# Probablity density function $f(\mathbf{x},\mathbf{v},t)$

The phase density function is usually denoted as $f(\mathbf{x},\mathbf{v},t)$ which gives probable number of particles moving with velocity $\mathbf{v}$ at position $\mathbf{x}$ at time t. Also we assume that position and velocities of particles are not correlated i.e. $\mathbf{x}$ and $\mathbf{v}$ are independent.

We use $f(\mathbf{x},\mathbf{v},t)$ as a basis Lattice Boltzmann functionto solve fluid dynamics equations. $$f_i(\mathbf {x} +\Delta \mathbf {x}, t + \Delta t) - f_i(\mathbf {x}, t ) = -\frac{\Delta t}{\tau} (f_i - f^{eq}_i)$$

The entire point of solving fluid dynamics equation is(If I am correct) to know velocity at a given point and time. How can we use $f(\mathbf{x},\mathbf{v},t)$ to proceed with lattice Boltzmann equation when $f(\mathbf{x},\mathbf{v},t)$ has multiple values of velocities at given point $\mathbf{x}$ ?

-

At a given point in space, there are several values of $f_i$, one for each discrete velocity $v_i$. So you have to apply the lattice-Boltzmann equation as many times. This is exactly the point of the lattice-Boltzmann method: each "element" is actually defined by its discrete value of the velocity. You apply the equation for each of these elements which have a given, fixed velocity.