  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.
