I would like to know if when you consider a system in which you have Brownian motion if it is considered a system in equilibrium or far from equilibrium and why. i.e., is Brownian motion considered as something in equilibrium?
$\begingroup$ Are you interested in a system of particles who are all individually undergoing Brownian motion? Or are you asking about a system whose parameters undergo Brownian motion? Only under the first of these is equilibrium a well-defined concept. $\endgroup$– probably_someoneDec 11, 2017 at 15:51
$\begingroup$ I am interested in this case: [link] (en.wikipedia.org/wiki/Brownian_motion#/media/…) Here we have one particle (the yellow one) performing a Brownian motion. The smaller particles (in black) are also performing Brownian motion due to the collisions with each others but I am interest in studying the yellow particle. $\endgroup$– pipitaDec 11, 2017 at 16:01
It would be in equilibrium, in particular a thermal equilibrium. A thermal equilibrium is one in which the things in contact with each other exchange no "heat energy." The particles in Brownian motion as they crash into each other are exchanging kinetic (motion) energy all the time. But "heat" is a statistical property, which can be characterized by the average kinetic energy. The average can be taken over time, look at one particle in Brownian motion, and its kinetic energy goes up and down as it crashes into other particles, but on average it maintains a particular value proportional to the temperature of the fluid it is in. Or the average can be taken over particles, at any instant, the average kinetic energy of a bunch of particles is some value which does not change (except for a tiny bit which can always be reduced by averaging over more particles) over time.
In a thermal equilibrium, the details of motion of any particular piece are quite complex and even fascinating. The brilliance and purpose of thermodynamics is recognizing how many of those details can be ignored in what circumstances to still correctly characterize the motions of systems including vast numbers of particles.
Per the comments under the answer, we are concerned with the system consisting of a Brownian particle in contact with solvent molecules. The position and momentum of the particle seems to fluctuate randomly and constantly, but the average position of the particle at long enough times is stationary. Therefore, thermodynamically the particle is in mechanical equilibrium, and since this is the only concerning balance. The average velocity is a constant non-zero value at long times, too. So, the system attains a constant temperature; it is in thermal equilibrium with the solvent as a heat bath. The observed random motions, which in most cases is the concerning phenomena, is one of fluctuation near thermodynamic equilibrium. See https://en.wikipedia.org/wiki/Thermal_fluctuations for details.