(sec:methods:particles)= # Particles ASPECT can, optionally, also deal with particles (sometimes called "tracers"). Particles can be thought of as point-like objects that are simply advected along with the flow. In other words, if $\mathbf u(\mathbf x,t)$ is the flow field that results from solving equations {math:numref}eq:stokes-1-{math:numref}eq:stokes-2, then the $k$th particle's position satisfies the equations {math} \begin{aligned} \frac{d}{dt} \mathbf x_k(t) = \mathbf u(\mathbf x_k(t),t). \end{aligned}  The initial positions of all particles also need to be given and are usually either chosen randomly, based on a fixed pattern, or are read from a file. Particles are typically used to track visually where material that starts somewhere ends up after some time of a simulation. It can also be used to track the *history* of the volume of the fluid that surrounds a particle, for example by tracking how much strain has accumulated, or what the minimal or maximal temperature may have been in the medium along the trajectory of a particle. To this end, particles can carry *properties*. These are scalar- or vector-valued quantities that are attached to each particle, that are initialized at the beginning of a simulation, and that are then updated at each time step. In other words, if we denote by $\mathbf p_{k,m}(t)$ the value of the $m$th property attached to the $k$th particle, then $\mathbf p_{k,m}(t)$ will satisfy a differential equation of the form {math} \begin{aligned} \frac{\partial}{\partial t} \mathbf p_{k,m}(t) = \mathbf g_m\left(\mathbf p_{k,m}, p(\mathbf x_k(t),t)), T(\mathbf x_k(t),t)), \varepsilon(\mathbf u(\mathbf x_k(t),t)), \mathfrak c(\mathbf x_k(t),t)\right). \end{aligned}  The exact form of $\mathbf g_m$ of course depends on what exactly a particular property represents. Like with compositional fields (see {ref}sec:methods:compositional-fields), it is possible to describe the right hand side $\mathbf g_m$ in ways that also allows for impulse (delta) functions in time. How particles are used in practice is probably best explained using examples. To this end, see in particular {ref}sec:cookbooks:using-particles. All particle-related input parameters are listed in {ref}parameters:Postprocess/Particles. The implementation of particles is discussed in great detail in {cite:t}gassmoller:etal:2018.