By Birkhoff G.D.
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Additional info for A Theorem on Series of Orthogonal Functions with an Application to Strum-Liouville Series (1917)(en)
4. 1 Fluctuation-dissipation theorem for a one-component ﬂuid The ﬁrst step in establishing the FDT for any thermodynamic system is to identify the relevant dissipative ﬂuxes from the corresponding expression ˙ or the dissipation function Ψ. For a onefor the entropy production S, component ﬂuid the dissipation function was given by Eq. 46) with two dissipative ﬂuxes: a vectorial heat ﬂow Q(r, t) and a second-rank (symmetric) deviatoric stress tensor Π(r, t). 4) . The next step in ﬂuctuating hydrodynamics is to identify the dissipation matrix from the phenomenological laws.
80) requires the velocities to be parallel to the interface, so that: v1,z = v2,z = 0, when z = L. 86) When the wall is solid (rigid) v2 = 0, so that, in accordance with Eq. 79), all components of the ﬂuid velocity at the wall must vanish (not just the vertical one). Hence for a ﬂat solid wall, dropping the index 1, we require for the ﬂuid velocity v: v = 0, when z = L. 87) As a consequence, the derivatives of the ﬂuid velocity along the boundary must also vanish: ∂x v = ∂y v = 0. The derivatives with respect to the vertical z-direction have to be determined from the balance of forces, Eq.
20 Chapter 2: Nonequilibrium thermodynamics The goal of nonequilibrium thermodynamics is to calculate the spatiotemporal evolution of ﬁelds like mass density ρ(r, t), center-of-mass velocity v(r, t), energy density ρu(r, t), etc. In the previous subsections we have shown how this evolution is governed by balance laws. In formulating the balance laws, we needed to introduce new ﬁelds that do not exist in equilibrium thermodynamics: diﬀusion ﬂuxes Jk (r, t) in Eq. 13), a deviatoric stress tensor Π(r, t) in Eq.