\section*{Symbols and abbreviations}

Here an exhaustive list of all mathematical symbols used in this text, and their definitions, is collected. Notation wise, boldface roman font, such as $\vU$ is used for vectors, and calligraphic font, $\sU$ is used for dimensionless quantities, except in the case of dimensionless state variables, which are denoted with a star, such as $T^*$. The length of a vector $\vU$ is denoted in serif font as $U$. Matrices are denoted with underlined boldface symbols, such as $\mB$.

The symbols $\varphi$ and $\psi$ are reserved to indicate arbitrary, or placeholder, variables and functions respectively. 

Due to the large volume of notation, a few symbols carry several definitions, these are summarised at the end of the table to facilitate differentiating them based on context.

\begingroup\centering

\begin{tabular}{c l c}
    \multicolumn{3}{c}{Symbols used in the text}\\
    \toprule
    Symbol & Description & Unit \\
    \midrule
        $A$ & Helmholtz energy & \si{\joule}\\
        \symbline{A_{pqr\ell},A_{pqrl}',A_{pqr\ell}'''}{Specific scalar values, see Section \ref{subsubsec:summational_expressions}}{-}
        \symbline{a_{pq}}{Specific integrals, see Eq. \ref{eq:apq_expanded}}{\cubic\meter\per\second}
        \symbline{\va_i^{(p)}}{Specific Sonine polynomials, see Eq. \eqref{eq:aip_poly_definitions}}{-}
        \symbline{b}{Impact parameter, see Figures \ref{fig:encounter_geometry} and \ref{fig:cyl_coord}}{\meter}
        \symbline{D_{ij}}{Diffusion coefficient}{\mol\per\square\meter\per\second}
        \symbline{\Hat{D}}{Predicted diffusion coefficient}{\mol\per\square\meter\per\second}
        \symbline{D_T}{Thermal diffusion coefficient}{\mol\per\square\meter\per\second}
        \symbline{\vD}{Velocity distribution response to $\Vec{d}_{12}$, see Eq. \eqref{eq:binary_phi}}{\meter}
        \symbline{\Vec{d}_{12}}{Specific driving force, see Eq. \ref{eq:d12_kingas_driving_force}}{\per\meter}
        \symbline{d_p}{Sonine polynomial expansion coefficient, see Eq. \eqref{eq:kin_D_poly_expans}}{\meter}
        \symbline{\Vec{F}}{Force}{\newton}
        $f$ & Exact velocity distribution function (vdf.) & \si{\cubic\second\meter}$^{-6}$\\
        $f^{(r)}$ & r-th approximation term to the vdf. & \si{\cubic\second\meter}$^{-6}$\\
        \symbline{\Vec{g}}{Relative particle velocity}{\meter\per\second}
        \symbline{g}{Relative particle speed}{\meter\per\second}
        \symbline{\sg}{Dimensionless relative particle speed}{-}
        \symbline{H}{Enthalpy}{\joule}
        \symbline{H_1^{(1)}, H_{12}^{(1)}, H_{12}^{(12)}}{Specific bracket integrals, see Eq. \eqref{eq:H_integrals_def}}{-}
        \symbline{h}{Molar enthalpy}{\joule\per\mol}
        \symbline{h_i}{Partial molar enthalpy of species $i$}{\joule\per\mol}
        \symbline{h_x, h_y}{Integration step, $h_x = N_x\delta_x$, $h_y = N_y\delta_y$}{-}
        \symbline{I}{Several specific integrals, see Eq. \eqref{eq:I_integrals_def}}{*}
        \symbline{\flux{i}{}}{Mass flux of species $i$}{\mol\per\square\meter\per\second}
        \symbline{\flux{q}{}}{Measurable heat flux}{\watt\per\square\meter}
        \symbline{J}{Several specific integrals, see Eq. \eqref{eq:def_J_integral}}{\per\meter}
        \symbline{k_B}{Boltzmann constant}{\joule\per\kelvin}
        \symbline{k_T}{Thermal diffusion ratio}{-}
        \symbline{k_{ij}, \ell_{ij}}{Modified Lorentz-Berthelot mixing rule parameters}{-}
    \bottomrule
    \multicolumn{3}{l}{* Various units}
\end{tabular}

\newpage

\begin{tabular}{c l c}
    \multicolumn{3}{c}{Continued} \\
    \toprule
    Symbol & Description & Unit \\
    \midrule
        \symbline{L}{Onsager phenomenological coefficient}{*}
        \symbline{\mathcal{L}}{Lagrange target function}{*}
        \symbline{l_p}{Sonine polynomial expansion coefficient, see Eq. \eqref{eq:kin_D_poly_expans}}{\meter}
        \symbline{M_i}{$m_i / m_0$}{-}
        $m_i$ & Molar or particle mass of species $i$ & \si{\gram\per\mol} or \si{\kilo\gram} \\
        \symbline{m_0}{Sum of particle masses}{\kilo\gram}
        & & \\
                            & Number of species & \si{-} \\
        $N$& Order of approximation & \si{-} \\
                            & Number of intervals & \si{-} \\
        & & \\
        \symbline{N_A}{Avogadros number}{\per\mol}
        \symbline{N_x,N_y}{Multiple of minimum integration step, $\delta_x$, $\delta_y$}{-}
        \symbline{n}{Total moles in system}{\mol}
        \symbline{n_i}{Moles of species $i$}{\mol}
        \symbline{\vn}{Vector of all mole numbers in system}{\mol}
        \symbline{p}{Pressure}{\pascal}
        \symbline{Q}{Canonical partition function}{-}
        & & \\
        \multirow{2}{*}{$R$} & Distance of closes approach between particles, see Figure \ref{fig:kin_dyn_closest_approach} & \si{\meter}\\
                            & Universal gas constant & \si{\joule\per\mol\kelvin} \\
        & & \\
        \symbline{\Vec{r}}{Positional vector}{\meter}
        \symbline{r}{Polar coordinate radial position}{\meter}
        \symbline{S}{Entropy}{\joule\per\kelvin}
        \symbline{S_T}{Soret coefficient}{\per\kelvin}
        \symbline{S_m^{(n)}}{Sonine polynomial}{*}
        \symbline{\mathcal{S}}{Integration domain}{-}
        \symbline{\mathcal{S}^*}{Discretised integration domain}{-}
        \symbline{\mathcal{S}_{ij}^*}{Continuous subdomain of $\mathcal{S}^*$, see Eq. \eqref{eq:integration_domains}}{-}
        \symbline{\mathcal{S}_{ij}^{\Delta}}{Discretisation of $\mathcal{S}_{ij}^*$, see Eq. \eqref{eq:integration_domains}}{-}
        \symbline{\mathcal{S}_{ij}^{x}, \mathcal{S}_{ij}^{xy}, \mathcal{S}_{ij}^{y}}{Specific domain mappings, see Eq. \eqref{eq:integration_refine_maps}}{-}
        \symbline{T}{Thermodynamic temperature}{\kelvin}
        \symbline{t}{Time}{\second}
        \symbline{U_i}{Peculiar speed of particles of species $i$}{\meter\per\second}
        \symbline{\vU_i}{Peculiar velocity of particles of species $i$, $\vu_i - \vu^n$}{\meter\per\second}
        \symbline{\Bar{U}_i}{Average peculiar speed of particles of species $i$}{\meter\per\second}
        \symbline{\sU}{Dimensionless peculiar speed of particles of species $i$}{-}
        \symbline{\vu^n}{Mole average velocity of system}{\meter\per\second}
        \symbline{u_i}{Speed of particles of species $i$}{\meter\per\second}
        \symbline{\Bar{u}_i}{Average speed of particles of species $i$}{\meter\per\second}
        \symbline{\vu_i}{Velocity of particles of species $i$}{\meter\per\second}
        \symbline{u_{p-HS}}{Pseudo hard-sphere potential, defined in Eq. \eqref{eq:pseudo_hs_potential}}{\joule}
        \symbline{u^{Mie}}{Mie potential, defined in Eq. \eqref{eq:def_mie_potential}}{\joule}
        \symbline{V}{Volume}{\cubic\meter}
        \symbline{v}{Molar volume}{\cubic\meter\per\mol}
    \bottomrule
    \multicolumn{3}{l}{* Various units}
\end{tabular}

\newpage

\begin{tabular}{c l c}
    \multicolumn{3}{c}{Continued}\\
    \toprule
        Symbol & Description & Unit \\
    \midrule
        \symbline{v_i}{Partial molar volume of species $i$}{\cubic\meter\per\mol}
        \symbline{W_{ij}^{(\ell)}(r)}{Dimensionless collision integral, see Eq. \eqref{eq:avslutter_2}}{-}
        \symbline{x_i}{Mole fraction of species $i$}{-}
        \symbline{\vx}{Vector of all mole fractions in the system}{-}
        \symbline{x, y, z}{Cartesian coordinates}{-}
        \symbline{Z}{Compressibility factor}{-}
        \symbline{\alpha_T}{Thermal diffusion factor}{-}
        $\alpha_q$ & Value of specific integrals, see Eq. \eqref{eq:alpha_q_definition} & m$^4$s$^{-1}$ \\
        \symbline{\Gamma}{Gamma function}{-}
        \symbline{\gamma, \phi}{Lagrange multipliers}{-}
        $\delta_q$ & Value of specific integrals, see Eq. \eqref{eq:delta_q_definition} & m$^4$s$^{-1}$ \\
        $\delta_{ij}$ & Kronecker delta & \si{-} \\
        \symbline{\delta_x, \delta_y}{Minimum integration step}{-}
        & & \\
        \multirow{2}{*}{$\epsilon$} & Angular collision coordinate, see Figure \ref{fig:cyl_coord} & \si{-} \\
        & Small number, identified by sub- and superscripts & * \\
        & & \\
        \symbline{\varepsilon}{Mie potential well depth}{\joule}
        \symbline{\hat{\epsilon}}{Descriptor for deviation}{-}
        \symbline{\eta}{Packing fraction}{-}
        \symbline{\theta}{Angular collision coordinate, see Figure \ref{fig:encounter_geometry}}{-}
        \symbline{\Lambda}{Thermal response function, see Eq. \eqref{eq:thermal_response_def}}{\kelvin\second}
        \symbline{\vA}{Velocity distribution function response to $\nabla \ln T$}{\kelvin\meter}
        \symbline{\Tilde{\vA}}{$\vA - k_T \vD$}{}
        \symbline{\lambda}{Thermal conductivity}{\watt\per\meter\per\kelvin}
        \symbline{\hat{\lambda}}{Predicted conductivity}{\watt\per\meter\per\kelvin}
        \symbline{\lambda_a}{Mie potential attractive exponent}{-}
        \symbline{\lambda_r}{Mie potential repulsive exponent}{-}
        \symbline{\mu}{Chemical potential}{\joule\per\mol}
        $\rho$ & Molar density or particle density & \si{\mol\per\cubic\meter} or \si{\per\cubic\meter} \\
        \symbline{\rho_m}{Mass density}{\kilo\gram\per\cubic\meter}
        \symbline{\rho_P}{Pearson correlation coefficient}{-}
        \symbline{\sigma}{Mie potential parameter}{\meter}
        \symbline{\sigma^{HS}}{Hard sphere diameter}{\meter}
        \symbline{\Bar{\sigma}_{ij}}{Average distance of closest approach, see Eq. \eqref{eq:def:sigma_bar_ij}}{\meter}
        \symbline{\Bar{\sigma}}{Mixture average of $\Bar{\sigma}_{ij}$, see Eq. \eqref{eq:def_sigma_bar_avg}}{\meter}
        \symbline{\tau}{Kempers target function, see Eq. \eqref{eq:kempers_ig_target}}{\joule\per\kelvin}
        \symbline{\Phi}{Perturbation to the velocity distribution function}{-}
        \symbline{\varphi}{Arbitrary variable}{*}
        \symbline{\chi}{Deflection angle, see Figures \ref{fig:encounter_geometry} and \ref{fig:kin_dyn_closest_approach}}{-}
        \symbline{\rdf}{Density correction factor}{-}
        \symbline{\psi}{Arbitrary function}{*}
        \symbline{\Omega_{ij}^{(\ell)}(r), \Omega_{i}^{(\ell)}(r)}{Collision integral, see Eq. \eqref{eq:avslutter_2} and \eqref{eq:single_comp_collision_integral}}{\cubic\meter\per\second}
    \bottomrule
    \multicolumn{3}{l}{* Various units}
\end{tabular}

\newpage

\begin{tabular}{c l c}
    \multicolumn{3}{c}{Superscripts}\\
    \toprule
        Symbol & Description & Unit \\
    \midrule
        \symbline{CS}{Carnahan-Starling value}{-}
        \symbline{E}{Enskog value}{-}
        \symbline{HS}{Hard sphere}{-}
        \symbline{ig}{Ideal gas state}{-}
        \symbline{\alpha, \beta}{Bulb indices}{-}
        \symbline{*}{Reduced state point}{-}
        \symbline{\infty}{Value at infinite dilution}{-}
    \midrule
    \midrule
        \multicolumn{3}{c}{Operators} \\
    \midrule
        \symbline{\text{Cov}[\varphi_1,\varphi_2]}{Covariance of $\varphi_1$ and $\varphi_2$}{*}
        \symbline{\sD}{Differential operator, see Section \ref{subsubsec:boltzmann_eq}}{-}
        \symbline{\text{Var}[\varphi]}{Variance of $\varphi$}{*}
        \symbline{\DeltaAB}{Difference between bulbs}{*}
        \symbline{\DeltaIG}{Difference between real and ideal gas state}{*}
        \symbline{\DeltaHS}{Difference between real and hard sphere state}{*}
        \symbline{\Delta^2_x}{Numeric second derivative along $x$-coordinate}{*}
        \symbline{\Delta^2_y}{Numeric second derivative along $y$-coordinate}{*}
        \symbline{\Delta^2_{xy}}{Numeric second cross derivative}{*}
        \symbline{\nabla}{Spatial gradient}{\per\meter}
        \symbline{\nabla_T}{Spatial gradient at constant temperature}{\per\meter}
    \bottomrule
    \multicolumn{3}{l}{* Various units}
\end{tabular}

\begin{tabular}{c l l l}
    \multicolumn{4}{c}{Multiply defined symbols}\\
    \toprule
    & \multicolumn{3}{c}{Context and definition}\\
    Symbol & Thermodynamics & Kinetic gas theory & Numerics\\
    \midrule
        A & Helmholtz energy & - & Interpolating coefficient \\
        I & - & Specific integrals & Numeric integral \\
        N & Number of species & Order of approximation & Number of intervals \\
        $R$ & Gas constant & Distance of closest approach& - \\
        $x$ & Mole fraction & Mole fraction & Cartesian coordinate axis \\
        $\epsilon$ & - & Angular coordinate & Small scalar \\
        $\rho$ & Molar density & Particle density & - \\
    \bottomrule
\end{tabular}

\begin{tabular}{c l}
    \multicolumn{2}{c}{Abbreviations}\\
    \toprule
    Abbreviation & Definition\\
    \midrule
        AAD & Average absolute deviation\\
        AD & Average deviation \\
        BH & Barker-Henderson\\
        CoV & Centre of volume\\
        CoM & Centre of mass \\
        CS & Carnahan-Starling\\
        EoS & Equation of State\\
        FoR & Frame of Reference\\
        HS & Hard-sphere\\
        LB & Lorenz-Berthelot\\
        MD & Molecular dynamics \\
        NDP & Number of data points \\
        rdf. & Radial distribution function\\
        SAFT & Statistical associating fluid theory\\
        vdf. & Velocity distribution function\\
    \bottomrule
\end{tabular}

\endgroup