# Inferring urban polycentricity from the variability in human mobility
patterns

Carmen Cabrera-Arnau  Department of Geography and Planning, University of
Liverpool, Liverpool, United Kingdom  Chen Zhong  Centre for Advanced Spatial
Analysis (CASA), University College London, London, United Kingdom  Michael
Batty  Centre for Advanced Spatial Analysis (CASA), University College London,
London, United Kingdom  Ricardo Silva  Department of Statistical Science,
University College London, London, United Kingdom  Soong Moon Kang  School of
Management, University College London, London, WC1E 6BT, United Kingdom

###### Abstract

The polycentric city model has gained popularity in spatial planning policy,
since it is believed to overcome some of the problems often present in
monocentric metropolises, ranging from congestion to difficult accessibility
to jobs and services. However, the concept ‘polycentric city’ has a fuzzy
definition and as a result, the extent to which a city is polycentric cannot
be easily determined. Here, we leverage the fine spatio-temporal resolution of
smart travel card data to infer urban polycentricity by examining how a city
departs from a well-defined monocentric model. In particular, we analyse the
human movements that arise as a result of sophisticated forms of urban
structure by introducing a novel probabilistic approach which captures the
complexity of these human movements. We focus on London (UK) and Seoul (South
Korea) as our two case studies, and we specifically find evidence that London
displays a higher degree of monocentricity than Seoul, suggesting that Seoul
is likely to be more polycentric than London.

Keywords: Urban spatial structure, Smart card data, Polycentricity, Urban
complexity, Mixture models

##  1 Introduction

###  1.1 Measuring monocentricity

People in cities interact with their environment by developing urban land for
different socioeconomic activities. The way in which land use is located and
arranged within a city is either a result of self-organising mechanisms over
the course of time or as a result of specific interventions through different
varieties of urban planning, at different spatial scales. In this context, the
configuration is usually referred to as urban structure and through the study
of such structures, we can learn more about the spatial behaviours of the
societies that, over time, have built them. Moreover, urban structure also
plays an important role in shaping the present and future, given its impact on
different socioeconomic features such as mobility, access to jobs, social
mixing, heterogeneity, segregation, deprivation, urban efficiency, and
sustainability.

The simplest form of urban structure corresponds to the monocentric city,
where socioeconomic activity is localised in a unique central region. In
practice, monocentric cities facilitate the accumulation of social
interactions and innovation, and consequently give rise to economies of
agglomeration characterised by increasing returns to scale [1, 2]. However,
monocentric cities are also subject to heavy tidal flows on the transport
facilities during peak hours, severe congestion and disproportionally high
rents close to the centre [3, 4, 5]. The monocentric structure of cities
prevailed until the industrial revolutions led to new forms of transport that
broke the bounds of compact cities. Consequently, monocentric cities have
gradually decentralized, transforming into complex hierarchies of different
kinds of centres, neighbourhoods and sprawling structures that are tied
together by a multiplicity of transport and information systems [6]. Yet,
explanatory models of urban structure based on a monocentric approach are
still used due to their simplicity and formal analytical elegance. Their
validity, however, should be questioned both in terms of the theoretical
assumptions needed for the formulation of the models [7, 8, 9] and from the
point of view of public policy [10], since most plans for future cities have
long abandoned the idea of the monocentre.

Polycentricity has therefore become the focus of much spatial policy [11],
since it is believed that urban dwellers in polycentric cities might benefit
from congestion relief in comparison with their monocentric counterparts [12]
and from increased accessibility to jobs and services, which may translate in
higher rent and housing prices all across the city, but also in more time-
efficient and cost-efficient travel. Despite the raise in popularity of the
idea of polycentric development, it remains a rather fuzzy concept as it seems
to mean different things to different actors and on different scales [13, 11,
14, 15, 16]. The lack of a concise and coherent definition raises an issue:
how to measure polycentricity? If we do not know what to measure, we simply
cannot measure it [11, 15, 16]. In this work, instead of attempting to answer
the ill-defined question ‘to what extent is a city polycentric?’, we provide
an approach to analysing departures from a well-defined concept of
monocentricity.

Despite the fuzziness in the definition of ‘polycentric city’, there is a long
tradition of theoretical research and empirical evidence surrounding the
debate on monocentricity versus polycentricity. We will simply indicate recent
work here such as that based on an analysis of data from US metropolitan areas
by Arribas-Bel and Sanz-Garcia [10], which shows that monocentricity still
retains a substantial influence on the intraurban structure of many
metropolitan areas. This is despite the general consensus in the literature
that modern cities above a certain size threshold become polycentric and that
monocentricity is an older concept more appropriate to the city in history
prior to the industrial revolution. In this sense, the concept might perhaps
be somewhat obsolete when dealing with the real world. Additionally, the
authors cited in [10] find that there is no clear evolutionary trend in US
cities towards polycentricity between 1990 and 2010.

By contrast, Alidadi and Dadashpoor [17] analyse data from Iran to find that a
monocentric model is not able to explain the spatial distribution of
employment in Tehran while the main core has been losing its importance with
the passage of time. Li [18] draws upon fine-grained LandScan population data
corresponding to 286 Chinese cities to find that in general, urban spatial
structure has become more polycentric as well as more concentrated (i.e. with
a higher share of their population living in the centres) while these changes
have usually resulted in population and economic productivity growth.

There are other studies that find evidence for mixed types of urban structure.
Hajrasouliha and Hamidi [19] base their study on three typologies of urban
structure: monocentricity, polycentricity, and generalized dispersion. When
analysing the spatial structure of employment data from 356 US metropolitan
regions, they find that mixed typologies of urban structure outnumber the
three “pure” ones by almost four to one. They also find that polycentricity is
somewhat more common than monocentricity. Similarly, in [20], Sweet et al use
cross-sectional data to estimate the relative strengths of monocentricity,
polycentricity, and dispersion for characterizing Canadian cities. Their
results indicate that elements of each model are evident, but each tends to
dominate in different contexts. When focusing on Montreal, Toronto, and
Vancouver, their results imply that accessibility, municipal competition, and
globalization play a role in shaping urban structure.

###  1.2 Using new forms of data

In the past, most empirical studies of urban structure based their conclusions
on data associated with the spatial distribution of employment or population,
obtained largely from traditional sources of direct observation by
questionnaires such as surveys, censuses and administrative records. The
rationale behind this choice of datasets is that they are comprehensive and
representative and have the potential to uncover where city dwellers conduct
most of their socioeconomic activity. It has only been in recent years that
the focus has been turned to alternative data sources, which can offer real-
time and easy-access records at very small spatial scales. In particular, the
locations that people choose to visit at different times of the day or week
are very much conditioned by the spatial structure of the city and at the same
time, the complexity of human movements shapes the usage of urban space and
the arrangement of resources [6]. Therefore, the study of patterns in human
mobility through alternative data sources can help us understand the travel
behaviour of city dwellers and it can also help us uncover the socio-economic
features of urban structure.

For example, recent studies have used data derived from social media platforms
as well as location tracker devices in mobile phones in order to understand
the spatial structure of cities based on the places visited by the users [21,
22, 23, 24, 25, 26, 27, 28, 29]. Taxi trajectory data is another alternative
source of data that has gained in popularity in recent years as a means to
uncover information about urban structure [30, 31, 32, 33]. Taxi trajectory
data not only has the potential to reveal the characteristics of human
movement within the city, but also real-time traffic status as well as
potential social inequalities. A third alternative source of data is that
derived from smart travel cards or simply, smart cards. Like the other sources
of alternative data already mentioned, smart card data offers information
regarding daily human activities at high resolution, both in the spatial and
temporal domains and consequently, it has been used to explore urban structure
[34, 35, 36, 37, 38, 39]. Here, we will focus on the latter type of new data
sources that record such movements. In our case, this is smart card data from
the automatic fare collection system in London’s and Seoul’s public transport,
which contains information about the origin, destination and time at which
each individual journey occurs.

###  1.3 Aim and contribution

Our aim is to provide a novel approach to model the extent to which a city
departs from the monocentric structure by considering the variability inherent
in human mobility patterns, and by avoiding the fuzziness in the concept of
polycentricity. To investigate the applications of this approach, we consider
two case studies using high spatiotemporal resolution data derived from smart
travel cards corresponding to London, United Kingdom, and Seoul, South Korea.

Our methodology first considers the frequency distribution of the length of
journeys terminating at each station in the public transport system of a given
city on a typical weekday. We define the “nucleus” of each city as the station
representing a hypothetical centre. We also consider the network structure of
the public transport system in order to measure the length of the journeys by
the network distance between stations. We then introduce Poisson mixture
models as a statistical approach to describe the frequency distribution of the
length of the journeys terminating at each station in the transport system.
The Poisson mixture models enable us to capture the variability in the human
mobility patterns reflected in urban structure, which in real cities includes
a blend of features which characterise both monocentric and polycentric
cities.

Next, we state what we call the monocentric hypothesis: “If a city was
perfectly monocentric, the expected length of the journeys taken to a given
station, except for the nucleus itself, would be equal to the length of the
shortest path between the nucleus and the destination station”. In this
hypothetical scenario, the nucleus would be the only centre for socioeconomic
activity in the city, and consequently, a typical journey terminating at a
given station other than the nucleus would have its origin at the nucleus.
Journeys whose destination is the nucleus would have their origin at various
locations across the public transport system.

In reality, cities and urban mobility patterns are more complex than stated by
the monocentric hypothesis, so quantifying deviations from this idealistic
behaviour enables us to understand the extent to which a city departs from
monocentricity, or in other words, it enables us to indirectly infer its
degree of polycentricity.

Therefore, the main contribution of this analysis is a solution to the problem
of quantitatively describing the degree of monocentricity of a city. Our data-
driven approach based on mixture models considers the complexity of urban
space since these models are able to capture the variability in human
movements that arises as a result of sophisticated forms of urban structure.
Instead of considering discrete typologies of urban structure (e.g.
monocentric and polycentric), the method proposed here conceptualises urban
structure typologies as a spectrum, where monocentricity is an idealistic
extreme. According to the observed patterns of human mobility, we specifically
find evidence that London displays a higher degree of monocentricity than
Seoul, suggesting that Seoul is likely to be more polycentric than London.

The rest of the paper is organized as follows. In Section 2, we describe in
detail the data sets used for the analysis and how the data has been
processed. Section 3 is dedicated to the methodology which we followed for the
analysis. We explain how we conceive of the public transport system as a
complex network. We also introduce the probabilistic modelling framework and
mixture models. In Section 4, we present the results of the analysis
corresponding to the two case studies of London and Seoul. We provide some
concluding remarks and points of discussion in Section 5. We also included
Supplementary Information with some additional results to support our findings
and conclusions.

##  2 Methodology

###  2.1 Data and notation

The Oyster card in London and the T-money card in Seoul are automatic fare
collection systems that record the place and time when a traveller enters and
exists the public transport system by tapping in and out with their card. In
2012, more than 80% of all journeys on public transport in London were made
using Oyster card whereas in Seoul 98.9% of all journeys on public transport
were made using T-money in 2013. Our study is based on tap-in and tap-out
records of Oyster and T-money card but we only use tap-outs because we
consider that these are a better indication of employment locations in the
morning peak whereas tap-outs in the evening peak are more likely to involve
trips that involve travel to entertainment. The taps that we used are thus
aggregated at a temporal resolution of 1 hour. For London, we used data
recorded 24 hours on 5 weekdays between January 20 and 24 in the year 2014,
and for Seoul, 4 weekdays between December 17 and 21 in the year 2012. We
excluded the data from Wednesday, December 19, because it was a presidential
election day in Korea and regular travel patterns were disrupted.

We process the data to obtain the number of journeys of a given length which
terminate at each station on a typical weekday. Data for a typical weekday is
obtained by averaging the number of counts across all the weekdays included in
the raw data set and rounding the average value to the closest integer. The
total daily average count of journeys was 3.22 million encompassing 382
stations in London and 5.96 million for 512 stations in Seoul.

For the subsequent parts of the analysis, we introduce the following notation.
Each of the $N$ stations in each city is symbolised by $S_{i}$, with
$i=1,...,N$. Station $S_{i}$ is the destination of $M_{i}$ journeys so it has
$M_{i}$ tap-outs. The length of the $l$th journey terminating at station
$S_{i}$ is symbolised by $L_{i}^{l}$, where $l$ is an index over the $M_{i}$
journeys to $S_{i}$ and therefore, $l=1,...,M_{i}$.

###  2.2 The transport system as a complex network

We define an undirected simple network $G=(V,E)$ as an abstract
conceptualisation of the public transport system in London or Seoul. The
network $G$ is formed by the set of $N$ vertices or nodes $V$ and the set of
edges $E$. The $i$th node of the network $G$ corresponds to station $S_{i}$ in
the public transport system. An edge is present between two nodes $i$ and $j$
if there is at least a line of transport that provides a direct connection
between the stations $S_{i}$ and $S_{j}$. The distance between stations
$S_{i}$ and $S_{j}$ is symbolised by $d_{ij}$ and is defined as the sum of the
distances associated with the minimum number of edges that need to be
traversed in order to travel from $S_{i}$ to $S_{j}$. The length of a journey
between two stations is defined as the distances associated with the number of
edges that are traversed from the origin to the destination nodes, but here we
assume that the length of a journey between stations $S_{i}$ and $S_{j}$ is
equal to the distance $d_{ij}$, i.e. we assume that, from all possible
trajectories from $S_{i}$ to $S_{j}$, passengers always choose the one
involving the fewest stops.

For the next steps of the analysis, we establish a hypothetical centre in the
network, which we call the “nucleus”. Different notions of centrality can be
considered, although not all of them are suitable for our analysis. For
example, if we considered that the nucleus is the closest station to the
geographical centroid of the city, then the definition of centre would depend
on the physical boundaries of the city region, which in turn, can be
established according to a variety of different criteria. If instead, we
considered a measure of centrality based on the topology of the network, such
as the betweenness centrality of each node, then the nucleus in Seoul would be
Wangsimni station, which does not necessarily represent what many Seoulites
would consider to be a central region of Seoul. Similarly, a measure of
centrality based on traffic flows may also not coincide with what most people
consider to be the centre. For these reasons, we opt for a somewhat arbitrary
choice of nucleus based on what is popularly considered to be a central area:
Piccadilly Circus station in London and City Hall station in Seoul. We assign
the index $i=1$ to the station corresponding to the nucleus in each city. To
counter the arbitrariness of these choices of nucleus, we provide a
sensitivity test of the results of our analysis. This can be found in the
Supplementary Information, in the section titled ‘Sensitivity analysis for
different choices of nucleus’.

Figure 1 shows the physical layout of London and Seoul’s public transport
system, with the lines and stations that are included in the data sets. The
transport lines are traced simply as straight lines to show the topology of
the network. The colour of the nodes corresponds to the average length of the
journeys terminating at $S_{i}$, symbolised by $\bar{L}_{i}$ and the size
represents the number of journeys $M_{i}$ reaching each station $S_{i}$.

Figure 1: Map of London’s and Seoul’s public transport systems.

###  2.3 Modelling the distribution of journey lengths

In this section, we introduce a probabilistic approach to model the frequency
distribution of journey lengths on a typical weekday. We regard $L_{i}$ as a
discrete random variable denoting the length of journeys whose destination is
station $S_{i}$. For each station, our data set gives $M_{i}$ realisations of
$L_{i}$, so the observed length of the $l$th journey, symbolised by
$L_{i}^{l}$, would correspond to the $l$th realisation of $L_{i}$. The true
probability distribution of random variable $L_{i}$ is unknown, however, its
empirical probability density function, denoted by $\hat{f}_{i}$, can be
obtained from the observed data as

$\hat{f}_{i}(L_{i}=h)=\frac{1}{M_{i}}\sum_{l=1}^{M_{i}}\mathbbm{1}_{L_{i}^{l}=h},$
(1)

with $h\in\mathbb{N}$. In equation (1), $\mathbbm{1}_{L_{i}^{l}=h}$ is an
indicator function that takes the value 1 when $L_{i}^{l}=h$ and zero
otherwise. Hence, the probability that the length of a journey with
destination at station $S_{i}$ is equal to $h$ is approximated by
$\hat{f}_{i}(L_{i}=h)$, computed as the number of observed counts of journeys
of length $h$ terminating at $S_{i}$, divided by the total number of journeys
terminating at $S_{i}$, i.e. $M_{i}$.

Under the monocentric hypothesis stated in section 1.3, if a city was
perfectly monocentric, the expected length of the journeys taken to a given
station, except for the nucleus itself, would be equal to the length of the
shortest path between the nucleus and the destination station. Hence, this
null hypothesis can be expressed mathematically as (2)

$E[L_{i}]=d_{1i}$ (2)

for $i=2,...,N$, where $E[L_{i}]$ is the expected value of random variable
$L_{i}$, which can be approximated by the sample mean
$\hat{\mu}_{i}=\frac{1}{M_{i}}\sum_{l=1}^{M_{i}}\mathbbm{1}_{L_{i}^{l}}$.
Taking this into account, the monocentric hypothesis can be expressed as
$\hat{\mu}_{i}=d{1i}$.

In reality, the data does not lie on the line given by $\hat{\mu}_{i}=d{1i}$,
as shown in Figure 2. In the Figure, the network distance from the nucleus to
the destination station is represented on the $x$-axis and the average length
of journeys arriving at a station is represented on the $y$-axis. Each bubble
in the plot represents one station. The solid line is the regression line
obtained via ordinary least squares. The results of the linear regression are
shown in Table 1. The red dotted line represents equation (2), i.e. the line
where points would lie if the monocentric hypothesis was satisfied. In fact,
there is a tendency for the average length of the journeys terminating at
station $S_{i}$ to be less than $d_{1i}$ as $d_{1i}$ gets larger. This
suggests that journeys which terminate at stations that are far from the
nucleus, tend to take place more locally. The effect is particularly obvious
in Seoul, showing that the observed patterns of mobility depart from the
monocentric hypothesis to a greater extent.

Figure 2: Relationship between the mean of the distribution of journeys terminating at each station, $\hat{\mu}^{p}_{i}$ and $\hat{\mu}^{d}_{i}$, and the distance $d_{1i}$ between the nucleus $S_{1}$ and the destination station $S_{i}$. Table 1: Results of linear regressions considering $d_{1i}$ as the explanatory variable and $\hat{\mu}_{i}$ as the response variables. Piccadilly Circus is considered to be the nucleus in London and City Hall in Seoul. | Intercept | Slope | $R$ |  $p$-value
---|---|---|---|---
London | $4.352\pm 0.283$ | $0.580\pm 0.027$ | $0.758$ | $<0.05$
Seoul | $8.203\pm 0.157$ | $0.152\pm 0.010$ | $0.585$ | $<0.05$

In addition, we observe not only that $\hat{\mu}_{i}=d{1i}$ is not satisfied,
but also that $L_{i}$ displays a high degree of variability for $i=1,...,N$.
This effect is captured in Figure 3, where each data point corresponds to an
individual journey, the $x$-coordinate represents the distance $d_{1i}$
between the nucleus and the destination station, and the $y$-coordinate
represents the length of each individual journey $L_{i}^{l}$, with
$l=1,...,M_{i}$ and $i=1,...,N$. Once again, the red dotted line represents
equation (2).

Figure 3: Distribution of the length of journeys terminating at any
destination station which is at a given distance from the nucleus. The solid
line indicates the median of the distribution for each value of $d_{1i}$. The
dashed line represents the line $L_{i}^{l}=d_{1i}$.

Figures 2 and 3 are thus a manifestation that real cities do not conform to
the hypothesised monocentric scenario. Next, explore the deviations from the
monocentric hypothesis by leveraging the observed variability in our data.

###  2.4 An approach based on mixture models

In order to describe the deviations from the hypothesised monocentric
behaviour, we introduce mixture models, which are probabilistic models with
the ability to represent the possible presence of different statistical sub-
populations within the overall population. In the context of this paper,
mixture models can be used to infer the possible presence of centres other
than the nucleus based only on the data for the number and length of journeys
terminating at each station. The approach that we propose here consists in
assuming that the true probability distribution for the number of journeys to
station $S_{i}$ is given by a mixture distribution of the following form

$f_{i}(L_{i}=h|\mathbf{w}_{i},\boldsymbol{\theta}_{i})=\sum_{j=1}^{K}w_{i}^{j}p_{i}^{j}(L_{i}=h|\boldsymbol{\theta}_{i}^{j}).$
(3)

In equation (3), the probability that a journey terminating at station $S_{i}$
has length $L_{i}=h$, is now conditional on the parameters of the true
distribution, $\mathbf{w}_{i}$ and $\boldsymbol{\theta}_{i}$. The probability
density function of the true distribution is given by a weighted sum of $K$
probability density functions corresponding to each of the components of the
mixture. The number of components in the mixture $K$ corresponds to the number
of centres assumed by the model. If $K=1$, then the only centre accounted for
in the model is the nucleus, but if $K>1$, the model assumes that there are
subcentres other than the nucleus. The weights of these components are given
by the column vector $\mathbf{w}_{i}$, with $K$ entries that satisfy
$\sum_{j=1}^{K}w^{j}_{i}=1$. Each component has an associated probability
density function given by $p^{j}_{i}(L_{i}=h|\boldsymbol{\theta}_{i}^{j})$,
with parameters $\boldsymbol{\theta}_{i}^{j}$ and so,
$\boldsymbol{\theta}_{i}$ is a matrix with $K$ rows and as many columns as the
number of parameters that characterise the probability density function
$p^{j}_{i}$.

For the purposes of our analysis, we set $p^{j}_{i}$ to be a Poisson
distribution with parameter $\mu_{i}^{j}$, so that
$p^{j}_{i}(L_{i}=h|\boldsymbol{\theta}_{i}^{j})$ is now
$p^{j}_{i}(L_{i}=h|\mu_{i}^{j})=\frac{1}{h!}(\mu_{i}^{j})^{h}\exp(-\mu_{i}^{j})$,
for $i=1,...,N$ and $j=1,...,K$. This choice of distribution is motivated by
the fact that the length of journeys terminating at station $S_{i}$
represented by random variable $L_{i}$ is in the form of count data, but also
by the mathematical simplicity of the Poisson distribution, which is
characterised by only one parameter.

In order to find the maximum likelihood estimates of $\mathbf{w}_{i}$ and
$\boldsymbol{\theta}_{i}$ given the observed data for station $S_{i}$, we
apply the expectation-maximisation algorithm. The number of components $K$ for
the Poisson mixtures is one of the hyperparameters of the algorithm and needs
to be determined before the learning process. In section 3, we discuss
different choices for the number of components $K$.

##  3 Experiments and results

The case where $K=1$ is equivalent to a simple Poisson distribution. The
maximum likelihood estimator for the Poisson parameter $\mu_{i}$ corresponding
to station $S_{i}$ is the average of the $M_{i}$ observations for $L_{i}$.
Therefore, a visualisation of the relation between the estimated Poisson
parameter corresponding to a station and the distance between the nucleus and
the station is provided in Figure 2.

###  3.1 Poisson mixture model with two components

To account for the presence of centres other than the nucleus, we introduce
Poisson mixture models with $K=2$. The parameters of a Poisson mixture with
two components are the weights $\mathbf{w}_{i}=(w^{1}_{i},w^{2}_{i})$ and the
distribution parameters for each component
$\boldsymbol{\theta}_{i}=(\mu^{1}_{i},\mu^{2}_{i})$, which are also the mean
of each component. We can obtain the maximum likelihood estimators for these
parameters by applying the expectation-maximisation algorithm to data
corresponding to $L_{i}$ and we denote them by $\hat{w}^{1}_{i}$,
$\hat{w}^{2}_{i}$, $\hat{\mu}^{1}_{i}$ and $\hat{\mu}^{2}_{i}$. We refer to
the component with the lowest estimated mean as the proximal component. We
denote its associated weight by $\hat{w}^{p}_{i}$ and its mean by
$\hat{\mu}^{p}_{i}$, so that
$\hat{\mu}^{p}_{i}=min(\hat{\mu}^{1}_{i},\hat{\mu}^{2}_{i})$. Similarly, we
call the distal component the component whose estimated mean is higher and
denote its associated weight by $\hat{w}^{d}_{i}$ and its mean by
$\hat{\mu}^{d}_{i}$, so that
$\hat{\mu}^{d}_{i}=max(\hat{\mu}^{1}_{i},\hat{\mu}^{2}_{i})$. The 2-component
Poisson mixture model enables us to classify each of the $M_{i}$ individual
observations of $L_{i}$ as belonging to the proximal or distal components with
probability given by the respective estimated weights. Therefore, journeys
belonging to the proximal component are those that take place at a local scale
and the journeys belonging to the distal component take place at a global,
city-wide scale.

Figure 4 shows the relationship between the proximal and distal means
corresponding to a given station $S_{i}$ and the network distance $d_{1i}$
between $S_{i}$ and the nucleus $S_{1}$. Both the proximal and distal means,
$\hat{\mu}^{p}_{i}$ and $\hat{\mu}^{d}_{i}$, are represented in the
$y$-coordinate. The size of the data points is proportional to the number of
journeys terminating at each station. The results of the regression are
displayed in Table 2. In the Supplementary Information, we discuss the
relationship between $d_{1i}$ and the weights corresponding to station
$S_{i}$, $\hat{w}^{d}_{i}$, $\hat{w}^{p}_{i}$.

Figure 4: Relationship between the proximal and distal mean of the
distribution of journeys terminating at each station, $\hat{\mu}^{p}_{i}$ and
$\hat{\mu}^{d}_{i}$, and the distance $d_{1i}$ between the nucleus $S_{1}$ and
the destination station $S_{i}$.

As $d_{1i}$ becomes larger, there is no significant increase in the proximal
mean $\hat{\mu}^{p}_{i}$, since it remains around 5 and never above 10. The
effect is strikingly obvious in the case of Seoul. These observations are
likely to be the consequence of the existence of other socioeconomic centres,
closer to the destination station $S_{i}$, where passengers prefer to travel
to carry out some socioeconomic activities at a more local level. In contrast,
the distal component displays a significant linear growth with $d_{1i}$. The
distal component captures long-distance, city-wide journeys from stations that
are possibly close to the nucleus, to stations that are in the peripheral
regions of the city.

Table 2: Results of linear regressions considering $d_{1i}$ as the explanatory variable and $\hat{\mu}_{i}^{p}$, $\hat{\mu}_{i}^{d}$ from the 2-component Poisson mixture model as the response variables. Piccadilly Circus is considered to be the nucleus in London and City Hall in Seoul. | Intercept | Slope | $R$ |  $p$-value
---|---|---|---|---
London | Proximal | $4.295\pm 0.437$ | $0.180\pm 0.041$ | $0.228$ | $<0.05$
Distal | $10.348\pm 0.435$ | $0.427\pm 0.041$ | $0.488$ | $<0.05$
Seoul | Proximal | $5.018\pm 0.116$ | $0.013\pm 0.007$ | $0.086$ | $0.06$
Distal | $10.826\pm 0.191$ | $0.403\pm 0.012$ | $0.844$ | $<0.05$

###  3.2 Poisson mixture model with three components

Similarly, with $K=3$, the parameters to be estimated are
$\mathbf{w}_{i}=(w^{1}_{i},w^{2}_{i},w^{3}_{i})$ and
$\boldsymbol{\theta}_{i}=(\mu^{1}_{i},\mu^{2}_{i},\mu^{3}_{i})$. The proximal
and distal components are defined as for $K=2$. We call the third component
remaining the medial component, and we denote its weight by $w^{m}_{i}$ and
its distribution parameter by $\mu^{m}_{i}$. The results of the linear
regression between the mean of each component and $d_{1i}$ are gathered in
Table 3. Figure 5 represents the relationship between the proximal, medial and
distal means corresponding to each station $S_{i}$, i.e. $\mu^{p}_{i}$,
$\mu^{m}_{i}$ and $\mu^{d}_{i}$, and the distance between $S_{i}$ and the
nucleus. In the Supplementary Information, we discuss the relationship between
$d_{1i}$ and the weights corresponding to the three-component Poisson mixture
model for station $S_{i}$, i.e. $\hat{w}^{d}_{i}$, $\hat{w}^{d}_{i}$,
$\hat{w}^{p}_{i}$.

The behaviour of the proximal and distal components is analogous to the $K=2$
case. However, adding a third component allows to capture the variability in
the data with even more detail. Theoretically, Poisson mixture models have no
limitation for the maximum number of components to be added in their
formulation, however, increasing $K$ indefinitely is not always sensible since
it could lead to overfitting and hinder the interpretability of outcomes. For
this reason, here we recommend keeping $K$ to 2 or 3 as a good trade-off
between capturing the detail in the data variability whilst keeping the
components meaningful without overcomplicating the model.

Table 3: Results of linear regressions considering $d_{1i}$ as the explanatory variable and $\hat{\mu}_{i}^{p}$, $\hat{\mu}_{i}^{m}$, $\hat{\mu}_{i}^{d}$ from the 3-component Poisson mixture model as the response variables. Piccadilly Circus is considered to be the nucleus in London and City Hall in Seoul. | Intercept | Slope | $R$ |  $p$-value
---|---|---|---|---
London | Proximal | $4.070\pm 0.407$ | $0.079\pm 0.038$ | $0.109$ | $<0.05$
Medial | $5.982\pm 0.381$ | $0.494\pm 0.036$ | $0.593$ | $<0.05$
| Distal | $12.683\pm 0.487$ | $0.558\pm 0.046$ | $0.546$ | $<0.05$
Seoul | Proximal | $3.876\pm 0.107$ | $0.008\pm 0.007$ | $0.052$ | $0.252$
Medial | $9.108\pm 0.246$ | $0.175\pm 0.015$ | $0.468$ | $<0.05$
| Distal | $15.678\pm 0.315$ | $0.430\pm 0.091$ | $0.713$ | $<0.05$

Figure 5: Relationship between the proximal, medial and distal mean of the
distribution of journeys terminating at each station, $\hat{\mu}^{p}_{i}$,
$\hat{\mu}^{m}_{i}$ and $\hat{\mu}^{d}_{i}$, and the distance $d_{1i}$ between
the nucleus $S_{1}$ and the destination station $S_{i}$.

##  4 Discussion and conclusions

Our work constitutes a novel approach to the study of urban structure that
makes use of a probabilistic modelling framework based on Poisson mixture
models and new forms of data. Simply by analysing data related to the length
of the journeys taken on the public transport system corresponding to London,
United Kingdom, and Seoul, South Korea, the proposed probabilistic modelling
framework enables us to disaggregate the journeys into several statistical
subpopulations according to their destination station and their length,
measured on network distances between stations. The methodology relies on the
monocentric hypothesis, a null hypothesis stating that in a perfectly
monocentric city, there is only one centre where all the socioeconomic
activity is concentrated. This centre is set to be at the station with the
largest number persons travelling to it associated with the number of tap-
outs, which we call the nucleus. Consequently, in the monocentric hypothesis,
all the journeys terminating at any station other than the nucleus should have
the nucleus as their origin station. Characterising deviations from the
monocentric hypothesis allows us to infer the degree of polycentricity of a
city by detecting the potential presence of centres other than the nucleus.

The analysis of data from London and Seoul leads us to the following key
findings. Firstly, the distribution of the length of journeys terminating at
any destination station displays a higher degree of variability than predicted
by the monocentric hypothesis in both London and Seoul. Secondly, by modelling
the length of journeys terminating at each station in the public transport
system with a single-component Poisson distribution, we observe that,
especially in the case of Seoul, the most frequent journey length terminating
at a given station is shorter than the distance between the nucleus and the
station, perhaps due to the presence of closer, more local urban centres.
Thirdly, by introducing the 2-component Poisson mixture model, we are able to
classify each of the individual journeys to a station into what we call the
proximal and distal components. The proximal component corresponds to journeys
that take place at a local scale and the distal component involves journeys
that take place at a global, city-wide scale. We see that regardless of the
distance between the destination station and the nucleus, the most frequently
observed journey length associated with the proximal component is around 5, as
measured by network distance or number of stops away. These observations are
particularly clear in the case of Seoul and are likely to be the consequence
of the existence of other socioeconomic centres, closer to the destination
station, where passengers may prefer to travel to carry out some socioeconomic
activities at a more local level. Conversely, the most frequently observed
journey length associated with the distal component is larger than the
distance from the nucleus to the destination station for destination stations
that are close to the nucleus, showing that passengers who terminate their
journey at one of these stations may be travelling not only from the nucleus,
but also from other origin stations that are further away. As the distance
from the nucleus to the destination station increases, the most frequently
observed journey length associated with the distal component increases fast,
indicating that the distal component captures long-distance, city-wide
journeys from origin stations that are possibly close to the nucleus, to
stations that are in the peripheral regions of the city. Finally, increasing
the number of components in the Poisson mixture model can help unpick the
detail in the variability of the data; however, it can also make the model too
complex and result in overfitting. After testing for other choices of nucleus
(e.g. station with highest between-ness centrality in the transport network),
we find that the observed patterns described above still hold.

Understanding urban structure from data related to the transportation system
of the cities has significant implications for urban policy. The modelling
framework outlined in this paper provides a quantitative way of characterising
the degree of monocentricity of a city and we suggest that this approach may
be useful in the context of making decisions for spatial strategic planning.
In particular, the recent quest for centralizing activities in more compact
cities where the emphasis has been upon inner and city center living, could be
much informed by this approach where the difficulty of moving towards more
compact urban structures might be measured by the different parameters
associated with the Poisson mixture distributions. In this sense, these
distributions are not only able to reveal how the centricity of cities might
change under different travel regimes but also how travel behavior itself
might be altered.

Additionally, our approach is also of relevance to those interested in the
most theoretical, and even historical, aspects of urban areas and it prompts
questions for future research. As our findings indicate, London’s case study
aligns more with the scenario depicted by the monocentric hypothesis than
Seoul’s. The construction of London’s transport network started at the end of
the 19th century while the construction of Seoul’s started in 1971,
approximately a century later, and in the course of all those years, several
studies have reported a tendency for cities to become more and more
polycentric. Assuming that the layout of the transport network and the
passengers’ travelling behaviours are yet another manifestation of urban
structure, then the fact that our findings suggest that London is more
monocentric than Seoul, should not come as a surprise. But, does this
assumption hold in general? Has London’s early construction of a public
transport network conditioned its urban structure and slowed its transition
towards a more polycentric arrangement like Seoul’s? There is considerable
scope for extending this type of analysis to the evolution of past cities,
developing where possible ways in which mixture models like these can reveal
how spatial behaviors can and have altered over decades. Public transport data
will always be an issue for such historical analysis but these methods can
easily be extended to other trip distributions such as the journey to work on
different modes and very different time intervals where there is data
available in the UK for example, for the last 6 decades.

This study is limited by the fact that the data set is relatively small, since
it only contains data for two cities and for five weekdays in the case of
London, and four in the case of Seoul. Similarly, the data is limited to some
modes of transport in the whole public transport system of both cities. For
example, it excludes bus journeys, although tap-out data for bus journeys is
not always recorded. Finally, our work only explores one temporal scale, but a
deeper understanding of urban structure could be obtained by studying data at
different times of the day as well as time periods of different lengths, and
by analysing the temporal evolution of the data.

## References

  * [1] Anas, A.  Agglomeration and taste heterogeneity: Equilibria, stability, welfare and dynamics.  _Regional Science and Urban Economics_ 18, 7–35, DOI: https://doi.org/10.1016/0166-0462(88)90003-8 (1988).
  * [2] Rosenthal, S. S. & Strange, W. C.  The determinants of agglomeration.  _Journal of Urban Economics_ 50, 191–229, DOI: https://doi.org/10.1006/juec.2001.2230 (2001).
  * [3] Ahlfeldt, G. & Wendland, N.  How polycentric is a monocentric city? Centers, spillovers and hysteresis.  _Journal of Economic Geography_ 13, 53–83 (2013).
  * [4] Anas, A., Arnott, R. & Small, K.  The panexponential monocentric model.  _Journal of Urban Economics_ 47, 165–179 (2000).
  * [5] Huai, Y., Lo, H. K. & Ng, K. F.  Monocentric versus polycentric urban structure: Case study in Hong Kong.  _Transportation Research Part A: Policy and Practice_ 151, 99–118, DOI: https://doi.org/10.1016/j.tra.2021.05.004 (2021).
  * [6] Zhong, C., Arisona, S. M., Huang, X., Batty, M. & Schmitt, G.  Detecting the dynamics of urban structure through spatial network analysis.  _International Journal of Geographical Information Science_ 28, 2178–2199, DOI: 10.1080/13658816.2014.914521 (2014).  https://doi.org/10.1080/13658816.2014.914521.
  * [7] Wheaton, W. C.  _Monocentric models of urban land-use: Contributions and criticism_ (Baltimore, MD: John Hopkins Press, 1979).
  * [8] Griffith, D. A.  Modelling urban population density in a multi-centered city.  _Journal of Urban Economics_ 9, 298–310, DOI: https://doi.org/10.1016/0094-1190(81)90029-2 (1981).
  * [9] Berry, B. J. L. & Kim, H.-M.  Challenges to the monocentric model.  _Geographical Analysis_ 25, 1–4, DOI: https://doi.org/10.1111/j.1538-4632.1993.tb00275.x (1993).  https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1538-4632.1993.tb00275.x.
  * [10] Arribas-Bel, D. & Sanz-Gracia, F.  The validity of the monocentric city model in a polycentric age: US metropolitan areas in 1990, 2000 and 2010\.  _Urban Geography_ 35, 980–997, DOI: 10.1080/02723638.2014.940693 (2014).  https://doi.org/10.1080/02723638.2014.940693.
  * [11] Green, N.  Functional polycentricity: A formal definition in terms of social network analysis.  _Urban Studies_ 44, 2077–2103, DOI: 10.1080/00420980701518941 (2007).  https://doi.org/10.1080/00420980701518941.
  * [12] Brinkman, J. C.  Congestion, agglomeration, and the structure of cities.  _Journal of Urban Economics_ 94, 13–31, DOI: https://doi.org/10.1016/j.jue.2016.05.002 (2016).
  * [13] Davoudi, S.  EUROPEAN BRIEFING: Polycentricity in European spatial planning: from an analytical tool to a normative agenda.  _European Planning Studies_ 11, 979–999, DOI: 10.1080/0965431032000146169 (2003).  https://doi.org/10.1080/0965431032000146169.
  * [14] Kloosterman, R. C. & Musterd, S.  The polycentric urban region: Towards a research agenda.  _Urban Studies_ 38, 623–633, DOI: 10.1080/00420980120035259 (2001).  https://doi.org/10.1080/00420980120035259.
  * [15] Meijers, E.  Measuring polycentricity and its promises.  _European Planning Studies_ 16, 1313–1323, DOI: 10.1080/09654310802401805 (2008).  https://doi.org/10.1080/09654310802401805.
  * [16] Rauhut, D.  Polycentricity – one concept or many?  _European Planning Studies_ 25, 332–348, DOI: 10.1080/09654313.2016.1276157 (2017).  https://doi.org/10.1080/09654313.2016.1276157.
  * [17] Alidadi, M. & Dadashpoor, H.  Beyond monocentricity: examining the spatial distribution of employment in Tehran metropolitan region, Iran.  _International Journal of Urban Sciences_ 22, 38–58, DOI: 10.1080/12265934.2017.1329024 (2018).  https://doi.org/10.1080/12265934.2017.1329024.
  * [18] Li, Y.  Towards concentration and decentralization: The evolution of urban spatial structure of Chinese cities, 2001–2016.  _Computers, Environment and Urban Systems_ 80, 101425, DOI: https://doi.org/10.1016/j.compenvurbsys.2019.101425 (2020).
  * [19] Hajrasouliha, A. H. & Hamidi, S.  The typology of the American metropolis: monocentricity, polycentricity, or generalized dispersion?  _Urban Geography_ 38, 420–444, DOI: 10.1080/02723638.2016.1165386 (2017).  https://doi.org/10.1080/02723638.2016.1165386.
  * [20] Sweet, M. N., Bullivant, B. & Kanaroglou, P. S.  Are major Canadian city-regions monocentric, polycentric, or dispersed?  _Urban Geography_ 38, 445–471, DOI: 10.1080/02723638.2016.1200279 (2017).  https://doi.org/10.1080/02723638.2016.1200279.
  * [21] Tu, W. _et al._ Coupling mobile phone and social media data: a new approach to understanding urban functions and diurnal patterns.  _International Journal of Geographical Information Science_ 31, 2331–2358, DOI: 10.1080/13658816.2017.1356464 (2017).  https://doi.org/10.1080/13658816.2017.1356464.
  * [22] Yin, J., Soliman, A., Yin, D. & Wang, S.  Depicting urban boundaries from a mobility network of spatial interactions: a case study of Great Britain with geo-located Twitter data.  _International Journal of Geographical Information Science_ 31, 1293–1313, DOI: 10.1080/13658816.2017.1282615 (2017).  https://doi.org/10.1080/13658816.2017.1282615.
  * [23] Yang, X. _et al._ Understanding the spatial structure of urban commuting using mobile phone location data: A case study of Shenzhen, China.  _Sustainability_ 10 (2018).
  * [24] Zhu, W., Ma, D., Zhao, Z. & Guo, R.  Investigating the complexity of spatial interactions between different administrative units in china using Flickr data.  _Sustainability_ 12, DOI: 10.3390/su12229778 (2020).
  * [25] Xiao, Y., Wang, Y., Miao, S. & Niu, X.  Assessing polycentric urban development in Shanghai, China, with detailed passive mobile phone data.  _Environment and Planning B: Urban Analytics and City Science_ 48, 2656–2674, DOI: 10.1177/2399808320982306 (2021).  https://doi.org/10.1177/2399808320982306.
  * [26] Liu, X. _et al._ Analysis of urban agglomeration structure through spatial network and mobile phone data.  _Transactions in GIS_ 25, 1949–1969, DOI: https://doi.org/10.1111/tgis.12755 (2021).  https://onlinelibrary.wiley.com/doi/pdf/10.1111/tgis.12755.
  * [27] Ponce-Lopez, R. & Ferreira, J.  Identifying and characterizing popular non-work destinations by clustering cellphone and point-of-interest data.  _Cities_ 113, 103158, DOI: https://doi.org/10.1016/j.cities.2021.103158 (2021).
  * [28] Miao, R., Wang, Y. & Li, S.  Analyzing urban spatial patterns and functional zones using sina weibo poi data: A case study of Beijing.  _Sustainability_ 13, DOI: 10.3390/su13020647 (2021).
  * [29] Song, Z. _et al._ Building-level urban functional area identification based on multi-attribute aggregated data from cell phones \- a method combining multidimensional time series with a som neural network.  _ISPRS International Journal of Geo-Information_ 11, DOI: 10.3390/ijgi11020072 (2022).
  * [30] Zhou, Y., Fang, Z., Thill, J.-C., Li, Q. & Li, Y.  Functionally critical locations in an urban transportation network: Identification and space–time analysis using taxi trajectories.  _Computers, Environment and Urban Systems_ 52, 34–47, DOI: https://doi.org/10.1016/j.compenvurbsys.2015.03.001 (2015).
  * [31] Nie, W.-P., Zhao, Z.-D., Cai, S.-M. & Zhou, T.  Understanding the urban mobility community by taxi travel trajectory.  _Communications in Nonlinear Science and Numerical Simulation_ 101, 105863, DOI: https://doi.org/10.1016/j.cnsns.2021.105863 (2021).
  * [32] Li, X., Ma, X. & Wilson, B.  Beyond absolute space: An exploration of relative and relational space in Shanghai using taxi trajectory data.  _Journal of Transport Geography_ 93, 103076, DOI: https://doi.org/10.1016/j.jtrangeo.2021.103076 (2021).
  * [33] Choi, J., No, W., Park, M. & Kim, Y.  Inferring land use from spatialtemporal taxi ride data.  _Applied Geography_ 142, 102688, DOI: https://doi.org/10.1016/j.apgeog.2022.102688 (2022).
  * [34] Roth, C., Kang, S. M., Batty, M. & Barthélemy, M.  Structure of urban movements: Polycentric activity and entangled hierarchical flows.  _PLOS ONE_ 6, 1–8, DOI: 10.1371/journal.pone.0015923 (2011).
  * [35] Maeda, T. N., Mori, J., Hayashi, I., Sakimoto, T. & Sakata, I.  Comparative examination of network clustering methods for extracting community structures of a city from public transportation smart card data.  _IEEE Access_ 7, 53377–53391, DOI: 10.1109/ACCESS.2019.2911567 (2019).
  * [36] Yang, Y., Heppenstall, A., Turner, A. & Comber, A.  A spatiotemporal and graph-based analysis of dockless bike sharing patterns to understand urban flows over the last mile.  _Computers, Environment and Urban Systems_ 77, 101361, DOI: https://doi.org/10.1016/j.compenvurbsys.2019.101361 (2019).
  * [37] Tang, L., Zhao, Y., Tsui, K. L., He, Y. & Pan, L.  A clustering refinement approach for revealing urban spatial structure from smart card data.  _Applied Sciences_ 10, DOI: 10.3390/app10165606 (2020).
  * [38] Aslam, N. S., Zhu, D., Cheng, T., Ibrahim, M. R. & Zhang, Y.  Semantic enrichment of secondary activities using smart card data and point of interests: a case study in London.  _Annals of GIS_ 27, 29–41, DOI: 10.1080/19475683.2020.1783359 (2021).  https://doi.org/10.1080/19475683.2020.1783359.
  * [39] Zhang, Y., Marshall, S., Cao, M., Manley, E. & Chen, H.  Discovering the evolution of urban structure using smart card data: The case of London.  _Cities_ 112, 103157, DOI: https://doi.org/10.1016/j.cities.2021.103157 (2021).

## Acknowledgements

We are very grateful to Prof HaeRan Shin from Seoul National University for
kindly helping us get access to T-money data. CZ, MB and RS are Turing Fellows
at the Alan Turing Institute. The data analysis was done within UCL.

## Competing interests

The authors declare that they have no competing interests.

## Author’s contributions

CCA: conceived the study, designed the methodology, carried out the data
analysis, contributed towards the interpretation of results, drafted the
manuscript; CZ: conceived the study, provided the London data, contributed
towards the interpretation of results; MB: contributed towards the
interpretation of results, participated in drafting and revising the
manuscript; RS: conceived the study, designed the methodology, contributed
towards the interpretation of results; SMK: conceived the study, provided the
Seoul data, contributed towards the interpretation of results. All authors
gave their final approval for publication.

## Data and code availability

The data and the code used in the methodology of this paper can be found on
GitHub through this link.

## Funding

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 949670), and from ESRC under JPI Urban Europe/NSFC (grant
No. ES/T000287/1).


# Size-Sieving Separation of Hard-Sphere Mixtures through Cylindrical Pores

Yue Yu  Division of Natural and Applied Sciences, Duke Kunshan University,
Kunshan, Jiangsu, 215300, China  Kai Zhang<EMAIL_ADDRESS>Division of Natural and Applied Sciences, Duke Kunshan University, Kunshan,
Jiangsu, 215300, China  Data Science Research Center (DSRC), Duke Kunshan
University, Kunshan, Jiangsu, 215300, China

###### Abstract

The collision dynamics of hard spheres and cylindrical pores is solved
exactly, which is the minimal model for a regularly porous membrane.
Nonequilibrium event-driven molecular dynamics simulations are used to show
that the permeability $P$ of hard spheres of size $\sigma$ through
cylinderical pores of size $d$ follow the hindered diffusion mechanism due to
size exclusion as $P\propto(1-\sigma/d)^{2}$. Under this law, the separation
of binary mixtures of large and small particles exhibits a linear relationship
between $\alpha^{-1/2}$ and $P^{-1/2}$, where $\alpha$ and $P$ are the
selectivity and permeability of the smaller particle, respectively. The mean
permeability through polydisperse pores is the sum of permeabilities of
individual pores, weighted by the fraction of the single pore area over the
total pore area.

###### pacs:

47.61.-k, 47.45.-n, 47.55.Mh, 47.56.+r, 83.10.Rs, 02.70.Ns

Membrane separation is an energy-efficient way to extract some substances from
others with a multitude of industrial applications such as water desalination,
ion exchange, carbon capture and protein purification Baker (2012). The goal
here is to maximize the acquisition rate of the desired species at the highest
possible purity by effectively filtering out undesired ones. However, an
intrinsic trade-off always exists between this pair of separation performance
characteristics – permeability and selectivity Park  _et al._ (2017). Although
different penetrants, being angstrom-sized (Å) ions/molecules, nanometer-sized
($n$m) macromolecules, or micrometer-sized ($\mu$m) bacteria, may have
different transport mechanisms, it is now generally believed that narrowing
down the size distribution of membrane pores can achieve higher separation
performance Moon  _et al._ (2020); Culp  _et al._ (2021). Apart from widely-
used microfiltration and ultrafiltration membranes Mehta and Zydney (2005),
tremendous efforts have been made to prepare regularly porous stuctures on
nano- or molecular scales in systems such as zeolites Smit and Maesen (2008),
metal organic frameworks Qiu  _et al._ (2014) (MOFs), and silicon nanochannels
Gruener and Huber (2008). Recent advances in self-assembly techniques further
enabled the synthesis of ceramic Kresge  _et al._ (1992), graphene-based Li
_et al._ (2013) or block copolymer Phillip  _et al._ (2006) molecular sieves,
which are featured with well-controlled pores as regular as parallel
cylinders. It is therefore needed to develop a quantitative understanding of
how the steric (size) exclusion effect imposed by regular pores determines the
separation of particles of different sizes.

In the early study of gas transport through long cylindrical tubes Knudsen
(1909), it was found that the bulk self diffusivity
$D_{s}=\frac{1}{3}\lambda\bar{v}$ of a gas with molecular mass $m$, mean free
path $\lambda$ and mean velocity $\bar{v}=\sqrt{\frac{8k_{B}T}{\pi m}}$ at
temperature $T$ becomes $D_{K}=\frac{1}{3}d\bar{v}$ inside the tube, when the
tube diameter $d\ll\lambda$. An implicit assumption of this Knudsen diffusion
mechanism is that the diameter of gas molecules $\sigma$ is much smaller than
$d$ so that they can be treated as point masses. If two gases were to be
separated under Knudsen diffusion, it is the mass dependence of
$D_{K}\sim\frac{1}{\sqrt{m}}$ that makes the heavier one exit the tube later,
regardless of their sizes. More theoretical analysis showed that the transport
diffusivity $D$ of rarefied gases flowing under a density gradient also
depends on the length of the tube $L$ through the ratio $L/d$ and $D$
approaches $D_{K}$ only when $L\gg d$ Pollard and Present (1948); Shi  _et
al._ (2012); Colson and Barlow (2019). But their conclusions are again limited
to the case of $\sigma\to 0$. In the context of molecular sieves, we are more
interested in the regime of $\sigma\lesssim d$ where size exclusion takes
effect. Although modifications to the Knudsen diffusion theory have been
proposed to replace $d$ with an effective diameter $d_{\rm eff}=d-\sigma$
expecting that $D_{K}\propto d-\sigma$ Gilron and Soffer (2002), it is not
clear whether this rule actually holds and whether the transport still remains
Knudsen-like.

Figure 1: (a) Simulation box setup of DCV-GCMD. A particle at position ${\bf
r}$ in the waiting zone (shaded) makes a solid angle $\Omega$ to the pore
opening. (b) Side and (c) top view of the three possible cases of particle-
pore collision events.

In this Letter, we study the transport and separation of low-density binary
hard-sphere mixtures through cylindrical pores under a chemical potential
gradient (Fig. 1). Each mixture consists of hard spheres $A$ (large) and $B$
(small) flowing from the upstream chamber $1$ at chemical potential $\mu_{1}$
to the downstream chamber $2$ at a lower chemical potential $\mu_{2}$,
implemented by the dual control volume grand canonical molecular dynamics
(DCV-GCMD) method Heffelfinger and Ford (1998); FORD and HEFFELFINGER (1998).
Previous molecular simulations on similar problems have mostly focused on
Lennard-Jones like particles under slit-shaped confinement Cracknell  _et al._
(1995); Furukawa  _et al._ (1996); Furukawa and Nitta (1997); Furukawa  _et
al._ (1997); Xu  _et al._ (1998, 2000); Takaba  _et al._ (1998), and in
consequence, were not able to distinguish size sieving from other effects.
Other methods resort to equilibrium and/or stochastic sampling Mon and Percus
(2002); Malek and Coppens (2003) that cannot genuinely reflect the
nonequilibrium transport dynamics driven by pressure gradients. To understand
size-sieving filtration through regularly porous membranes, it is imperative
to faithfully simulate the entrance of particles into pores. We analytically
resolve the collision dynamics of hard spheres and cylindrical pores by
considering three possibilities (Fig. 1(b-c)). A hard sphere can hit the
membrane surface and bounce back (case 1), or hit the circular edge of the
pore and then bounce away (case 2), or directly fly into the pore and hit its
interior wall (case 3). We can compute the particle-pore collision time by
considering the excluded volume of a sphere over the pore surface (red shaded
area in Fig. 1(b)). The final solution is about finding the intersection
between a linear path and a torus as in the problem of ray tracing in computer
graphics Lengyel (2012); sup .

We set the diameter $\sigma_{A}$ of $A$ particles as the unit of length. The
simulation box has dimensions of $Z=60$ and $X=Y=42$ with membrane thickness
$L=20$ (Fig. 1(a)). Periodic boundary conditions are imposed along all
directions such that particles can fly from chamber $1$ to $2$ through the two
membranes in the box with opposite streaming velocities $u$ and $-u$. We first
pack monodisperse cylindrical pores in parallel on a square lattice with
diameters ranging from $d=1.05$ to $3.5$. The number of pores on the membrane
are adjusted to keep the total pore area $S$ (or porosity) constant. The
diameter of the smaller particle $B$ spans over a wide range from
$\sigma_{B}=0.95$ down to $0.005$. We use $\mu_{1}=-5$ and
$\mu_{2}=-5.5,-\infty({\rm particle~{}sink})$, which maintain a number density
$\rho_{1}\lesssim 0.007$ and $\rho_{2}\lesssim 0.004$ for each of the two
components. The systems under study are thus gaseous for which ideal gas law
about pressure holds, i.e. $p=\rho k_{B}T$. Estimation to the mean free path
$\lambda=\frac{k_{B}T}{\sqrt{2}\pi\sigma^{2}p}$ and Knudsen number ${\rm
K_{n}}=\lambda/d$ reveals that our systems are within or close to the free
molecular flow regime.

Figure 2: Permeability of hard spheres of size $\sigma$ (indicated by
different symbols) through pores of diameter $d=1.05-3.5$, following the
hindered diffusion law $P=P_{0}(1-\frac{\sigma}{d})^{2}$ (black dashed). (a)
Exponential law $P=P_{0}\exp(-2\sigma/d)$ is approximately obeyed for small
$\sigma/d$ (red dotted-dashed). Renkin’s law does not apply here (green
solid). (b) $P$ shows apparent linearity for small $\frac{d}{\sigma}-1$ but
saturates at $P_{0}$ as $\sigma/d\to 0$. Vertical dotted lines mark the
boundaries at ${\rm K_{n}}\sigma^{3}=0.1,1.0,10$.

The particle flux $J$ (the number of particles going through a membrane per
unit pore area per unit time) is calculated from the difference between the
net insertion/removal numbers in the two chambers Heffelfinger and Ford (1998)
or from the number of particle crossing events in each pore. Instead of using
the transport diffusivity $D$ defined by the Fick’s law
$J=-D\frac{\Delta\rho}{L}$, we report the transport permeability $P$ defined
by $J=-P\frac{\Delta p}{L}=-P\nabla p$ and at the low densities of this work
$D\approx Pk_{B}T$. We find that, for all the particle sizes $\sigma$ and pore
sizes $d$ under study, $P(d,\sigma)$ collapse onto a universal curve when
plotted as a function of the ratio $\sigma/d$ (Fig. 2). It can be numerically
verified that the hindered diffusion law Beck and Schultz (1970)

$\displaystyle P(d,\sigma)=P_{0}\left(1-\frac{\sigma}{d}\right)^{2}$ (1)

is obeyed with a fitting coefficient $P_{0}\approx 8$. The geometric factor
$\left(1-\frac{\sigma}{d}\right)^{2}$ simply originates from the accessible
area $\pi(d-\sigma)^{2}/4$ for a particle to enter the circular opening.
Usually, hindered diffusion describes the transport of large particles in
liquid solvent Deen (1987), and due to hydrodynamic effect, an extra factor
needs to be included as in the Renkin’s law
$P=P_{0}(1-\frac{\sigma}{d})^{2}\left(1-2.104\frac{\sigma}{d}+2.09\frac{\sigma^{3}}{d^{3}}-0.95\frac{\sigma^{5}}{d^{5}}\right)$
Renkin (1954). For solvent-free gaseous particles, the geometric factor alone
can determine the permeability (Fig. 2(a)). When viewed as a function of
$d-\sigma$ or $\frac{d}{\sigma}-1$, $P$ shows apparent linearity when $\sigma$
is comparable with $d$. But such linearity is just the numerical consequence
of the hindered diffusion law, i.e.
$(1-\sigma/d)^{2}=\left[1-\frac{1}{1+(d/\sigma-1)}\right]^{2}=\frac{(d/\sigma-1)^{2}}{\left[1+(d/\sigma-1)\right]^{2}}$.
Around $\frac{d}{\sigma}-1\to 0$, the trend of $P$ is actually curved as
$\sim(d/\sigma-1)^{2}$. In the limit of $\sigma\ll d$, instead of having a
Knudsen-like diffusion with $P\sim d$, we observe that $P$ saturates at a
constant $P_{0}$ (Fig. 2(b)).

The front factor $P_{0}$ in our result can be understood by considering the
net flux of point particles through the circular opening at each pore. On
average, half of the particles move from chamber 1 to 2 with a streaming
velocity $u$ and it takes $\tau=L/u$ for all the $N$ particles inside a pore
to be emitted to chamber 2. Meanwhile, there must be another $N$ particles
entering that pore from chamber 1 at steady state, which are originally in the
“waiting” zone of thickness $L$ next to the pore opening (red shaded area in
Fig. 1(a)). We can choose the waiting zone to be a cylinder of volume $V$ with
a height $L$ and a radius $\infty$. The probability that a particle at a given
point ${\bf r}$ in this region to enter the pore is determined by the solid
angle $\Omega({\bf r})$ made from ${\bf r}$ to the circular pore opening of an
area $s=\pi d^{2}/4$. If we assume $\rho_{1}=\rho$ and $\rho_{2}=0$ for
simplicity, then $N=\rho\int\limits_{V}{\rm d}{\bf r}\frac{\Omega({\bf
r})}{4\pi}=\frac{\rho}{4\pi}I(d,L)$, where the integral
$I(d,L)=\int\limits_{0}^{\infty}r{\rm d}r\int\limits_{0}^{2\pi}{\rm
d}\theta\int\limits_{0}^{L}{\rm d}z\Omega(r,\theta,z)$ can be evaluated
numerically for given $d$ and $L$ Paxton (1959). The permeability or
diffusivity for point particles can thus be estimated as
$P_{0}k_{B}T=D_{0}=\frac{N}{s\tau}\left/\frac{\Delta\rho}{L}\right.=\frac{Iu}{\pi^{2}d^{2}}$.
For $L=20$, we find $\frac{I}{\pi^{2}d^{2}}=10.0$ and if we make the
approximation that $u\approx\langle|v_{z}|\rangle=\sqrt{\frac{2k_{B}T}{\pi
m}}\approx 0.798$, $P_{0}\approx 7.98$ which is close to the result of
numerical fitting in Eq. (1) .

Having confirmed the hindered diffusion mechanism, we can predict the
separation curves for binary gaseous mixtures through cylindrical pores as

$\displaystyle\alpha=\frac{1}{\left[\frac{\sigma_{A}}{\sigma_{B}}-\sqrt{P_{0}}\left(\frac{\sigma_{A}}{\sigma_{B}}-1\right)P_{B}^{-1/2}\right]^{2}}$
(2)

where $\alpha\equiv\frac{P_{B}}{P_{A}}>1$ is the (ideal) selectivity of the
more permeable gas $B$ (smaller) with respect to the less permeable one $A$
(Fig. 3). In industry, majority of the gas separation membranes are rubbery or
glassy polymers, in which gases transport with an activated diffusion
mechanism $D\sim\exp(-\frac{a\sigma^{2}}{k_{B}T})$ Ghosal and Freeman (1994);
Gusev  _et al._ (1993). As a result, the “upper bound” of selectivity versus
permeability empirically satisfies the linear relationship
$\log\alpha=-\lambda_{\rm AB}\log P_{B}+\kappa$, whose slope was shown to be
$\lambda_{\rm AB}=\lambda_{\rm
F}\equiv\left(\frac{\sigma_{A}}{\sigma_{B}}\right)^{2}-1$ Robeson (1991);
Freeman (1999). Although there is no strict linear relationship on log-log
scale from Eq. (2), it can be shown that, for small $\sigma/d$,
$\log\alpha\approx-\lambda_{\rm AB}\log P_{B}+\lambda_{\rm AB}\log P_{0}$ with
$\lambda_{\rm AB}=\frac{\sigma_{A}}{\sigma_{B}}-1$ because $P\approx
P_{0}\exp(-2\sigma/d)$. The true linear relationship over the entire regime of
$\sigma/d$ is between $\alpha^{-1/2}$ and $P_{B}^{-1/2}$, i.e.

$\displaystyle\alpha^{-1/2}=\frac{\sigma_{A}}{\sigma_{B}}-\sqrt{P_{0}}\left(\frac{\sigma_{A}}{\sigma_{B}}-1\right)P_{B}^{-1/2}.$
(3)

Figure 3: Selectivity $\alpha$ versus permeability $P_{B}$ for binary hard
spheres of various size ratios $\sigma_{B}/\sigma_{A}$. Inset shows the linear
relationship between $\alpha^{-1/2}$ and $P_{B}^{-1/2}$.

Pore sizes in real membranes are inevitably polydisperse. We consider
variation of $d$ subject to the log-normal distribution $f(d;\bar{b},\delta)$
with mean $\bar{d}$ and standard deviation $\delta\bar{d}$ ($\delta$ is called
polydispersity)

$\displaystyle
f(d;\bar{d},\delta)=\frac{1}{d\sqrt{2\pi\ln(1+\delta^{2})}}e^{-\frac{\left[\ln\left(d\sqrt{1+\delta^{2}}\left/\overline{d}\right.\right)\right]^{2}}{2\ln(1+\delta^{2})}}.$

At a pressure gradient $\nabla p$, the total number $N$ of particles going
through all pores during time $\tau$ should equal to the sum of number $N_{i}$
of particles going through each pore $i$ with an area $s_{i}$, i.e.
$N=\overline{P}\nabla pS\tau=\sum\limits_{i}P_{i}\nabla
ps_{i}\tau=\sum\limits_{i}N_{i}$. If we assume the the permeability for each
pore of size $d_{i}$ is $P_{i}=P_{0}(1-\sigma/d_{i})^{2}$, then $N_{i}\propto
P_{i}s_{i}=(1-\sigma/d_{i})^{2}d_{i}^{2}=(d_{i}-\sigma)^{2}$. Simulations
using polydisperse membranes with $\delta=0.2$ and $\bar{d}=1.05$–$3.5$ show
that $N_{i}$ is indeed proportional to $(d_{i}-\sigma)^{2}$ (Fig. 4(a)).

Figure 4: (a) Number $N$ of particles (of size $\sigma_{A}=1.0$ and
$\sigma_{B}=0.8,0.5$) going through pores of different diameter $d$ during
simulations using polydisperse membranes. Horizontal line-bars enclose the
range of $d$’s for a given $N$ and $\bar{d}$ is their average. $N$ is
proportional to $(\bar{d}-\sigma)^{2}$ (black dashed line which is a guide to
the eye). (b) The average permeability $\bar{P}_{\delta}$ of the three types
of particles going through cylinderical pores with polydispersity $\delta$.
Solid line is calculated using the ansatz that the contribution from a pore of
size $d$ should be weighted by $d^{2}$. (c) Calculated permeability
enhancement as a function of selectivity and (d) calculated selectivity
enhancement as a function of the mean pore size $\bar{d}$ of the polydisperse
membrane, to separate binary mixtures of $\sigma_{A}=1$ and
$\sigma_{B}=0.8,0.5$.

The mean permeability
$\overline{P}=\sum\limits_{i}P_{i}\frac{s_{i}}{S}=\sum\limits_{i}P_{i}\frac{d_{i}^{2}}{\sum\limits_{i}d_{i}^{2}}$,
for a continuous distribution $f(d;\bar{d},\delta)$ of $d$, can thus be
calculated as

$\displaystyle\overline{P}_{\delta}(\bar{d},\sigma)=\frac{\int_{\sigma}^{\infty}f(d;\bar{d},\delta)P(d,\sigma)d^{2}{\rm
d}d}{\int_{\sigma}^{\infty}f(d;\bar{d},\delta)d^{2}{\rm d}d},$ (4)

after knowing the monodisperse permeability $P(d,\sigma)$ as a function of $d$
and $\sigma$ (Fig. 4(b)). Here, the weight in the average calculation is
$d^{2}$, which differs from $d^{4}$ in viscous (Hagen-Poiseuille) flows Mehta
and Zydney (2005). Eq. (4) can be used to predict permeability enhancement
$\overline{P}_{0.0}/\overline{P}_{\delta}$ at a given selectivity or
selectivity enhancement $\alpha_{0.0}/\alpha_{\delta}$ at a given mean pore
size $\bar{d}$, when we narrow down the pore size distribution from $\delta>0$
to $0.0$ (monodisperse) as shown in Fig. 4(c)-(d).

The interests in transport and separation are not limited to free molecular
flows (${\rm K_{n}}>10^{1}$), but have also been directed to continuum (${\rm
K_{n}}<10^{-3}$), slip ($10^{-3}<{\rm K_{n}}<10^{-1}$) and transition flows
($10^{-1}<{\rm K_{n}}<10^{1}$) Welty  _et al._ (2015). Decades ago, it was
thought that molecular simulations are computationally too expensive to
address these regimes when the number of particles is large. Approximate
theoretical models or computational methods were thus developed to tackle
these problems, such as the dusty-gas model (DGM) Mason  _et al._ (1967), the
direct simulation Monte Carlo (DSMC) Bird (1963); Alexander and Garcia (1997)
and the finite element method Roy  _et al._ (2003). With the ever-increasing
computer power, it now becomes more and more promising to investigate dense
flows on particle level Plimpton (1995); Sheng  _et al._ (2020). It would be
interesting to extend the current methodolgy to low-${\rm Kn}$ porous systems.
Caution should be taken when applying the model used in this work to molecular
systems, because the intrinsic roughness of atomic packings, which can induce
back reflection and dissipation, are not captured by smooth cylinderical walls
Malek and Coppens (2003); Arya  _et al._ (2003).

###### Acknowledgements.

We thank M. Ronen Plesser for helpful discussions on solid angle calculation.
We acknowledge support from Duke Kunshan startup funding. This work also
benefited from resources made available at the Duke Compute Cluster (DCC) and
the Kunshan Supercomputing Center (KSSC).

## References

  * Baker (2012) R. W. Baker,  _Membrane Technology and Applications_ (John Wiley & Sons, 2012).
  * Park  _et al._ (2017) H. B. Park, J. Kamcev, L. M. Robeson, M. Elimelech, and B. D. Freeman, Science 356 (2017).
  * Moon  _et al._ (2020) J. D. Moon, B. D. Freeman, C. J. Hawker, and R. A. Segalman, Macromolecules 53, 5649 (2020).
  * Culp  _et al._ (2021) T. E. Culp, B. Khara, K. P. Brickey, M. Geitner, T. J. Zimudzi, J. D. Wilbur, S. D. Jons, A. Roy, M. Paul, B. Ganapathysubramanian, _et al._ , Science 371, 72 (2021).
  * Mehta and Zydney (2005) A. Mehta and A. L. Zydney, J. Membr. Sci. 249, 245 (2005).
  * Smit and Maesen (2008) B. Smit and T. L. Maesen, Chem. Rev. 108, 4125 (2008).
  * Qiu  _et al._ (2014) S. Qiu, M. Xue, and G. Zhu, Chem. Soc. Rev. 43, 6116 (2014).
  * Gruener and Huber (2008) S. Gruener and P. Huber, Phys. Rev. Lett. 100, 064502 (2008).
  * Kresge  _et al._ (1992) C. Kresge, M. Leonowicz, W. J. Roth, J. Vartuli, and J. Beck, Nature 359, 710 (1992).
  * Li  _et al._ (2013) H. Li, Z. Song, X. Zhang, Y. Huang, S. Li, Y. Mao, H. J. Ploehn, Y. Bao, and M. Yu, Science 342, 95 (2013).
  * Phillip  _et al._ (2006) W. A. Phillip, J. Rzayev, M. A. Hillmyer, and E. Cussler, J. Membr. Sci. 286, 144 (2006).
  * Knudsen (1909) M. Knudsen, Annalen der Physik 333, 75 (1909).
  * Pollard and Present (1948) W. Pollard and R. D. Present, Phys. Rev. 73, 762 (1948).
  * Shi  _et al._ (2012) Y. Shi, Y. T. Lee, and A. S. Kim, Transport in Porous Media 93, 517 (2012).
  * Colson and Barlow (2019) F. Colson and D. A. Barlow, Phys. Rev. E 100, 062125 (2019).
  * Gilron and Soffer (2002) J. Gilron and A. Soffer, J. Membr. Sci. 209, 339 (2002).
  * Heffelfinger and Ford (1998) G. S. Heffelfinger and D. M. Ford, Mol. Phys. 94, 659 (1998).
  * FORD and HEFFELFINGER (1998) D. M. FORD and G. S. HEFFELFINGER, Mol. Phys. 94, 673 (1998).
  * Cracknell  _et al._ (1995) R. F. Cracknell, D. Nicholson, and N. Quirke, Phys. Rev. Lett. 74, 2463 (1995).
  * Furukawa  _et al._ (1996) S.-i. Furukawa, T. Shigeta, and T. Nitta, J. Chem. Eng. Japan 29, 725 (1996).
  * Furukawa and Nitta (1997) S.-i. Furukawa and T. Nitta, J. Chem. Eng. Japan 30, 116 (1997).
  * Furukawa  _et al._ (1997) S.-i. Furukawa, K. Hayashi, and T. Nitta, J. Chem. Eng. Japan 30, 1107 (1997).
  * Xu  _et al._ (1998) L. Xu, M. G. Sedigh, M. Sahimi, and T. T. Tsotsis, Phys. Rev. Lett. 80, 3511 (1998).
  * Xu  _et al._ (2000) L. Xu, M. Sahimi, and T. T. Tsotsis, Phys. Rev. E 62, 6942 (2000).
  * Takaba  _et al._ (1998) H. Takaba, K. Mizukami, M. Kubo, A. Fahmi, and A. Miyamoto, AIChE Journal 44, 1335 (1998).
  * Mon and Percus (2002) K. Mon and J. K. Percus, J. Chem. Phys. 117, 2289 (2002).
  * Malek and Coppens (2003) K. Malek and M.-O. Coppens, J. Chem. Phys. 119, 2801 (2003).
  * Lengyel (2012) E. Lengyel,  _Mathematics for 3D game programming and computer graphics_ (Cengage Learning PTR, Boston, 2012).
  * (29) See Supplemental Material for simulation details.
  * Beck and Schultz (1970) R. E. Beck and J. S. Schultz, Science 170, 1302 (1970).
  * Deen (1987) W. Deen, AIChE Journal 33, 1409 (1987).
  * Renkin (1954) E. M. Renkin, J. Gen. Physiol. 38, 225 (1954).
  * Paxton (1959) F. Paxton, Rev. Sci. Instrum. 30, 254 (1959).
  * Ghosal and Freeman (1994) K. Ghosal and B. D. Freeman, Polym. Adv. Technol. 5, 673 (1994).
  * Gusev  _et al._ (1993) A. A. Gusev, S. Arizzi, U. W. Suter, and D. J. Moll, J. Chem. Phys. 99, 2221 (1993).
  * Robeson (1991) L. M. Robeson, J. Membr. Sci. 62, 165 (1991).
  * Freeman (1999) B. D. Freeman, Macromolecules 32, 375 (1999).
  * Welty  _et al._ (2015) J. Welty, G. L. Rorrer, and D. G. Foster,  _Fundamentals of momentum, heat, and mass transfer_ (John Wiley & Sons, 2015).
  * Mason  _et al._ (1967) E. Mason, A. Malinauskas, and R. Evans Iii, J. Chem. Phys. 46, 3199 (1967).
  * Bird (1963) G. Bird, Phys. Fluids 6, 1518 (1963).
  * Alexander and Garcia (1997) F. J. Alexander and A. L. Garcia, Comput. Phys. 11, 588 (1997).
  * Roy  _et al._ (2003) S. Roy, R. Raju, H. F. Chuang, B. A. Cruden, and M. Meyyappan, J. Appl. Phys. 93, 4870 (2003).
  * Plimpton (1995) S. Plimpton, J. Comput. Phys. 117, 1 (1995).
  * Sheng  _et al._ (2020) Q. Sheng, L. Gibelli, J. Li, M. K. Borg, and Y. Zhang, Phys. Fluids 32, 092003 (2020).
  * Arya  _et al._ (2003) G. Arya, H.-C. Chang, and E. J. Maginn, Phys. Rev. Lett. 91, 026102 (2003).


# The Square Kilometre Array

Naval Research Laboratory and Square Kilometre Array Program Development
Office, 4555 Overlook Ave. SW, Washington, DC 20375-5351 USA
E-mail Basic research in radio astronomy at the NRL is supported by 6.1 Base
funding.

###### Abstract:

The Square Kilometre Array (SKA) is intended as the next-generation radio
telescope and will address fundamental questions in astrophysics, physics, and
astrobiology. The international science community has developed a set of Key
Science Programs: (1) Emerging from the Dark Ages and the Epoch of
Reionization, (2) Galaxy Evolution, Cosmology, and Dark Energy, (3) The Origin
and Evolution of Cosmic Magnetism, (4) Strong Field Tests of Gravity Using
Pulsars and Black Holes, and (5) The Cradle of Life/Astrobiology. In addition,
there is a design philosophy of “exploration of the unknown,” in which the
objective is to keep the design as flexible as possible to allow for future
discoveries. Both a significant challenge and opportunity for the SKA is to
obtain a significantly wider field of view than has been obtained with radio
telescopes traditionally. Given the breadth of coverage of cosmic magnetism
and galaxy evolution in this conference, I highlight some of the opportunities
that an expanded field of view will present for other Key Science Programs.

##  1 Introduction

In the $20^{\mathrm{th}}$ Century, we discovered our place in the Universe. We
learned that it was much bigger than we imagined and much more exotic. Beyond
our Milky Way, the Universe is filled with galaxies—each their own island
universe. They range in size from dwarf galaxies barely able to survive near
their larger neighbors to giant elliptical galaxies, orders of magnitudes
larger than the Milky Way. These galaxies of stars also contain a multitude of
other components including gas with a wide range of temperatures; compact
objects including white dwarfs, neutron stars, and black holes; and planets.
Over the course of the century, black holes moved from a theoretical curiosity
to a well-recognized endpoint of stellar evolution and a likely fundamental
component of the centers of galaxies, with the potential to power immense jets
of relativistic particles that affect their surroundings. By the end of the
century, we were beginning to unveil the basic structure and processes of the
Universe in which these objects are embedded, including evidence of its origin
and the still mysterious properties of dark matter and dark energy.

Our probes of the Universe have expanded dramatically as well. Electromagnetic
radiation has been detected from celestial objects at frequencies below 1 MHz
($\lambda\sim 300$ m) to energies exceeding 1 TeV. Moreover, the range of
possible signals has expanded beyond just electromagnetic radiation. Cosmic
rays rain down on the Earth, some with energies approaching those of
macroscopic objects. Gravitational radiation has been detected indirectly, and
numerous potential classes of sources have been suggested, with the
expectation that the Earth is awash in gravitational waves. Neutrinos have
been detected from both the Sun and supernova 1987A, and many of the processes
that generate high energy cosmic rays should also produce a spectrum of high-
energy neutrinos.

In the $21^{\mathrm{st}}$ Century, we seek to understand the Universe we
inhabit. To do so will require a suite of powerful new instruments, on the
ground and in space, operating across the entire electromagnetic spectrum and
for multiple decades. Observations at centimeter- to meter wavelengths have
provided deep insight to a wide range of phenomena ranging from the solar
system to the most distant observable celestial emission. This long and rich
record of important discoveries in the radio spectrum, including 3 Nobel
prizes, has been possible since many of the relevant physical phenomena can
only be observed, or understood best, at these wavelengths. These phenomena
include the cosmic microwave background (CMB), quasars, pulsars, gravitational
waves, astrophysical masers, magnetism from planets through galaxies, the
ubiquitous jets from black holes and other objects, and the spatial
distribution of hydrogen gas, the predominant baryonic constituent of the
Universe. Moreover, through the invention of aperture synthesis, also
recognized by the Nobel committee, radio astronomy has reached unprecedented
levels of imaging resolution and astrometric precision, providing the fuel for
further discovery.

With only a handful of exceptions, radio telescopes and arrays have been
limited to apertures of about $10^{4}$ m2, constraining, for instance, studies
of the 21-centimeter hydrogen emission to the nearby Universe ($z\sim 0.2$)
[4]. Contemporaneous with the astronomical discoveries in the latter half of
the $20^{\mathrm{th}}$ Century have been technological developments that offer
a path to substantial improvements in future radio astronomical measurements.
Among the improvements are mass production of centimeter-wavelength antennas
enabling apertures potentially 100 times larger than previously available,
fiber optics for the transmission of large volumes of data, high-speed digital
signal processing hardware for the acquisition and analysis of the signals,
and computational improvements leading to massive processing and storage.
These new technologies, combined with dramatically improved survey speeds and
the other advances, can open up an enormous expanded volume of discovery
space, providing access to many new celestial phenomena and structures,
including 3-dimensional mapping of the web of hydrogen gas through much of
cosmic history ($z\sim 2$).

The realization that radio astronomy was on the doorstep of a revolutionary
age of scientific breakthrough has led the international community to
investigate this opportunity in great detail over the last decade. That
coordinated effort, involving a significant fraction of the world’s radio
astronomers and engineers, has resulted in the Square Kilometre Array (SKA)
Program (Figure 1), an international roadmap for the future of radio astronomy
over the next two decades and one for which access to a wide field of view is
an integral part of the science.

Figure 1: An artist’s impression of the core of the SKA illustrating the
various technologies over the frequency range 70 MHz to 10 GHz. All of these
technologies would enable various levels of wide-field imaging.

From its inception, development of the SKA Program has been a global endeavor.
In the early 1990s, there were multiple, independent suggestions for a “large
hydrogen telescope.” It was recognized that probing the fundamental baryonic
component of the Universe much beyond the local Universe would require a
substantial increase in collecting area. The IAU established a working group
in 1993 to begin a worldwide study of the next generation radio observatory.
Since that time, the effort has grown to comprise 19 countries and more than
50 institutes, including about 200 scientists and engineers.

##  2 Key SKA Science

Over the past several years, there has been extensive activity related to
developing a detailed science case for the SKA, culminating in the SKA Science
Book [4]. Highlighting the SKA Science Case are Key Science Projects (KSPs),
which represent unanswered questions in fundamental physics, astrophysics, and
astrobiology. Furthermore, each of these projects has been selected using the
criterion that it represents science that is either unique to the SKA or in
which the SKA will provide essential data for a multi-wavelength analysis [6].
The KSPs are

Emerging from the Dark Ages and the Epoch of Reionization

The ionizing ultra-violet radiation from the first stars and galaxies produced
a fundamental change in the surrounding intergalactic medium, from a nearly
completely neutral state to the nearly completely ionized Universe in which we
live today. The most direct probe of this Epoch of Re-ionization (EoR), and of
the first large-scale structure formation, will be obtained by imaging neutral
hydrogen and tracking the transition of the intergalactic medium from a
neutral to ionized state. Moreover, as the first galaxies and AGN form, the
SKA will provide an unobscured view of their gas content and dynamics via
observations of highly redshifted, low-order molecular transitions (e.g., CO).

Galaxy Evolution, Cosmology, and Dark Energy

Hydrogen is the fundamental baryonic component of the Universe. The SKA will
have sufficient sensitivity to the 21-cm hyperfine transition of H i to detect
galaxies to redshifts $z>1$. One of the key questions for $21^{\mathrm{st}}$
Century astronomy is the assembly of galaxies; the SKA will probe how galaxies
convert their gas to stars over a significant fraction of cosmic time and how
the environment affects galactic properties. Simultaneously, baryon acoustic
oscillations (BAOs), remnants of early density fluctuations in the Universe,
serve as a tracer of the early expansion of the Universe. The SKA will
assemble a large enough sample of galaxies to measure BAOs as a function of
redshift to constrain the equation of state of dark energy.

The Origin and Evolution of Cosmic Magnetism

Magnetic fields likely play an important role throughout astrophysics,
including in particle acceleration, cosmic ray propagation, and star
formation. Unlike gravity, which has been present since the earliest times in
the Universe, magnetic fields may have been generated essentially ab initio in
galaxies and clusters of galaxies. By measuring the Faraday rotation toward
large numbers of background sources, the SKA will track the evolution of
magnetic fields in galaxies and clusters of galaxies over a large fraction of
cosmic time. The SKA observations also will seek to address whether magnetic
fields are primordial and dating from the earliest times in the Universe or
generated much later by dynamo activity.

Strong Field Tests of Gravity Using Pulsars and Black Holes

With magnetic field strengths as large as $10^{14}$ G, rotation rates
approaching 1000 Hz, central densities exceeding $10^{14}$ g cm-3, and
normalized gravitational strengths of order 0.4, neutron stars represent
extreme laboratories. Their utility as fundamental laboratories has already
been demonstrated through results from observations of a number of objects.
The SKA will find many new millisecond pulsars and engage in high precision
timing of them in order to construct a Pulsar Timing Array for the detection
of nanohertz gravitational waves, probing the space-time environment around
black holes via both ultra- relativistic binaries (e.g., pulsar-black hole
binaries) and pulsars orbiting the central supermassive black hole in the
centre of the Milky Way, and probe the equation of state of nuclear matter.

The Cradle of Life

The existence of life elsewhere in the Universe has been a topic of
speculation for millennia. In the latter half of the $20^{\mathrm{th}}$
Century, these speculations began to be informed by observational data,
including organic molecules in interstellar space, and proto-planetary disks
and planets themselves orbiting nearby stars. With its sensitivity and
resolution, the SKA will be able to observe the centimeter-wavelength thermal
radiation from dust in the inner regions of nearby proto-planetary disks and
monitor changes as planets form, thereby probing a key regime in the planetary
formation process. On larger scales in molecular clouds, the SKA will search
for complex prebiotic molecules. Finally, detection of transmissions from
another civilization would provide immediate and direct evidence of life
elsewhere in the Universe, and the SKA will provide sufficient sensitivity to
enable, for the first time, searches for unintentional emissions or “leakage.”

In addition to the KSPs listed, and recognizing the long history of discovery
at radio wavelengths (pulsars, cosmic microwave background, quasars, masers,
the first extrasolar planets, etc.), the international science community also
recommended that the design and development of the SKA have “Exploration of
the Unknown” as a philosophy. Wherever possible, the design of the telescope
is being developed in a manner to allow maximum flexibility and evolution of
its capabilities to probe new parameter space (e.g., time-variable phenomena
that current telescopes are not well-equipped to detect). This philosophy is
essential as many of the outstanding questions of the 2020–2050 era—when the
SKA will be in its most productive years—are likely not even known today.

##  3 Opportunities for Panoramic SKA Science

Many of the papers in this volume illustrate far better than I could the
opportunities for Panoramic SKA Science, particularly in the areas of cosmic
magnetism and galaxy structure and evolution via H i observations.
Consequently, and similar to my approach in the conference itself, I shall
focus on opportunities for wide-field observations as they concern some of the
other SKA KSPs.

###  3.1 Emerging from the Dark Ages and the Epoch of Reionization

The primary focus of this KSP is tracking the transition from the Universe’s
largely neutral state to its currently nearly completed ionized state. Wide-
field observations will be both important and natural as the key observations
will be of the highly-redshifted H i line at frequencies below 200 MHz, which
will be carried out using dipole-based sparse aperture arrays. Dipoles have
fields of view that can exceed $\pi$ sr easily, but the relevant frequencies
are below the nominal focus of this conference.

A potential and important secondary observation that could be conducted near 1
GHz, however, would be of the synchrotron radiation from the first galaxies
[12]. Copious numbers of massive stars would have likely formed within these
first galaxies and then exploded soon thereafter as supernovae. If the
interstellar magnetic fields of these galaxies have developed sufficiently,
the galaxies will emit synchrotron radiation as a result of cosmic rays
accelerated by the supernova remnants from these first massive stars. While it
is not yet known if these galaxies will be detectable, the radio-far infrared
correlation for star-forming galaxies is now known to hold at least out to a
redshift $z\approx 3$ [14]. If it continues to hold to $z\approx 6$, then
radio observations would be a powerful means of probing dust-enshrouded first
galaxies and wide-field observations would naturally allow for large volumes
of the Universe to be sampled quickly.

###  3.2 Fundamental Physics Using Observations of Pulsars and Black Holes

Wide-field capabilities that enable the SKA to access a substantial solid
angle will be important for pulsar studies, even though pulsars are point
sources so that “panoramic imaging” per se of them is unlikely to be
profitable.

Fundamental physics constraints are derived from pulsar observations via long-
term timing programs that measure precisely the times of arrival of the
pulses. A significant constraint on the utility of radio pulsars is the
scarcity of “useful” pulsars. For instance, until recently, the most
significant constraints on the nuclear equation of state derived from radio
pulsars resulted from the first millisecond pulsar, PSR B1937$+$21, discovered
in the _early 1980s_. Similarly, many of the tests of theories of gravity and
for gravitational wave emission rely on one or a few objects. Recent surveys
have begun to demonstrate the potential for vastly increasing the number of
radio pulsars and thereby increasing the number of “useful” systems.

Perhaps the best example of the impact of increasing field of view for pulsar
surveys is the Parkes Multibeam Survey [10]. By installing a multiple feed
horn system on the Parkes antenna, the effective field of view was increased
by a factor of 13. The resulting survey essentially doubled the total number
of pulsars. (See also §3.4.) Future field-of-view expansion technologies
(e.g., phased array feeds or dense aperture arrays) coupled with the vastly
increased sensitivity of the SKA offer promise for an even larger yield.

The impact of a wide field of view for pulsar timing and monitoring programs
is less clear. In principle, a telescope with a sufficiently wide field of
view could time multiple pulsars simultaneously, yielding an improved
“throughput.” In practice, the current estimates of the density on the sky of
“useful” pulsars is sufficiently low that only dense aperture arrays are
likely to have a field of view that could be large enough to time multiple
pulsars simultaneously, except perhaps in special regions of the sky.
Moreover, in order to mitigate interstellar propagation effects, timing
observations have to be carried out over a relatively large frequency range
(e.g., 0.8–3 GHz), wider than what dense aperture arrays are currently thought
to be able to achieve.

###  3.3 Cradle of Life/Astrobiology

One of the key assumptions in the search for life elsewhere in the Universe,
particularly in searches for life within the solar system, is that other life
is likely to be based on carbon chemistry (i.e., “organic”), like life on
Earth is. Prime support for this approach is that the vast majority of multi-
atom molecules in interstellar space contain carbon, including a number of
complex organic species [15, 2, 1, and references within]. As the number of
atoms increases, the rotational and vibrational transitions tend to shift to
lower frequencies, and searches for and studies of complex organic molecules
have relied upon observations below 2 GHz. Consequently, a wide field-of-view
at frequencies around 1 GHz could be quite valuable for conducting surveys of
molecular clouds for complex organic molecules; such observations would find a
natural complement in ALMA observations.

Direct evidence for life elsewhere in the Universe would be the detection of
signals from another technological civilization. Two examples from our own
civilization are cell phone transmissions and aeronautical navigation, both of
which make use of frequencies around 1 GHz, though neither are strong enough
to be detectable over interstellar distances (even with the SKA!). More
generally, the “waterhole” between 1.4 and 1.7 GHz has been a focus of
numerous previous searches for extratestrial transmissions, as it has been
argued that any technological civilization capable of trying to communicate
over interstellar distances would certainly know about the H i line at 1.4 GHz
and the OH lines around 1.7 GHz.

One approach for searching for extraterrestial intelligence (SETI) is to
monitor a “habstar,” a star that might be orbited by a terrestrial planet(s)
within the star’s habitable zone [16]. Much like pulsar timing, being able to
monitor multiple habstars would increase the throughput of SETI observations;
the key contrast between pulsar and habstar observations is that the density
on the sky of suitable main sequence stars is sufficiently high that most, if
not all, fields of view will include more than one habstar.

###  3.4 The Dynamic Radio Sky

A series of discoveries over the past decade have both illustrated and
emphasized that the time domain has been explored only poorly at radio
wavelengths [3, 7, 9, 11]. Although time resolutions approaching 1 ns have
been achieved [8], typically these have been obtained only on relatively
narrow fields of view. The challenge and opportunity for the SKA, and
consistent with the “exploration of the unknown” design philosophy, is to
obtain both high time resolution and access to a significant solid angle.

Some of these observations might naturally happen in the course of pulsar
surveys (§3.2); indeed, the discovery of rotating radio transients, a new
class of radio-emitting neutron stars, resulted from the novel processing of a
pulsar survey [11]. Other types of transient surveys and exploration programs
might utilize wide fields of view in different manners, however.

We provide two examples to illustrate the potential range of applications of a
wide field of view:

  1. 1.

Extreme scattering events (ESEs) are a class of dramatic flux density
variations ($\sim 50$%) of extragalactic sources caused by intervening plasma
lenses [5]. The initial surveys for ESEs observed only a relatively small
number of the strongest, most compact sources on the sky. Yet within even a
modest field of view, if the full field of view can be imaged, are potentially
tens to hundreds of sources. A potential ESE search program could be conducted
by surveying a significant solid angle with a regular cadence and constructing
light curves of all of the sources within the survey region. Clearly an
expanded field of view would determine the total number of sources that could
be monitored.

  2. 2.

Many low-mass stars (spectral types K and M) show significant “radio
activity,” often with radio flares or bursts on short time scales [13]. This
radio emission is thought to be linked to coronal processes on the stars,
likely closely coupled to the magnetic field structure. Study of the coronal
processes in these extreme cases may provide understanding of solar processes,
which could impact not only astrophysics but aspects of the Earth-Sun
connection as well. Most, if not all, of the strongly “radio active” stars in
the solar neighborhood are known, but the typical separation on the sky is
fairly large. Similar to the case for pulsar timing, monitoring a large number
of low-mass stars for radio bursts would have a much higher throughput if
access to a wide field of view becomes possible.

We emphasize that these are only two possible examples, chosen to illustrate
the possible range of transient survey programs. The actual impact of the
field of view on any transient program will also depend upon the temporal
characteristics of the transients being targeted, their luminosity function,
and distribution on the sky, to the extent that these parameters are known.

## References

  * [1] Belloche, A., Garrod, R. T., Müller, H. S. P., Menten, K. M., Comito, C., & Schilke, P. 2009, Astron. & Astrophys. 499, 215.
  * [2] Belloche, A., Menten, K. M., Comito, C., Müller, H. S. P., Schilke, P., Ott, J., Thorwirth, S., & Hieret, C. 2009, Astron. & Astrophys. 482, 179.
  * [3] Bower, G. C. et al. 2008, Astrophys. J. 666, 346.
  * [4] C. L. Carilli and S. Rawlings, Science with the Square Kilometer Array, New Astron. Rev., 48, Elsevier, Amsterdam, 2004
  * [5] Fiedler, R. L., Dennison, B., Johnston, K. J., & Hewish, A. 1987, Nature 326, 675.
  * [6] Gaensler, B. M. 2004, “Key Science Projects for the SKA,” SKA Memorandum 44
  * [7] Hallinan, G. et al. 2007, Astrophys. J. 663, L25.
  * [8] Hankins, T. H., Kern, J. S., Weatherall, J. C., & Eilek, J. A. 2003, Nature 422, 141.
  * [9] Hyman, S. D. et al. 2005, Nature 434, 50.
  * [10] Manchester, R. N. et al. 2001, Mon. Not. R. Astron. Soc. 328, 17
  * [11] McLaughlin, M. A. et al. 2006, Nature 439, 817.
  * [12] Murphy, E. 2009, Astrophys. J., submitted
  * [13] Osten, R. A., & Bastian, T. S. 2008, Astrophys. J. 674, 1078.
  * [14] Seymour, N., Huynh, M., Dwelly, T., et al. 2009, Mon. Not. R. Astron. Soc., in press; arXiv:0906.1817
  * [15] Snyder, L. E., Hollis, J. M., Jewell, P. R., Lovas, F. J., & Remijan, A. 2006, Astrophys. J. 647, 412.
  * [16] Turnbull, M. C. ,& Tarter, J. C. 2003, Astrophys. J. Supp. 145, 181.