Multi-Scale Simulation

Deterministic Differential Equation

Before starting the description of the evolution equation of stochastic quantity, we give some basic equations which describe the motion of small particle. Lets consider fluid particle with mass , the friction force which acts on it can be expressed as:

F_c = - \alpha v

Therefore the equation of the motion of the particle is given by:

m \dot{v}+ \alpha v = 0 \\ or\\ \dot{v} + \gamma v = 0 ; \quad \gamma = \alpha /m = 1/ \tau

Thus an initial velocity decreases to zero with the relaxation time according to:

v(t) = v(0) e^{-t\tau} = v(0) e^{-t\gamma}

The above differential equation is a deterministic equation, the velocity at time is completely determined by its initial value.

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