PX284 - I2 - significance

considering two systems that can exchange particles, with temperatures and numbers of particles: left $(T_{1}, N_{1})$ and right $(T_{2}, N_{2})$
if $N = N_{1} + N_{2}$ is fixed:

\frac{d N_{2}}{d N_{1}} = - 1

the entropy will change on exchange of particles, and from the second law of thermodynamics it must increase:

d S = {(\frac{\partial S_{1}}{\partial N_{1}})}_{E, V} d N_{1} + {(\frac{\partial S_{2}}{\partial N_{2}})}_{E, V} d N_{2} \geq 0

{(\frac{\partial S}{\partial N})}_{U, V} = - \frac{μ}{T}

∴ d S = (\frac{μ_{1}}{T_{1}} - \frac{μ_{2}}{T_{2}}) d N_{2} \geq 0

thus, if $T_{1} = T_{2}$ and $μ_{1} > μ_{2} : d N_{2} > 0$ and particles flow from left to right
flow will continue until equilibrium: $μ_{1} = μ_{2}$
ie. entropy causes particle flow