The Einstein-Boltzmann Relation for Thermodynamic and Hydrodynamic Fluctuations

When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the fluctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic fluctuations must be included, the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, $\delta^{2} S$ will depend on velocity variations. Some authors do not include velocity variations in $\delta^{2} S$, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At first 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 fluctuation-dissipation theorem which shows some differences as compared with the non-convective case. We also show that $\delta^{2} S$ is a Liapunov function when it includes velocity fluctuations.