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How can I give an object a motion like the particles' Brownian motion, keyframing the object location value for each frame? Is there a better way than just using a random value?

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You could add keyframe modifiers.

  1. Add keyframe to the x,y,z positionsn at the beginning of the animation
  2. Go to the Graph Editor -> right panel (press N to show it) -> "Modifiers" label -> Add Modifier -> Noise.
  3. Do the operation 2 to the three keyframes (X, Y and Z positions).
  4. In the Modifier panel (where you just added the modifier) of each keyframe (X, Y and Z) modify the noise properties in order to make them different each other and to set the Brownian motion as you want.
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