We live in a world out of equilibrium – a nonequilibrium world. We are surrounded by phenomena occurring in nature, in industrial and technological processes and in controlled experiments that we can only understand with the aid of a theoretical framework that encompasses nonequilibrium processes. Our understanding of these phenomena is largely based on a macroscopic theory that starts with the balance equations for the densities of mass, momentum, energy and other macroscopic quantities. To solve these equations, it is necessary to introduce relationships based on experiments that relate the observable properties of materials to the variables that define their macroscopic state. These relationships may describe equilibrium or locally equilibrium states of the material and in this case they are called equations of state. But we also need other relationships that relate the fluxes of properties to the property gradients that drive them. These are called constitutive or transport equations. The main subject of this book is the study of these transport equations and the material properties, such as the transport coefficients that account for the differences in the behaviour of different substances, using molecular dynamics simulation methods.

The molecular dynamics (MD) simulation method was developed soon after the Monte Carlo (MC) method, for the purpose of studying relaxation and transport phenomena [9]. Both MC and MD employed periodic boundary conditions, in which the system of interest is assumed to be replicated periodically in all directions, to limit (but not totally eliminate) the effects of the finite system size. At first, applications of this new technique focused on the structure, dynamics and equations of state of equilibrium systems [10–12]. The development in the 1950s of the Green–Kubo formalism, relating linear transport coefficients to equilibrium fluctuations in the corresponding fluxes [13, 14], made it possible to use equilibrium simulations to study nonequilibrium properties. However these methods, based on the computation of time correlation functions, were difficult to apply to all of the transport properties except self-diffusion due to their large computational requirements in comparison to the computing power available at that time. In addition, they could only address transport processes in the linear regime, i.e. where the flux is directly proportional to the thermodynamic driving force. These factors motivated the development of nonequilibrium molecular dynamics (NEMD) methods.

In this chapter, we introduce homogeneous nonequilibrium molecular dynamics simulation techniques by discussing the theoretical background to the SLLOD equations of motion. When these equations of motion are used in conjunction with compatible periodic boundary conditions and a homogeneous thermostat, they provide a very robust, reliable and well-understood method for studying fluids subjected to homogeneous flows. Here, we introduce the SLLOD equations of motion for the simple case of atomic fluids. This provides the groundwork for our discussion of methods for simulating homogeneous flows of molecular fluids in Chapter 8.

The SLLOD Equations of Motion

Background

To conduct microscopic simulations of flows driven by boundaries, mimicking real physical systems (e.g. Couette or elongational flows) we must explicitly include the walls. This inevitably induces density inhomogeneities into the fluid. If one is interested in nano-confined flow, then this is an appropriate simulation strategy since spatial inhomogeneity needs to be explicitly included in the simulation. However, if one is concerned with computing bulk properties such as mass, momentum and heat transport coefficients that we do not want to be distorted by surface effects, then the explicit use of boundaries is inappropriate.

An alternative to using atomistic wall boundaries is to generate flow through a suitable implementation of periodic boundary conditions. The first and most popular method of inducing flow through the periodic boundary conditions employs the so called Lees-Edwards boundary conditions [15] to generate planar shear flow. In such a scheme, a simulation box is replicated in all directions by periodic images.

So far we have largely considered NEMD simulation methods for fluids undergoing homogeneous flows. The exceptions to this were the “method of planes” and volume averaging techniques, in which expressions were derived that allow us to compute densities and fluxes for inhomogeneous fluids for various geometries (see Chapter 4). In this chapter we will use some of these expressions to compute relevant properties of highly confined fluids under several different flow conditions. In addition, we will show how to implement appropriate equations of motion to faithfully model fluids subject to spatially periodic fields and fluids under extreme confinement. It is with the latter application in mind that NEMD techniques are particularly important in the field of nanofluidics.

For over 150 years classical Navier-Stokes hydrodynamics [315] has been a wonderfully successful theoretical tool for predicting the properties of fluids and gases under a large variety of conditions. Its success extends from describing the dynamics of galactic motion [316], the aerodynamics of flight [317], the hydrodynamics of substances from liquid water to dense polymer melts [28], and right down to the flows of fluids on the microscale [318, 319]. It has even been shown to be accurate down to nanoscale dimensions [96], as long as certain conditions are maintained. In a numerical study on the Lennard-Jones fluid, it was first clearly demonstrated that, for an atomic fluid confined by atomistic walls, the Navier-Stokes equations were valid down to confinement spaces as low as around 5 to 10 atomic diameters [96, 320]. Below this spacing the fluid becomes highly inhomogeneous in space so the assumptions of constant density, constant viscosity, etc. break down, as do the Navier-Stokes equations.

At such small length scales another significant problem arises: the transport properties of fluids become nonlocal in nature. Although this effect has implicitly been built into the theory of generalised hydrodynamics [321] (see also Chapter 11) it has only recently been validated when the spatial extent of variations in the velocity gradient of the fluid are of the order of the width of the viscosity kernel [322]. The kernel itself is a nonlocal material property of the system with both wavevector and frequency dependence [321, 323] and has been accurately computed and parameterised for the Lennard- Jones fluid [324].

Most of the world's people are now living in cities and urbanization is expected to keep increasing in the near future. The resulting challenges are complex, difficult to handle, and range from increasing dependence on energy, to the emergence of socio-spatial inequalities, to serious environmental and sustainability issues. Understanding and modeling the structure and evolution of cities is then more important than ever as policy makers are actively looking for new paradigms in urban and transport planning.

The recent advances obtained in the understanding of cities have generated increased attention to the potential implication of new theoretical models in agreement with data. Questions such as urban sprawl, effects of congestion, dominant mechanisms governing the spatial distribution of activities and residences, and the effect of new transportation infrastructures are fundamental questions that we need to understand if we want a harmonious development of cities in the future, from both social and economic points of view.

Cities were for a long time the subject of numerous studies in a large number of fields. Discussion of the ideal city can be traced back at least to the Renaissance, and more recently scientists have tried to describe quantitatively the formation and evolution of cities. Regional science and then quantitative geography addressed various problems such as the spatial organization of cities, the impact of infrastructures, and transport. It is remarkable to note that as early as the 1970s quantitative geographers realized the crucial importance of networks in these systems, and produced visionary studies about networks, their evolution, and the complexity of cities (Haggett et al. 1977).

These studies were further developed mathematically by economists who discussed the interplay between space and economic aspects in cities. Many important models find their origin in the seminal paper of Von Thunen and describe isolated, monocentric cities in terms of utility maximization subject to budget constraints. These models allowed spatial economics to get a grasp of the relations between space, income, and transportation; for example. Japanese economists Fujita and Ogawa discussed the impact of agglomeration effects between firms in a general model that deals with the location choice for individuals and companies.

In the previous chapter, we discussed the analysis and modeling of mobility patterns in cities. However, as cities expand, their transportation networks are also growing, with increasing interconnections between different transportation modes. In large cities, we can now choose the transportation mode to travel from one point to another, and this trip can even involve several different modes. This multimodality is a new aspect of large cities and brings new questions and problems. From the users, point of view, it becomes difficult to deal with the huge amount of information needed for describing the different transportation networks and their interconnections. From the transport agencies point of view, the managing task becomes harder because the different modes are usually run by separate agencies; this renders optimization difficult owing to the large number of aspects that have to be taken into account (Guo and Wilson 2011).

In particular, an important problem concerning multimodality is the synchronization between different modes. For example, on average in the UK, 23% of travel time is lost in connections for trips with more than one mode (Gallotti and Barthelemy, 2014). This lack of synchronization between modes induces differences between the theoretical quickest trip and the “time-respecting” path, which takes into account waiting times at interconnection nodes.

In order to address these problems and more generally to understand the impact of the coupling between modes, we need new tools in order to identify the main factors that govern their efficiency. The multilayer network approach seems to be the most convenient framework for studying these systems (Kivel ä et al. 2014; Boccaletti et al. 2014). In this framework, each layer represents a mode and intermodal connections are represented by inter-layer links. In this chapter we discuss some aspects of multimodality and present tools for measuring and characterizing these coupled networks and their efficiency as a whole.

A multilayer network view of urban navigation

Empirical observations of multimodality

We first describe empirical results obtained by Gallotti and Barthelemy (2014) from timetables for the whole UK and for all transport modes. We note that these results were not obtained for traffic data (that are usually difficult to get). Instead we used these timetable data and the assumption of uniform demand.

What is our “understanding”?

As we discussed in Chapter 1, “understanding” has many definitions, with a variable amount of quantitative input. For a physicist, understanding does not mean having a story consistent with reality only, but also having mathematical tools and models able to describe real phenomena and to predict the outcome of experiments. Even if a qualitative description of processes is somewhat satisfying, it is not enough for constructing a science of cities. Indeed, we would like to identify the most important parameters, not only to understand the past, but also to be able to construct a model that gives, with a reasonable confidence, the future evolution of a city and to test the impact of various policies.

At this point, we certainly have a number of pieces of the puzzle, and we have discussed some of them in this book. It doesn't however mean that we have solved the full puzzle. New data sources and large datasets allow us to get a precise idea of what is happening in cities. We are currently experiencing an exciting time during which we can challenge the purely theoretical developments made these last decades. In many empirical studies, the identification of relevant factors was essentially done statistically, and we can now hope to go beyond and to have a more mechanistic approach, where a model based on simple processes is able to reproduce empirical observations.

Concerning the spatial structure of cities, new data sources give us a real-time, high-resolution picture of mobility. The structure of mobility flows that come out from these datasets departs from the usual image of a monocentric city where flows converge towards the central business district. Instead, for large cities, the main flows appear to be far from the localization between centers of residence and activities, that we could have na ïvely expected. This massive amount of data also allows us to quantitatively assess the degree of polycentricity of an urban system. A simple model showed that congestion is a crucial factor in understanding the evolution of polycentricity and mobility patterns with the population size.

Statistical physics of complex systems

The beginning of statistical physics can be traced back to thermodynamics in the nineteenth century. The field is still very active today, with modern problems occurring in out-of-equilibrium systems. The first problems (up to c. 1850) were to describe the exchange of heat through work and to define concepts such as temperature and entropy. A little later many studies were devoted to understanding the link between a microscopic description of a system (in terms of atoms and molecules) and a macroscopic observation (e.g., the pressure or the volume of a system). The concepts of energy and entropy could then be made more precise, leading to an important formalization of the dynamics of systems and their equilibrium properties.

More recently, during the twentieth century, statistical physicists invested much time in understanding phase transitions. The typical example is a liquid that undergoes a liquid-to-solid transition when the temperature is lowered. This very common phenomenon turned out, however, to be quite complex to understand and to describe theoretically. Indeed, this type of “emergent behavior” is not easily predictable from the properties of the elementary constituents and as Anderson (1972) put it: ”… the whole becomes not only more than but very different from the sum of its parts.” In these studies, physicists understood that interactions play a critical role: without interactions there is usually no emergent behavior, since the new properties that appear at large scales result from the interactions between constituents. Even if the interaction is “simple,” the emergent behavior might be hard to predict or describe. In addition, the emergent behavior depends, not on all the details describing the system, but rather on a small number of parameters that are actually relevant at large scales (see for example Goldenfeld 1992).

Statistical physics thus primarily deals with the link between microscopic rules and macroscopic emergent behavior and many techniques and concepts have been developed in order to understand this translation – among them the notion of relevant parameters, but also the idea that at each level of description of a system there is a specifically adapted set of tools and concepts.

Mobility is obviously a crucial phenomenon in cities. In fact, it is probably one of the most important mechanisms that govern the structure and dynamics of cities. Indeed, individuals go to cities to buy, sell or exchange goods, to work, or to meet with other individuals and for this they need different transportation means. This is where technology enters the problem through the (average) velocity of transportation modes. This average velocity increased during the evolution of technology and modified the structure and organization of cities. For example, we see in Fig. 5.1 that the “horizon” of an individual depends strongly on her transportation mode. For a walker, the horizon is essentially isotropic and small, while the car allows for a wider exploration but one which is anisotropic and follows transportation infrastructures. This correlation between the spatial structure of the city and the available technology at the moment of its creation is clearly illustrated by Anas et al. (1998) for US cities. Many major cities, such as Denver or Oklahoma City, developed around rail terminals that triggered the formation of central business districts. In contrast, automobile-era cities that developed later, such as Dallas or Houston, have a spatial organization that is essentially determined by the highway system.

In terms of mobility, the city center is also the location that mimimizes the average distance to all other locations in the city. Very naturally, it is then the main attraction for businesses and residences, which leads to competition for space between individuals or firms, giving rise to the real-estate market. There is also a well-known relation between land-use and accessibility, as was discussed some time ago by Hansen (1959), and new, extensive datasets will certainly enable us in the future to characterize precisely the relation between these important factors.

It is of course very difficult to make an exhaustive review about all studies on mobility and we will focus in this chapter on several specific points. We will mostly describe the general features of mobility and will leave the discussion about multimodal aspects for Chapter 6.

The locations of homes, activities, and businesses shape a city, and identifying the mechanisms that govern these spatial distributions is crucial for our understanding of these systems. We present here some recently discussed aspects, which may provide a basis for further insights.We will begin with a discussion on the location of stores and facilities, which are very likely governed by optimal considerations.

We will then discuss the polycentric aspects of cities by starting with their identification and measures. We will describe how to characterize and measure an activity center – a “hotspot” – defined as a local maximum of the activity density. The important empirical result is that the number of these hotspots scales sublinearly with the population size.

We continue by describing two classical theoretical models for polycentricity: the Fujita–Ogawa model, proposed in the 1980s, which relies on the idea that agglomeration effects are responsible for polycentricity, and the edge-city model proposed by Krugman. As we shall see, these models cannot, however, explain the scaling of the number of hotspots with population and this leads us to reconsider the classical Fujita–Ogawa model in order to derive a result in agreement with empirical observations.

Optimal locations

Distribution of public facilities

Public facilities such as airports, post offices, and hospitals have to be distributed according to the local population density in order to optimize their efficiency. These facilities constitute an important part of the urban structure and help to shape the spatial distribution of population. It is therefore important to understand the organization of these particular places.

We can measure these spatial distributions, and the natural null model to compare against these empirical observations is the optimal case where the average distance from an individual to the nearest facility is minimized (Gastner and Newman 2006), and we follow here the derivation given by Gusein-Zade (1982).

Infrastructure such as transportation networks for individuals and freight, power grids or distribution systems is crucial for our societies and the good functioning of cities. All these networks are embedded in space: nodes have a position and links have a certain length, and hence a cost associated with their formation and maintenance. These “spatial networks” (Barthelemy 2011) necessitate specific tools for their characterization and in this chapter we first review some of the most important ones. We then focus on transportation networks, starting with the road and street network, followed by subway networks. We end this chapter with a digression on the railroad network and we discuss the importance of the spatial scale and the main differences between subways and railroads.

Roads and streets: patterns

An important component of cities is their street network (in the following we will not make the distinction between streets and roads). These networks are made of nodes that represent the intersections and the links are segments of roads between consecutive intersections. These networks can be thought as a simplified view of cities, that captures a large part of their structure and organization (Southworth and Ben-Joseph 2003), and contains a large amount of information about underlying and universal mechanisms at play in their formation and evolution. Identifying the main mechanisms in these systems is not a new task (Haggett and Chorley 1969; Xie and Levinson 2011), but the recent availability of digitized maps, historical or contemporary (see Chapter 1), allows us to test ideas and models on large-scale cross-sectional and historical data.

Street networks are approximatively planar graphs and are now fairly well characterized (Jiang and Claramunt 2004; Rosvall et al. 2005; Porta et al. 2006b,a; Lämmer et al. 2006; Crucitti et al. 2006; Cardillo et al. 2006; Xie and Levinson 2007; Jiang 2007; Masucci et al. 2009; Chan et al. 2011; Courtat et al. 2011).

In this first chapter, we propose a rough, synthetic view of cities, by retaining what we believe to be some salient features from a quantitative point of view. There are many books and reviews, giving countless details and figures about cities (in particular, the reader can consult the updated version of reports produced by the UN – see for example “World urbanization propects, the 2014 revision”), and instead of offering a long list of various properties of cities (which can be found in different books and reliable sources such as the Census Bureau for the US, the UN, the OECD, or the World Bank), we focus here on a small set of key figures and discuss important scales manifesting in cities.

Cities are complex objects with many different temporal and spatial scales, related to a large number of processes. While a small set of numbers is certainly not enough to describe the full complexity of cities, such numbers can nevertheless allow for quantitative studies and for a large-scale characterization of urban systems. There is much variety among cities in terms of morphology, population, density distribution, and also functions, yet despite these differences, we observe statistical regularities for some observables. Indeed, we can expect that large systems composed of a large number of constituents lead to collective behaviors characterized by statistical regularities. Another reason for this “universality” is the existence of fundamental processes common to all cities: spatial organization of activities and residences, mobility of individuals, and so on. One of the most challenging problems of a science of cities is then to identify the minimal set of mechanisms that describe the evolution of cities.

A science of cities

The nature of the problem

A central issue in understanding urbanization is the large number of entangled, time-varying processes that generate cities. Many disciplines such as quantitative geography or urban economics have addressed some aspects of this problem and produced either very abstract models, or, at the other extreme, simulations with very large numbers of parameters designed for specific locales.