Mandelstam problem

The theory of thermal (cid:29)uctuations in open hydrodynamic steady states (OHSS) is presented exclusively within the framework of hydrodynamics. The history of studies of (cid:29)uctuations in a continuous medium with a stationary (cid:29)ux is described. It is shown that the application the (cid:29)uctuation-dissipation theorem (FDT) to the OHSS with the requirement of ful(cid:28)lling the Onsager’s reciprocal relations ((cid:29)uctuating hydrodynamics), is erroneous. The reason is that the (cid:29)ux, changing the dynamics and initial values of the (cid:29)uctuations, violates the detailed balance existing in equilibrium. This is demonstrated by the example of the Mandelstam problem on (cid:29)uctuations in a medium with a heat (cid:29)ux. For this problem, the structure dynamic factor is calculated for an isotropic solid and a liquid. The loss of time symmetry by the correlation functions of (cid:29)uctuations and the asymmetry of their spectral representations in this problem is due to the spatial temperature variation, which determines the (cid:29)ux. In order to show the generality of this result for all OHSS with spatial heterogeneity, the Kelvin problem on thermal (cid:29)uctuations of the interface displacements between two liquids is also considered. The upper moving liquid has velocity potential changes as the temperature in the Mandelstam problem. Reciprocal relations for both the Mandelstam and the Kelvin problems are pointed out