Encyclopedia of Wireless Networks

Living Edition
| Editors: Xuemin (Sherman) Shen, Xiaodong Lin, Kuan Zhang

Brownian Motion

  • Mohammad Upal MahfuzEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-32903-1_231-1



Brownian motion is the random motion of particles, e.g., molecules, suspended in the fluid medium, e.g., liquid and gas, that results from a large number of collisions those particles experience with the fast-moving particles of the fluid medium.



Brownian motion


Molecular communication


Molecule-receptor binding process


Receiving nanomachine


Transmitting nanomachine


Virtual reception volume

Historical Background

Brownian motion (BM) is an important phenomenon that is the basis of diffusion-based propagation of molecules in molecular communication (MC) and, therefore, is the fundamental principle behind diffusion-based MC in the field of nanoscale communication networks, also known as nanonetworks. In the field of natural and applied sciences, BM is also popularly known as random walk motion of particles. The history of BM is quite old. Random walk motion of particles was first observed by Scottish botanist Robert Brown (1773–1858) who found with a microscope that pollen particles jiggled at a very high speed while they were suspended in water medium (Brown 1828) and hence the name Brownian motion. However, it was Albert Einstein who explained BM with the theory that it is due to an extremely large number of collisions that particles under investigation experience by interacting with the particles of the fluid medium (Einstein 1905). In the field of nanoscale communication networks (Akyildiz et al. 2008; Bush 2010), the concept of BM received a huge amount of interest among researchers (Ahmadzadeh et al. 2017, 2018; Eckford 2007; Gohari et al. 2016; Haselmayr et al. 2017; Jamali et al. 2018; Kadloor et al. 2012; Koo et al. 2016; Nakano et al. 2012, 2013, 2014; Pierobon and Akyildiz 2010, 2011) who referred to the theory of BM as the fundamental principle of the diffusion process (Berg 1993) to investigate the performance of MC systems among nanomachines. Thus, MC based on BM was considered as an option for communications among nanomachines forming nanonetworks (Akyildiz et al. 2008). Some of the works in this field later contributed to the knowledge base while developing the standards for nanoscale communication networks (IEEE 2016).


Physical Process

Random walk motion is the fundamental principle of BM. When molecules are suspended in a fluid medium, due to thermal excitation, the molecules collide with one another and thus they propagate in three dimensions (Berg 1993). Such movements of molecules could be either in a truly random direction when there is no external force present in the system or towards a given direction determined by the presence of an external force in the system (Berg 1993). The collisions between molecules occur on a very tiny timescale on the scale of picoseconds (Höfling and Franosch 2013), and thus billions of such collisions can take place within a small duration of time. When all the factors that can impact the movement of molecules are considered absent, these collisions are truly random and, therefore, molecules undergo diffusion process (Berg 1993; Berg and Purcell 1977; Crank 1975; Einstein 1905) and thus move in a random direction in the fluid (propagation) medium. Random walk of molecules can be characterized by the two major types, namely, continuous-time and discrete-time random walks, distinguished by the step size of the particle between two states and the time the particle remains in one state before it moves to the next state (Metzler and Klafter 2000).

Normal diffusion process is based on uncorrelated random walk model of the movement of the molecule in the fluid medium where the direction of the new movement of the molecule is independent of the direction of its previous movements and can be equally likely in any of the directions in the three-dimensional space. On contrary, anomalous diffusion is based on correlated random walk model of the molecule in the fluid medium where the next movement of the molecule is likely to continue in the current direction (Mahfuz et al. 2016). In the field of MC and nanonetworks, while most researchers have considered normal diffusion process as a result of random walk motion of molecules between a transmitting nanomachine (TN) and a receiving nanomachine (RN), a few researchers have considered anomalous diffusion of molecules and presented investigative results in the aspects of MC systems (Cao et al. 2015; Mahfuz et al. 2016; Mai et al. 2017). While normal diffusion considers random BM of molecules between TN and RN, anomalous diffusion considers the abnormalities involved in the normal diffusion process, for instance, when information molecules are confined within a small space, e.g., intra-cellular space, or when information molecules (particles) interact with one another, e.g., charged particles (Mahfuz et al. 2016). Current literature has also investigated into the role of anomalous diffusion on molecular propagation and availability of molecules at the RN as well as the performance of MC receivers (Cao et al. 2015; Mahfuz et al. 2016; Mai et al. 2017).

Molecular Propagation and Communication

BM is the fundamental principle that is the basis of molecular propagation between a pair of nanomachines in diffusion-based MC systems. As shown in Fig. 1, in the aspects of MC, when a TN, located at the origin (0,0,0) of the axes, emits information-carrying molecules in the fluid medium, these information-carrying molecules experience BM and, after a huge number of collisions with the molecules of the fluid medium, finally reach an RN located at (x, y, z). As shown in the inset in Fig. 1, the RN is located in the center of a small virtual reception volume (VRV) (Atakan and Akan 2010), which is a small unit volume surrounding the RN where the RN can sense information-carrying molecules, shown as blue circles, that become available to RN and come into contact with the receptors on its surface. Although information molecules can bind with the surface receptors with some probability as governed by the molecule-receptor binding process (MRBP), when a sufficiently large number of surface receptors are assumed, this ensures that the available information molecules at the RN come into contact with the RN and be received by the RN. Thus, the RN receives the information molecules and detects the message by processing the received information-carrying molecules. Here, the RN is assumed to be a transparent receiver that does not affect the BM of the information molecules in the fluid medium. RN is also assumed to perfectly count the number of information molecules that become available within the VRV at a given time.
Fig. 1

Diffusion of information-carrying molecules in three dimensions in the unbounded propagation medium (Mahfuz et al. 2014)

In the three-dimensional propagation medium denoted by x, y, and z axes, these information-carrying molecules propagate independently in three dimensions and reach the RN located at a distance r from the TN, where r2 = x2 + y2 + z2. The diffusion of information molecules in three dimensions and time (t), in second (s), can be expressed as Eq. (1) below (Berg 1993)
$$ \frac{\partial U}{\partial t}=D{\nabla}^2U $$
where U denotes the concentration of information-carrying molecules, i.e., the number of information-carrying molecules per unit volume of reception space, D (in μm2/s) denotes the diffusion coefficient of information molecules propagating in the given fluid medium, and \( {\nabla}^2=\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {y}^2}+\frac{\partial^2}{\partial {z}^2} \). The value of D is given by \( D=\frac{k_BT}{6\pi \eta {R}_H} \) where kB = 1.38 × 10−23 J/K is the Boltzman constant, T is the temperature (in K), η is the viscosity of the fluid medium, and RH is the hydraulic radius of the molecule (Nakano et al. 2013). Diffusion coefficient values of commonly used information molecules can be found in Nakano et al. (2013) and Freitas (1999). Often, it is necessary to measure the values of diffusion coefficient experimentally for which there have been several methods to follow. For instance, several static and dynamic methods to obtain diffusion coefficient values experimentally have been explored in Culbertson et al. (2002). A simple and inexpensive experimental method to measure diffusion coefficient when temperature and pressure vary has been shown in Delgado et al. (2005).
When the TN located at (0,0,0) is assumed to transmit Q0 molecules, all at a time in an impulsive manner at time t = 0, the average concentration of information-carrying molecules, in number of information-carrying molecules per unit volume of the reception space, available at the RN, located at (x, y, z) in an unbounded three-dimensional propagation medium, can be found from the solutions of the diffusion equation in Eq. (1) as follows (Berg 1993):
$$ U\left(r,t\right)=\frac{Q_0}{{\left(4\pi Dt\right)}^{3/2}}\exp \frac{-{r}^2}{4 Dt} $$

For the transparent receiver mentioned earlier, Eq. (2) shows the average number of information-carrying molecules per unit volume, i.e., concentration of information-carrying molecules, for a given number of transmitted molecules, Q0. The concentration of information-carrying molecules U(r, t) depends on distance (r) to the RN and the time elapsed after the release of the molecules from the TN. Apart from the transparent receiver mentioned in this entry, for other types of MC receivers, interested readers are referred to the following entry of this book: “Receiver Mechanisms for Synthetic Molecular Communication Systems with Diffusion” by Ahmadzadeh, A., Jamali, V., Wicke, W., and Schober, R.

There are several simulation frameworks currently available that help researchers to simulate BM in the study of MC-based nanonetworks. For instance, N3Sim uses diffusion-based MC to provide a simulation framework for nanonetworks with transmitters, receivers, and harvesters (Llatser et al. 2014). N3Sim not only uses BM dynamics to model the movements of molecules but also includes inertia of and interaction among the molecules in the system. Another simulation framework named AcCoRD (Actor-based Communication via Reaction Diffusion), which is an open-source project written in C programming language, is designed and used for MC analysis and reaction diffusion solutions (Noel et al. 2017). NanoNS, built on the widely used network simulator NS-2, is another simulation framework that simulates the diffusion-based MC in nanonetworks (Gul et al. 2010). To simulate the behavior of diffusion-based MC with drift effect inside blood vessels, a software simulation platform named BiNS2 has also been introduced (Felicetti et al. 2013).

Key Applications

The applications of BM include all scenarios where particle movement is used. One of the notable applications of BM would be in the diffusion-based MC and nanonetworks fields, for instance, in drug delivery and disease treatment as well as environmental monitoring, protection, and control (Akyildiz et al. 2008; Nakano et al. 2013).



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Wisconsin-Green BayGreen BayUSA

Section editors and affiliations

  • Adam Noel
    • 1
  1. 1.University of Warwick, UKWarwickUK