5. Hydrodynamic Fluctuations from the Boltzmann Equation

In the notation in [8, 18, 19], the distribution function is written as a sum over \(n\)-fold contracted products of \(n\)th-rank tensors \(f(\mathbf{{c}})=f_0(c)\sum _{k,n=0}^\infty \left\langle {\phi }_k^n\right\rangle \odot ^n{\phi }_k^n(\mathbf{{c}})\) with \(\left\langle {\phi }_k^n\right\rangle =\int f(\mathbf{{c}}){\phi }_k^n\ \mathrm{{d}}\mathbf c \) and base functions , where \(L_k^n\) are the associated Laguerre (\(k\)th-order) polynomials [20], \(\otimes ^n\mathbf{{c}}\) denotes the \(n\)-fold tensor product, and denotes the irreducible part of a tensor \(\mathbf a \). For the explicit construction of \(n\)th-rank irreducible tensors , see [18, p. 160]. The normalization coefficients evaluate as \(l_k^n=(\sqrt{\pi }k!(1+2n)!!/[2(k+n+1/2)!n!])^{1/2}\). The base function \({\phi }_k^n(\mathbf{{c}})\) is thus a \((2k+n)\)th-order polynomial in \(c\). The lowest-order base functions read \({\phi }_0^0 =1\), \({\phi }_0^1 =\sqrt{2}\mathbf{{c}}\), \({\phi }_1^0 =\sqrt{2/3}(3/2-c^2)\), \({\phi }_1^1 =(2/\sqrt{5})(5/2-c^2)\mathbf{{c}}\), and . Density, velocity, temperature, heat flux, and stress tensor are related to the moments as follows: \(\tilde{n}=\left\langle {\phi }_0^0\right\rangle \), \(\tilde{\mathbf{u}}=\left\langle {\phi }_0^1\right\rangle /\sqrt{2}\), \(\tilde{T}=\left\langle {\phi }_1^0\right\rangle \sqrt{3/2}\), \({\mathbf{{q}}}=\left\langle {\phi }_1^1\right\rangle \), and \({\sigma }=\left\langle {\phi }_0^2\right\rangle /\sqrt{2}\). The distribution function is then split into (orthogonal) parts as \(f(\mathbf{{c}})=f^\mathrm{{LM}}(\mathbf{{c}})+\delta f^\mathrm{{Grad}}(\mathbf{{c}})+\delta f^\mathrm{{rest}}(\mathbf{{c}})\) with \(f^\mathrm{{LM}}(\mathbf{{c}})\equiv f_0(c)(\left\langle {\phi }_0^0\right\rangle {\phi }_0^0 + \left\langle {\phi }_0^1\right\rangle {\phi }_0^1 + \left\langle {\phi }_1^0\right\rangle {\phi }_1^0)\) and \(\delta f^\mathrm{{Grad}}(\mathbf{{c}}) \equiv f_0(c)(\left\langle {\phi }_1^1\right\rangle {\phi }_1^1 + \left\langle {\phi }_0^2\right\rangle {\phi }_0^2)\), while the sum in \(\delta f^\mathrm{{rest}}(\mathbf{{c}})=\sum _{k,n} \left\langle {\phi }_k^n\right\rangle \odot ^n {\phi }_k^n(\mathbf{{c}})\) extends over the remaining \((k,n)\)-pairs. Number density, velocity, and temperature are therefore determined by \(f^\mathrm{{LM}}\) alone, and \(\delta f\) automatically obeys constraints such as the orthogonality requirement \(\int \delta f(\mathbf{{c}}) {\phi }_1^0\ \mathrm{{d}}\mathbf c =0\) and also \(\int \delta f(\mathbf{{c}})\xi (\mathbf{{c}})\mathrm{{d}}\mathbf c =0\), as mentioned in the text. These conditions become redundant if calculations are performed using the particular basis \({\phi }_k^n\). For Maxwell molecules, the dependence on the polar angle \(\phi \) can be included by replacing \(P_l(z)\) by \(e^{im\phi } P_l^m(z)\), involving the associated Legendre polynomials [20], and the eigenvalues are independent of \(m\). Then these base functions reduce to the eigenfunctions \(\psi _{r,l}(c,z)\) (5.29) of the Maxwell gas.

The integrals listed in Table 5.2 obey the following decoupling rules: