variation of hot and cold dumbbells has been
plotted separately to find the position of interface. From the plot, the
interface seems to appear somewhere near x/L=0.3. We can see that density
$\rho*$(x) of cold dumbbells before the interface(x/L=0.3) is much higher than
that of average density 0.8, which indicates that cold dumbbells form densely
packed clusters. Also the density of hot dumbbells (0.4<x/L<0.8) is much lower
than that of average density 0.8.
Each atom in a dumbbell has three translational degrees of freedom. By
equipartition theorem each degree of freedom contributes $\frac{1}{2}k_{b}T$
of energy. Hence temperature in ith sub-volume is given by

$\frac{3}{2}k_{b}T(i)=\frac{1}{n(i)}\sum_{j=1}^{n(i)}\frac{1}{2}mv^{2}_{j}$
(12)

Fig:12 gives plot of temperature variation of dumbbells across the interface
for different values of Th*. We can see from temperature variation that
temperature of hot dumbbells is not uniform but decrease as they move towards
the interface indicating energy transfer from hot to cold dumbbells. Pressure
profile along the interface is calculated from diagonal components of stress
tensor.

$P(i)=\frac{1}{3*V(i)}[\sum_{j=1}^{n(i)}\frac{1}{2}mv^{2}_{j}+\sum_{j=1}^{n(i)}\vec{r_{j}}.\vec{f_{j}}]$
(13) $P(i)=P_{kin}(i)+P_{vir}(i)$ (14)

Here $\vec{r}_{j}$ is the position vector of jth atom and $\vec{f}_{j}$ is
force on jth atom due to all other atoms(includes force due to interaction
with other dumbbells and the force due to bonding). The first term is the
kinetic part of pressure Pkin and second term is the virial part of pressure
Pvir. Fig:13 gives the pressure variation across the interface when
$\rho$*=0.8 at Th*=80 . We have also plotted the kinetic and virial
contribution to the pressure separately. We can see that the pressure is
almost constant along the length of simulation box. We can see that in the
regions where the cold dumbbells dominate (x/L<0.3) the virial pressure Pvir
is very high and kinetic pressure Pkin is low because of the low temperature
of cold dumbbells. In the region where the hot dumbbells dominate (x/L>0.3)
the kinetic pressure Pkin is very high because of the high temperature of hot
dumbbells. So, in the phase separated system the kinetic pressure of hot
dumbbells is balanced by the virial pressure of cold dumbbells.

####  3.2.5 Bond orientation order parameter

Fig. 14: The g(r) of all dumbbells when Tc*=Th*=2(blue) at $\rho*$=0.8 shows a
liquid like structure with damping oscillations. The g(r) of the cold
dumbbells(black) at $\rho*$=0.8 and Th*=80 indicate solid like order but g(r)
of hot dumbbells(red) at $\rho*$=0.8 and Th*=80 indicate gaseous phase. Fig.
15: Distribution of bond orientation order parameters q4 and q6 of atoms of
cold dumbbells when Th*=80 at $\rho*$=0.8. From the distribution, we can see
that the cold dumbbells have predominantly fcc and hcp arrangement

In Fig:14 we have radial distribution of all dumbbells(blue) at $\rho*=0.8$
and Tc*=Tc*=2 which shows liquid like structure with damping oscillations. In
the plot in Fig:14 we have radial distribution function g(r) between cold
dumbbells(black) and radial distribution function g(r) between hot
dumbbells(red) for $\rho*=0.8$ and Th*=80. The g(r) of cold dumbbells show
crystalline order but g(r) of hot dumbbells indicate a gaseous phase. From the
density variation(shown in figure 11) , we can see that the cold dumbbells are
forced to form dense clusters and from the radial distribution function (shown
in figure 14) we can see that the clusters show crystalline order. So in the
system of hot and cold dumbbells which is in liquid state when Tc*=Tc*=2, as
the temperature of hot dumbbells is increased, the cold dumbbells have
crystalline order while the hot dumbbells are in gaseous phase.

In order to identify the local crystalline order we calculate Steinhardt bond
order parameters. Two atoms of different dumbbells are considered to be
neighbors if their distance is below 1.2$\sigma$. The bond order parameters
are defined as

$q_{l}(i)=\sqrt{\frac{4\pi}{2l+1}\sum_{m=-l}^{l}q_{lm}q^{*}_{lm}}$ (15)

where,

$q_{lm}(i)=\frac{1}{N_{b}(i)}\sum_{j=1}^{N_{b}(i)}Y_{lm}(\theta_{ij},\phi_{ij})$
(16)

Nb(i) corresponds to the number of bonds for particle i, and Ylm is the
respective spherical harmonic function. The ideal values of these parameters
for various crystalline systems is given in table 1 .

| q4 | q6
---|---|---
fcc | 0.19 | 0.57
hcp | 0.097 | 0.48
Table 1: Value of q4 and q6 for fcc and hcp crystals. Fig. 16: Snapshots of
the non equilibrium system of hot and cold dumbbells at different fractions of
hot dumbbells fh when $\rho*$=0.8 and Th*=80 . We can see that hot and cold
dumbbells phase separate at all given fractions of hot dumbbells fh.

In Fig:15 distribution of bond order parameters q4 and q6 when Th*=80 at
$\rho*$=0.8 is given . We can see that the cold dumbbells form crystalline
solid where the atoms of dumbbells are arranged so that they have
predominantly face-centered cubic (fcc) and hexagonal-close packing (hcp)
structure. Furthermore, each dumbbells point in different directions and does
not posses orientational order. In an equilibrium system of dumbbells, at
temperature T*=2, above density $\rho*=1.08$ the dumbbells form disordered
solid60 where atoms of dumbbells are in face centered cubic arrangement but
the dumbbells have no orientational order and dumbbells point in different
direction. In our non-equilibrium system of hot and cold dumbbells, we can see
that the cold dumbbells form clusters of density much greater than
$\rho*=1.08$ (Refer Fig:11) but the atoms of dumbbells show both hcp and fcc
arrangements as obtained from bond order parameters. But the cold dumbbells of
the solid cluster in the non-equilibrium system also do not posses
orientational order like dumbbell crystal in the equilibrium system.

In the supplementary information (SI), we have compared velocity distribution
of the whole hot subsystem with the velocity distribution of the hot dumbbells
lying in middle of phase separated hot subsystem(refer figure S1 in
supplementary information). We also have discussed the effect of damping
parameter of thermostat on the non-equilibrium system of hot and cold
dumbbells. Figure S2 gives plots of effective temperature of cold dumbbells
$T^{c*}_{eff}$ and activity $\chi$ versus temperature of hot dumbbells
$T^{h*}$. We have also compared the pressure of different sub-cells with their
equilibrium counter-parts which have same density and temperature.

####  3.2.6 Effect of ratio of hot and cold dumbbells on phase separation

All the results we have discussed so far correspond to 50:50 mixture of hot
and cold dumbbells. We have also simulated the non-equilibrium system at
different ratios of hot and cold dumbbells. The fraction of hot dumbbells fh
in the system is given by

$f_{h}=\frac{Number\,of\,hot\,dumbbells}{Total\,number\,of\,dumbbells}$ (17)

Following the same procedure mentioned in section 3.2, we have simulated the
mixture of hot and cold dumbbells for different fraction of hot dumbbells fh
0.1, 0.3, 0.7 and 0.9 at density $\rho$* 0.8. The Fig:16 shows the snapshots
of the system at Th*=80 at fh 0.1, 0.3, and 0.9. We observe that for all given
fraction of hot dumbbells fh the hot dumbbells and cold dumbbells phase
separate. The effective temperature of cold dumbbells increases with increase
in fraction of hot dumbbells in the system (See Fig:S4 and S5 in the
supplementary information). We plot the probability distribution P($\psi$) of
the parameter $\psi$ defined in equation 9 for different fraction of hot
dumbbells fh to identify the critical activity at which the phase separation
of hot and cold dumbbells becomes macroscopic(See Fig:S6, S7, S8 and S9 in the
SI for the plot of P($\psi$) at different fraction of hot dumbbells fh). In
Table:2 we give the value of critical activity for different fraction of hot
dumbbells fh.

fh | critical activity
---|---
0.1 | 15.28
0.3 | 8.14
0.5 | 5.80
0.7 | 3.56
0.9 | 3.30
Table 2: The critical activity of system of hot and cold dumbbells at
different fraction of hot dumbbells fh. We can see that as the fraction of hot
dumbbells fh increases the value of critical activity decreases

From Table:2 we can see that as the fraction of hot dumbbells fh increases the
value of critical activity decreases. So we can conclude that as the fraction
of hot dumbbells is increased in the system, the hot and cold dumbbells start
phase separating at smaller values of activity.

##  4 Conclusion

In a system of hot and cold dumbbells, phase separation is observed at all
densities for sufficiently high activity. The extent of phase separation is
more at high density. As the temperature of hot dumbbells is increased, the
phase separated cold dumbbells form a cluster with crystalline order while the
hot dumbbells are in gaseous phase. We also determine the critical activity
from the distribution of parameter $\psi$ defined in equation 9 and find that
the critical activity of dumbbells is higher than that of Lennard-Jones
monomer. The critical activity also increases with decrease in density. The
entropy of hot dumbbells in the non-equilibrium system increases with the
activity due to increase in effective volume on phase separation. The density
of cold dumbbells is greater than average density of the system while the
density of hot dumbbells is less than the average density of the system . The
crystalline nature of cluster of cold dumbbells is studied by bond-orientation
order parameter and is found to be predominantly fcc and hcp. We also find
that critical activity decreases as we increase the fraction of hot dumbbells.
The representation of ’activity’ by temperature helps in understanding the
physics of non-equilibrium systems.

## Conflicts of interest

There are no conflicts to declare.

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## Supplementary Information

### 1.Comparison with the equilibrium system

Even though, the hot and cold subsystem were maintained at fixed temperatures
at each simulation run, we see that the temperature varies as we move along
interface even within the hot subsystem of the phase separated system(See
Fig:12 in the main article). So we have plotted the velocity distribution of
the whole hot subsystem and compared it with velocity distribution of hot
dumbbells lying in the middle of phase separated hot subsystem. In Fig:S1, the
red plot shows velocity distribution of the whole hot subsystem and the blue
plot shows velocity distribution of the hot dumbbells within the hot sub
system(away from interface) in range x/L=0.5 to x/L=0.7(refer Fig:11 and 12 in
main article) when Th*=80 and density $\rho$*=0.8. The velocity distribution
of the whole hot sub-system(red) is clearly not Maxwellian. But the velocity
distribution of hot dumbbells away from interface (blue) resembles a
Maxwellian distribution.

S1: The red plot shows velocity distribution of the whole hot subsystem and
the blue plot shows velocity distribution of the hot dumbbells within the hot
sub system in range x/L=0.5 to x/L=0.7 when Th*=80.

We have the density, temperature and pressure of the sub-volumes along the
x-axis(refer Fig:11, 12, 13 in main article) at different Th*. So we choose
few sub-volumes, some of which are near interface and the rest away from
interface and compare the pressure of the sub-volumes with the pressure
obtained in a equilibrium system which has the same density and temperature as
that of the sub-volume.

In Supplementary Table 1, x/L refers to the position of the sub-volume, whose
density($\rho$*), temperature(T*) and pressure(P*(neq)) is given. x/L =0.3
indicates the sub-volume is located near interface. x/L=0.6 indicates sub-
volume is present away from interface within the phase separated hot
subsystem. P*(eq) refers to the pressure of an equilibrium system with the
same density and temperature as that of corresponding sub-volume. $\Delta$P
refers to the difference between P*(neq) and P*(eq). We can see from the table
that for sub-volume near interface(x/L=0.3) the difference $\Delta$P is larger
compared to sub-volumes away from interface(x/L=0.6). So as we move away from
the interface, the sub-volumes behave more closely to their equilibrium
counter-parts.

x/L | Th* |  $\rho$* | T* | P*(neq) | P*(eq) |  $\Delta$P
---|---|---|---|---|---|---
0.3 | 80 | 0.80 | 32.46 | 57.85 | 59.45 | 1.6
0.3 | 60 | 0.78 | 26.22 | 45.93 | 47.64 | 1.71
0.3 | 40 | 0.78 | 17.69 | 33.90 | 35.00 | 1.1
0.3 | 20 | 0.87 | 6.34 | 19.01 | 20.71 | 1.7
0.6 | 80 | 0.43 | 139.09 | 58.64 | 58.21 | 0.43
0.6 | 60 | 0.44 | 107.17 | 47.47 | 47.88 | 0.41
0.6 | 40 | 0.44 | 73.75 | 34.27 | 33.82 | 0.45
0.6 | 20 | 0.47 | 35.36 | 20.1 | 19.77 | 0.33
Supplementary Table 1: Here pressure of non-equilibrium system P*(neq) is
compared with pressure of corresponding equilibrium system P*(eq). We can see
from the table that for sub-volume near interface(x/L=0.3) the difference
$\Delta$P is larger compared to sub-volumes away from interface(x/L=0.6).

### 2\. Effect of thermostat parameters on simulation.

In section 2 in the main article, it is mentioned that Nosé-Hoover thermostat
is used to maintain the temperature of dumbbells. The damping parameter T-damp
is 50$\delta$t, where $\delta$t is the time step used in simulation. T-damp
determines how quickly the temperature of the system relaxes to temperature
value specified by the thermostat. Larger the value of T-damp, slower the
system reaches the temperature specified by the thermostat. In the non-
equilibrium system, for cold subsystem the value of T-damp determines the rate
at which it removes the energy transferred from the hot subsystem. To study
the effect of T-damp we perform simulation of the non-equilibrium system at
$\rho$* = 0.8 varying the damping parameter T-damp=50$\delta$t, 100$\delta$t,
200$\delta$t. In Fig:S2, plot of effective temperature of cold dumbbells
$T^{c*}_{eff}$ versus temperature of hot dumbbells $T^{h*}$ and plot of
activity $\chi$ versus temperature of hot dumbbells $T^{h*}$ at different
T-damp is given. We can see that as the value of T-damp is increased, the
effective temperature of cold dumbbells $T^{c*}_{eff}$ also increases. This is
because large value of T-damp implies a slow rate at which energy is removed
from cold system as a result of which the cold-dumbbells reach steady state at
high effective temperature. The increase in $T^{c*}_{eff}$ with T-damp results
in decrease in activity $\chi$ with increase in T-damp. So the value of T-damp
determines the activity of the non-equilibrium system for given values of Tc*
and Th*.

S2: a)Plot of effective temperature of cold dumbbells $T^{c*}_{eff}$ versus
$T^{h*}$ at different T-damp. b) Plot of activity $\chi$ versus $T^{h*}$ at
different T-damp. We can see that $T^{c*}_{eff}$ increases with increase in
T-damp. Conversely,activity $\chi$ decreases with increase in T-damp.

### 3.Additional Supplementary figures

S3: Probability distribution P($\psi$) of $\psi$ at density $\rho*$=0.5 and
0.2 for different values of temperature of hot dumbbells Th*. The bimodality
in distribution P($\psi$) of $\psi$ occurs at Th*=20 for both density
$\rho*$=0.5 and 0.2 respectively. These temperature of hot dumbbells Th*
correspond to the critical activity $\chi$ of 8.41 and 8.74 respectively. S4:
Plot of effective temperature of cold dumbbells $T_{eff}^{c*}$ versus
temperature of hot dumbbells Th* at density $\rho*$=0.8 for different
fractions of hot dumbbells fh. We can see that as the fraction of hot
dumbbells fh increases the effective temperature of cold dumbbells
$T_{eff}^{c*}$ also increases.  S5: Plot of activity $\chi$ versus temperature
of hot dumbbells Th* at density $\rho*$=0.8 for different fractions of hot
dumbbells fh. We can see that as the fraction of hot dumbbells fh increases
the activity $\chi$ decreases.  S6: Probability distribution P($\psi$) of
$\psi$ at density $\rho*$=0.8 for fraction of hot dumbbells fh=0.1 at
different Th*. Due to unequal number of hot and cold dumbbells the
distribution tends to be towards the majority species. The bimodality in
distribution P($\psi$) of $\psi$ occurs at Th*=35 for fh=0.1. The temperature
of hot dumbbells Th*=35 corresponds to activity $\chi$ of 15.28.  S7:
Probability distribution P($\psi$) of $\psi$ at density $\rho*$=0.8 for
fraction of hot dumbbells fh=0.3 at different Th*. Due to unequal number of
hot and cold dumbbells the distribution tends to be towards the majority
species. The bimodality in distribution P($\psi$) of $\psi$ occurs at Th*=20
for fh=0.3. The temperature of hot dumbbells Th*=20 corresponds to activity
$\chi$ of 8.14.  S8: Probability distribution P($\psi$) of $\psi$ at density
$\rho*$=0.8 for fraction of hot dumbbells fh=0.7 at different Th*. Due to
unequal number of hot and cold dumbbells the distribution tends to be towards
the majority species. The bimodality in distribution P($\psi$) of $\psi$
occurs at Th*=10 which corresponds to activity $\chi$ of 3.56.  S9:
Probability distribution P($\psi$) of $\psi$ at density $\rho*$=0.8 for
fraction of hot dumbbells fh=0.9 at different Th*. Here since the hot
dumbbells exceedingly dominate cold-dumbbells. So we have magnified the region
-1$<\psi<$0.5. The bimodality in distribution P($\psi$) of $\psi$ occurs at
Th*=10 which corresponds to activity $\chi$ of 3.30.