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A new derivation of the relationship between diffusion coefficient and entropy in classical Brownian motion by the ensemble method
by Yi Liao and XiaoBo Gong
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Authors (as registered SciPost users):  Yi Liao 
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Preprint Link:  scipost_202010_00022v2 (pdf) 
Date accepted:  20210510 
Date submitted:  20210313 17:42 
Submitted by:  Liao, Yi 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
The diffusion coefficienta measure of dissipation, and the entropya measure of fluctuation are found to be intimately correlated in many physical systems. Unlike the fluctuation dissipation theorem in linear response theory, the correlation is often strongly nonlinear. To understand this complex dependence, we consider the classical Brownian diffusion in this work. Under certain rational assumption, i.e. in the bicomponent fluid mixture, the mass of the Brownian particle $M$ is far greater than that of the bath molecule $m$, we can adopt the weakly couple limit. Only considering the firstorder approximation of the mass ratio $m/M$, we obtain a linear motion equation in the reference frame of the observer as a Brownian particle. Based on this equivalent equation, we get the Hamiltonian at equilibrium. Finally, using canonical ensemble method, we define a new entropy that is similar to the KolmogorovSinai entropy. Further, we present an analytic expression of the relationship between the diffusion coefficient $D$ and the entropy $S$ in the thermal equilibrium, that is to say, $D =\frac{\hbar}{eM} \exp{[S/(k_Bd)]}$, where $d$ is the dimension of the space, $k_B$ the Boltzmann constant, $\hbar $ the reduced Planck constant and $e$ the Euler number. This kind of scaling relation has been wellknown and welltested since the similar one for single component is firstly derived by Rosenfeld with the expansion of volume ratio.
Author comments upon resubmission
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 pointlike 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 KolmogorovSinai 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 KolmogorovSinai 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 KolmogorovSinai entropy and make it possible to calculate some more complex model. The KolmogorovSinai 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}
\begin{equation} \begin{aligned} &\frac{\partial}{\partial\beta }<\Phi>=<\Phi><H_{0}><\Phi H_{0}>\ &=<\Phi><U>+<\Phi><\Phi><\Phi>^{2}<\Phi U>\ &=<\Phi><\Phi><\Phi>^{2}\ \end{aligned} \end{equation}
\begin{equation} \begin{aligned} &<\phi(\textbf{x}\textbf{r}{i}) U(\textbf{r}\textbf{r}{j})> \ &=<\phi><U>(first ~integral ~with~\textbf{r}).\ \end{aligned} \end{equation} So, there is a quantitative measure of it possible but still hard if one can know the values of $<\Phi>$ and $<U>$.
\begin{itshape} 3. Finally, the organization of the paper is somewhat confusing. Earlier attempts (Eqs. 16 \& 17) should not come in the results section but should be moved to the introduction. \end{itshape}
\textbf{Answer:} We thank the referee for the very comprehensive suggestions. These suggestions contribute to improving the paper. We have turn the Eqs. 16 \& 17 into Eqs. 1 \& 2 in the Introduction in the revised manuscript.
\closing{Thanks and best regards}
List of changes
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 KolmogorovSinai entropy can be connected to the conventional thermodynamic entropy\cite{1998PhRvL..81..1762D}. The derivation of Eq.(\ref{eq:Hamiltonian}) based on the KolmogorovSinai 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 KolmogorovSinai 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.
Published as SciPost Phys. Core 4, 015 (2021)
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I have gone through the revised version of the manuscript and the authors' response to my earlier comments.
I think the manuscript have been considerably revised that it takes into account the earlier concerns raised. Therefore, I recommend publication.