Physics Asked by bubucodex on March 16, 2021
Spontaneous thermal fluctuations occur at microscopic level in liquids. It is said that hydrodynamic description is valid in the long wavelength and low frequency limit. So, to depict the thermal fluctuations occurring at microscopic level an extension of hydrodynamics is done by retaining the basic structure of hydrodynamic equations which is known as Molecular Hydrodynamics([https://www.sciencedirect.com/science/article/pii/B0122274105.004580#a0005])[1]. Are the wavelength and frequency that is being talked about here of the thermal fluctuations ? If that is so, what is actually a thermal fluctuation ? Is it a visible quantity in itself, or simply a manifestation of fluctuation of temperature through other quantities like velocity or number density of the liquid ? How can the wavelength and frequency of thermal fluctuation be determined ?
The linearised hydrodynamic equations for a fluid (no internal dofs), include the continuity equation, the Navier-Stokes equation and the energy transport equation. Respectively, they represent conservation of mass, momentum and kinetic energy. I assume you have seen these equations and in the second of these equations there contains the term
$$frac{{c_0}^2}{gamma} nabla rho_1(x,t)$$
where the constant $c_0$ is called the adiabatic sound speed.
Since you are dealing with pressure wave propagation through the fluid, the long wavelength or small frequency condition refers to that for sound waves (travelling in the fluid as pressure waves at the speed of sound).
Answered by joseph h on March 16, 2021
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