On the fluctuation-dissipation theorem for convective processes

When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the uctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic uctuations must be included; the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, δ²S will depend on velocity variations. Some authors do not include velocity variations in δ²S, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At rst sight, it seems that the introduction of such a function requires a generalisation of the Einstein-Boltzmann relation to be invoked. We review the reason why it is not necessary to introduce such a function, and therefore why there is no need to generalise the Einstein-Boltzmann relation in this way. We then obtain the uctuation-dissipation theorem, which shows some differences as compared with the non-convective case. We also show that δ²S is a Liapunov function when it includes velocity uctuations.