The diffusion coefficient--a measure of dissipation, and the entropy--a 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 non-linear. To understand this complex dependence, we consider the classical Brownian diffusion in this work. Under certain rational assumption, i.e. in the bi-component 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 first-order 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 Kolmogorov-Sinai 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 well-known and well-tested since the similar one for single component is firstly derived by Rosenfeld with the expansion of volume ratio.

中文翻译：

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经典布朗运动中扩散系数与熵关系的新推导

在许多物理系统中，扩散系数（耗散的度量）和熵（波动的度量）被发现密切相关。与线性响应理论中的波动耗散定理不同，相关性通常是强非线性的。为了理解这种复杂的依赖关系，我们考虑了这项工作中的经典布朗扩散。在一定的理性假设下，即在双组分流体混合物中，布朗粒子的质量$M$远大于浴分子的质量$m$，我们可以采用弱耦合极限。仅考虑质量比$m/M$的一阶近似，我们在观察者的参考系中获得了作为布朗粒子的线性运动方程。基于这个等价方程，我们得到平衡时的哈密顿量。最后，使用规范集成方法，我们定义了一个类似于 Kolmogorov-Sinai 熵的新熵。进一步，我们给出了热平衡中扩散系数 $D$ 和熵 $S$ 之间关系的解析表达式，即 $D =\frac{\hbar}{eM} \exp{[S /(k_Bd)]}$，其中 $d$ 是空间的维度，$k_B$ 是玻尔兹曼常数，$\hbar $ 是约化的普朗克常数，$e$ 是欧拉数。这种比例关系已广为人知并得到充分验证，因为单组分的类似比例关系首先由 Rosenfeld 随着体积比的扩展推导出来。$D =\frac{\hbar}{eM} \exp{[S/(k_Bd)]}$，其中 $d$ 是空间的维度，$k_B$ 是玻尔兹曼常数，$\hbar $ 是约化的普朗克常数和 $e$ 欧拉数。这种比例关系已广为人知并得到充分验证，因为单组分的类似比例关系首先由 Rosenfeld 随着体积比的扩展推导出来。$D =\frac{\hbar}{eM} \exp{[S/(k_Bd)]}$，其中 $d$ 是空间的维度，$k_B$ 是玻尔兹曼常数，$\hbar $ 是约化的普朗克常数和 $e$ 欧拉数。这种比例关系已广为人知并得到充分验证，因为单组分的类似比例关系首先由 Rosenfeld 随着体积比的扩展推导出来。