If a gold bar and a silver bar are tied together firmly and left for years, some of the gold particles will be found in the silver bar due to

Diffusion is the net movement of particles from a region of higher concentration to a region of lower concentration. It occurs in gases, liquids, and solids, though it is slowest in solids because the particles are closely packed and vibrate in fixed positions.

When a gold bar and a silver bar are pressed firmly together and left for years, gold atoms gradually migrate into the silver bar (and vice versa). This happens because the metal atoms, though fixed in a lattice, vibrate continuously. Over very long periods, some atoms acquire enough energy to move into neighbouring lattice positions, slowly spreading through the other metal. This is solid-state diffusion.

Brownian movement describes the random, erratic motion of microscopic particles suspended in a fluid (liquid or gas) caused by collisions with the surrounding fluid molecules. It does not apply to atoms within a solid lattice. Displacement is a chemical reaction in which a more reactive element replaces a less reactive one in a compound. Osmosis is the movement of water molecules through a semipermeable membrane from a dilute to a concentrated solution. Neither of these describes the mixing of atoms between two solid metals in contact.