English: The left panel shows the time evolution of the phase portrait of a harmonic oscillator for different temperatures. The right panel captures the resulting probability distributions at thermal equilibrium. We observe that for k_B T= 0, the velocity rapidly decreases from its initial value (the red dot) to zero, due to the damping force. However, for non zero temperatures, thermal fluctuations can kick the velocity to values higher than the initial velocity and the particle never comes to rest, even at long times.
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