According to the invention, in each unit (such as 201) of the machine, a tank (212) contains a rotary displacer (220) defining chambers (222) containing a working gas. The rotation of the displacer consecutively places the gas in a thermal exchange relation with heat-carrying exchange surfaces, i.e. a hot one for cooling regeneration and heat insulation and a cold one for heating regeneration and thermal insulation. At each moment, the tank is at an equal pressure, all the chambers of a same housing are at a same stage of the thermodynamic cycle, with the machine carrying out two cycles per revolution of the displacer in the present embodiment. The pressure in the tank is applied to a power piston (230). A gear (425) synchronises the piston (230) and the displacer (220). The invention can be used for improving the yield and enabling operation at a relatively high speed and/or under small temperature differences.