A gaseous fluid compression device includes: a first enclosure within which there is a movable first piston delimiting a first chamber and a second chamber; a second enclosure within which there is a movable second piston delimiting a third chamber and a fourth; a first exchange circuit connecting the first chamber and the fourth chamber, with a heat exchanger linked to a heat sink; a second exchange circuit connecting the second chamber and the third chamber, with a second heat exchanger linked to a heat source; and a transfer passage connecting the first chamber and the second chamber with an anti-backflow device. A back-and-forth movement of the interconnected pistons results in a compression of the gaseous fluid in the direction of the outlet.