2.1 The Particle State Vector

The particle state vector is characterized by a set of "external coordinates'' ( ${\vec x}$), which denote the spatial position of the particle, and "internal coordinates'' ( $\phi$), which could include particle size, composition, and temperature. From these coordinates, a number density function $n({\vec x}, \phi,t)$ can be postulated where $\phi \in \Omega_{\phi}$, ${\vec x} \in \Omega_{\vec x}$. Therefore, the average number of particles in the infinitesimal volume $dV_{\vec x}dV_{\phi}$ is $n({\vec x}, \phi,t) dV_{\vec x}dV_{\phi}$. In contrast, the continuous phase state vector is given by ${\vec Y}\equiv [Y_1({\vec x},t),Y_2({\vec x},t),\ldots,Y_c({\vec x},t)]$

The total number of particles in the entire system is then

The local average number density in physical space (i.e., the total number of particles per unit volume) is given by

The total volume fraction of all particles is given by

where $V(\phi)$ is the volume of a particle in state $\phi$.