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Let $q$ be heat dissipation to midium, $F$ be the force to a particle, and $\dot{x}$ is the velocity of it.

According to the equation (8) in Seifert 2005, $\dot{q} = F \dot{x}$ holds. How does this relation holds?

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This is just the typical result that the instantaneous rate at which a force does work (i.e. the power of the force) is given by the dot product of that force and the velocity of the body the force acts on. This can be seen by looking at the definition of work: $$\text dW=\mathbf F\cdot\text d\mathbf x$$ and then dividing by the infinitesimal time element $\text dt$: $$P=\frac{\text d W}{\text dt}=\mathbf F\cdot\frac{\text d \mathbf x}{\text dt}=\mathbf F\cdot \mathbf{\dot x}$$

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