## 2 Theoretical Approach
### Correlation of the Fluctuating Diffusion Fluxes  
The generalized fluxes can formally be expressed by [1; 14]  
\[\dot{x}_{\eta}=-\sum_{\xi}\gamma_{\eta\xi}X_{\xi}+y_{\eta}. \tag{7}\]  
The indices \(\eta\) and \(\xi\) correspond to various transport processes in all three spatial directions that may be present in the solution, where the fluctuating fluxes correlations are given by [14; 15]  
\[\langle y_{\eta}(t_{1})y_{\xi}(t_{2})\rangle=(\gamma_{\eta\xi}+\gamma_{\zeta\eta })\delta(t_{1}-t_{2}). \tag{8}\]  
The angular brackets \(\langle\dots\rangle\) denote ensemble averaging [12; 14]. Note that \(\langle y_{\eta}(t_{1})\rangle=0\). We can write the \(\eta\)-component of Equation (6) in the form  
\[i_{\eta}=-\sum_{\xi}\biggl{(}\frac{\alpha k_{B}T}{\Delta V}\delta_{\eta\xi} \biggr{)}\biggl{(}\frac{1}{k_{B}T}\frac{\partial\mu}{\partial r_{\xi}}\Delta V \biggr{)}+\delta i_{\eta} \tag{9}\]  
A comparison of Equation (9) with Equation (7) leads to the following expressions for the coefficient \(\gamma_{\eta\xi}\) and the fluctuating flux \(\delta i_{\eta}\)  
\[\gamma_{\eta\xi}=\frac{\alpha k_{B}T}{\Delta V}\delta_{\eta\xi},\;y_{\eta}= \delta i_{\eta}, \tag{10}\]  
where again \(\delta i_{\eta}\) denotes the fluctuation of the \(\eta\) component of the mass flux. The symbol \(\delta_{\eta\xi}\) is the Kronecker delta [16]. Combining Equations (7)-(10) and noting that \(\gamma_{\eta\xi}=\gamma_{\zeta\eta}\)[15] allows obtaining  
\[\langle\delta i_{\eta}(\mathbf{r}_{1},t_{1})\delta i_{\xi}(\mathbf{r}_{2},t_{2 })\rangle=\frac{2\alpha k_{B}T}{\Delta V}\delta_{\eta\xi}\delta(t_{1}-t_{2}). \tag{11}\]  
Taking the limit \(\Delta V\to 0\) transforms Equation (11) into  
\[\langle\delta i_{\eta}(\mathbf{r}_{1},t_{1})\delta i_{\xi}(\mathbf{r}_{2},t_{ 2})\rangle=2\alpha k_{B}T\delta_{\eta\xi}\delta(t_{1}-t_{2})\delta(\mathbf{r} _{1}-\mathbf{r}_{2}). \tag{12}\]  
Let us consider a two-component system where component \(1\) is the solute and component \(2\) is the solvent. The fundamental thermodynamic equation for the internal energy \(E\) reads [12; 14]  
\[dE=TdS-pdV+\mu_{1}dN_{1}+\mu_{2}dN_{2} \tag{13}\]  
where \(T\) is temperature, \(p\) is pressure, \(V\) is volume and \(N_{1}\) and \(N_{2}\) are the number of molecules of type \(1\) and \(2\). The chemical potentials of the solute and solvent read  
\[\mu_{1}=\mu_{1}^{0}+k_{B}T\ln N_{1},\;\text{and}\;\mu_{2}=\mu_{2}^{0}+k_{B}T \ln N_{2}. \tag{14}\]  
If the corresponding molecular masses are \(m_{1}\) and \(m_{2}\), then we can write total mass as  
\[M_{t}=N_{1}m_{1}+N_{2}m_{2} \tag{15}\]  
Combining Equations (13) and (15), and introducing a new variable \(c=N_{1}m_{1}\), leads to  
\[dE=TdS-pdV+\biggl{(}\frac{\mu_{1}}{m_{1}}-\frac{\mu_{2}}{m_{2}}\biggr{)}dc. \tag{16}\]  
The diffusion coefficient for component \(1\) is defined as  
\[\langle y_{\eta}(t_{1})y_{\xi}(t_{2})\rangle=(\gamma_{\eta\xi}+\gamma_{\zeta\eta })\delta(t_{1}-t_{2}). \tag{8}\]0where \(\rho\) is the solution density (see Equations (13) and (16))  
\[\langle y_{\eta}(t_{1})y_{\xi}(t_{2})\rangle=(\gamma_{\eta\xi}+\gamma_{\zeta\eta })\delta(t_{1}-t_{2}). \tag{8}\]1  
and \(n_{1}\) is the mass fraction of component 1, or  
\[\langle y_{\eta}(t_{1})y_{\xi}(t_{2})\rangle=(\gamma_{\eta\xi}+\gamma_{\zeta\eta })\delta(t_{1}-t_{2}). \tag{8}\]2  
Assuming \(n_{1}\ll 1\) (or \(N_{1}\ll N_{t}\), where \(N_{t}\) is the total number of molecules), and taking the derivative of the combined chemical potential \(\mu=\mu_{1}/m_{1}-\mu_{2}/m_{2}\) with respect to \(N_{1}\) leads to  
\[\langle y_{\eta}(t_{1})y_{\xi}(t_{2})\rangle=(\gamma_{\eta\xi}+\gamma_{\zeta\eta })\delta(t_{1}-t_{2}). \tag{8}\]3  
where \(c_{1}=N_{1}/V\) is the number concentration of component 1. Solving for \(\alpha\) results in  
\[\langle y_{\eta}(t_{1})y_{\xi}(t_{2})\rangle=(\gamma_{\eta\xi}+\gamma_{\zeta\eta })\delta(t_{1}-t_{2}). \tag{8}\]4  
It is helpful to introduce fluctuation fluxes per unit mass as \(\delta\mathbf{j}=\delta\mathbf{i}/m_{1}\) and \(\delta\hat{j}_{\eta}=\delta\hat{i}_{\eta}/m_{1}\).