A new derivation of the relationship between diffusion coefficient and entropy in classical Brownian motion by the ensemble method

Thanks for your kind considerations and referee’s detailed comments and suggestions to improve this work.

We thank the referee's approvement. All the revised parts are written in bold.

~~~~\textbf{Point by point: Reply to the referee comment}\ \ \begin{itshape} The authors report a derivation of a relation between entropy and diffusion coefficient of Brownian point-like particles using Canonical ensemble.

This is a well studied problem, as the authors have said. Similar scaling relations are mentioned in Eqs. 16 \& 17. But they have proposed a new way in deriving such scaling, which could be interesting.

Before recommending, I expect the authors to address the following points: \end{itshape}

\textbf{Answer:} We thank the referee's approvement. In the revised paper, we have performed more details to address the three points based on the referee's suggestions.

\begin{itshape} 1. What is, if any, the advantage of this method over the other methods? Is there any experimental comparison where this method seem to work better? \end{itshape}

\textbf{Answer:} The derivation of Eq.(13) based on the Kolmogorov-Sinai entropy would be showed in APPENDIX A. in APPENDIX B, the formula of the thermodynamic entropy of Brownian particle is derived, but it is hard to analytically solve. Fortunately, Dzugutov et al. have point that Kolmogorov-Sinai entropy, when expressed in terms of the atomic collision frequency, is uniquely related to the thermodynamic excess entropy by a universal linear scaling law\footnote{Dzugutov M., Aurell E., and Vulpiani A. \ 1998, Phys. Rev. Lett. 81, 1762.}. The linear law is not influence the exponential relationship between the diffusion coefficient and the entropy. Our method can give the analytic formula of Kolmogorov-Sinai entropy and make it possible to calculate some more complex model. The Kolmogorov-Sinai entropy is regarded as a measure, for the loss of information about the state of the system, per unit of time. This quantity is more mathematical than physical, so there are not any experimental comparison.

\begin{itshape} 2. The only comparison made so far is in the case of hard sphere model (Eq. 18), where it is argued that for massive Brownian particles (compared to that of the bath molecules), the expressions of entropies are comparable. Is there a quantitative measure of it possible (say, with typical parameter values)?

\end{itshape} One can assume that a system labelled as System $1$ with the volume $V$, only includes $N$ bath particles,which Hamiltonian reads, \begin{equation} \begin{aligned} H= \frac{\textbf{p}^{N}\cdot\textbf{p}^{N}}{2m}+U(\textbf{r}^{N}). \end{aligned} \end{equation} The partition function of this system under canonical ensemble is \begin{equation} \begin{aligned} Z_{1}=\frac{1}{N!h^{dN}}\int e^{-\beta H}d\textbf{p}{1}... d\textbf{p}d\textbf{r}{1}...d\textbf{r}.\ \end{aligned} \end{equation} When one introduces a heavier Brownian particle to join in the system, it is labelled as System $2$, which partition function is \begin{equation} \begin{aligned} Z_{2}=\frac{1}{N!h^{dN}h^{d}}\int e^{-\beta H_{s}}d\textbf{p}{1}... d\textbf{p}d\textbf{r}{1}...d\textbf{r}d\textbf{p}d\textbf{x} \end{aligned} \end{equation} one can define the entropy of Brownian particle which equals the difference of entropy of System 2 and System 1. One can obtain $\Delta \ln Z\equiv \ln Z_{2}-\ln Z_{1}$, based on the formula of the thermodynamic entropy, the thermodynamic entropy of Brownian particle $S_{T}$ reads \begin{equation} \begin{aligned} S_{T}=\frac{kd}{2}[\ln(\frac{2\pi M}{h^{2}\beta})+1]- k\ln[\frac{<e^{\beta\Phi}>}{V}]+k\beta \frac{\partial}{\partial\beta }\ln(<e^{\beta\Phi}>). \label{eq:ss} \end{aligned} \end{equation} Because \begin{equation} \begin{aligned} <e^{A}>&=<1+A+\frac{1}{2}A^{2}+\frac{1}{6}A^{3}+...> \ &=e^{+\frac{1}{2}(<A^{2}>-^{2})+O(A^{3})}. \end{aligned} \end{equation}

All the revised parts are written in bold.
1. We have turn the Eqs. 16 \& 17 into Eqs. 1 \& 2 in the Introduction in the revised manuscript.

2. In section III，we supplement these sentences “{\bf Dzugutov, Aurell and Vulpiani have made the assumption that the Kolmogorov-Sinai entropy can be connected to the conventional thermodynamic entropy\cite{1998PhRvL..81..1762D}. The derivation of Eq.(\ref{eq:Hamiltonian}) based on the Kolmogorov-Sinai entropy would be showed in APPENDIX \ref{sect:Defin}. in APPENDIX \ref{sect:Therm}, the formula of the thermodynamic entropy of Brownian particle is derived, but it is hard to analytically solve. Fortunately, Dzugutov et al. have point that Kolmogorov-Sinai entropy, when expressed in terms of the atomic collision frequency, is uniquely related to the thermodynamic excess entropy by a universal linear scaling law\cite{1998PhRvL..81..1762D}. The linear law is not influence the exponential relationship between the diffusion coefficient and the entropy.}”

3. we add the sections of APPENDIX A and APPENDIX B.