I apologize for disagreeing with some of the comments above. The diffusion coefficient (D) is not a thermodynamic property. It is correct to call it  a transport property, but it really is a hybrid property that characterizes kinetic effects. 

If a system is out of equilibrium, then transport coefficients give the time course of how a system evolves... the "direction" of a process is determined by thermodynamics, but the rate at which the process occurs is kinetic in nature. In part, D characterizes the response of systems out of equilibrium (for instance, non-uniform concentration or temperatures) and how they evolve over time. Thermodynamic properties do contribute to the diffusion coefficient, but factors such as molecular arrangements and activation energies (which are related to kinetics, not thermodynamics) also contribute.

For instance, D depends on the temperature. A simple example is the Einstein-Smoluchowski equation, which explicitly depends on the temperature. However, this explicit dependence is due to the relationship between the temperature and the molecular kinetic energy of the diffusing material. There is also an implicit exponential temperature dependence due to the viscosity term, which is used to characterize the influence of the medium on the diffusing species migration.

That being said, non-equilibrium thermodynamics and relaxation thermodynamic models, etc., include time in the formulation (equilibrium thermo never talks about time), and relates the transport processes to entropy generation, etc.  While these models are appealing theoretically because they give insight to the inner workings of existing diffusion equations, they add little to practical experiments or predictions compared to equations such as Fick's 1st and 2nd laws (or Fourirers laws for heat conduction), among others. 

It is interesting to me that diffusion can be modeled using activation energies in most cases. (The notable exception is diffusion in gases for which molecular interactions provide negligible energy barriers to molecular migration compared to the molecular kinetic energy). These activation energies are the result of a combination of properties of the materials involved and the molecular arrangements. (Even in solids, these arrangements are not truly constant due to vibrations, etc., but the idea still applies.)

Also, diffusion has often been modeled probabilistically using arguments related to random walks, etc. This works because of the Boltzmann relationship between energies and probabilities, and the large number of molecules involved, even when the diffusing species is present in a low concentration in another medium.  For other systems that are rarefied (including gas phase diffusion), other models apply which may be more molecularly based apply. (In those cases, the arguments are based on kinetic theory, and macroscopic properties are obtained by averaging.)I apologize for the long answer... it is a great question on a fun topic!

The diffusion coefficient is a transport property!

The "diffusion coefficient" is a thermodynamical property and lead to describe motion of species in media

As it was mentioned, it is a transport property. If you consider an isotherm system (e.g. an NVT-ensemble), then it is constant and therefore not a dynamical property. Since you can define it (by Stokes-Einstein equation) as a function of temperature, in general, it depends on thermodynamical state of the system.

Diffusion Coefficient is a non-equilibrium thermodynamics property. As many other transport properties, it can be qualitatively correlated with the residual entropy.

Diffusion coefficient is a thermodynamic property. Is related to the random movement (Brownian). The mean square displacement conducting particles Brownian motion has a resultant proportional to the time constant of proportionality is the diffusion coefficient. Einstein proposed a mathematical relationship where the average displacement is equal to the square root of twice the diffusion coefficient times time.

The diffusion coefficient is a very important thermodynamic property in colloidal systems, if you put a colloidal particle in a system, for example water, and measure the diffusion coefficient and then you add a protein together and measures the diffusion coefficient and this coefficient is the same it is a proof that the protein does not interact with the particle, but if the diffusion coefficient change, it can be said that there is interaction between the two. The diffusion coefficient is one of the most sensitive measures to determine whether there exists intermolecular interactions

I apologize for disagreeing with some of the comments above. The diffusion coefficient (D) is not a thermodynamic property. It is correct to call it  a transport property, but it really is a hybrid property that characterizes kinetic effects. 

If a system is out of equilibrium, then transport coefficients give the time course of how a system evolves... the "direction" of a process is determined by thermodynamics, but the rate at which the process occurs is kinetic in nature. In part, D characterizes the response of systems out of equilibrium (for instance, non-uniform concentration or temperatures) and how they evolve over time. Thermodynamic properties do contribute to the diffusion coefficient, but factors such as molecular arrangements and activation energies (which are related to kinetics, not thermodynamics) also contribute.

For instance, D depends on the temperature. A simple example is the Einstein-Smoluchowski equation, which explicitly depends on the temperature. However, this explicit dependence is due to the relationship between the temperature and the molecular kinetic energy of the diffusing material. There is also an implicit exponential temperature dependence due to the viscosity term, which is used to characterize the influence of the medium on the diffusing species migration.

That being said, non-equilibrium thermodynamics and relaxation thermodynamic models, etc., include time in the formulation (equilibrium thermo never talks about time), and relates the transport processes to entropy generation, etc.  While these models are appealing theoretically because they give insight to the inner workings of existing diffusion equations, they add little to practical experiments or predictions compared to equations such as Fick's 1st and 2nd laws (or Fourirers laws for heat conduction), among others. 

It is interesting to me that diffusion can be modeled using activation energies in most cases. (The notable exception is diffusion in gases for which molecular interactions provide negligible energy barriers to molecular migration compared to the molecular kinetic energy). These activation energies are the result of a combination of properties of the materials involved and the molecular arrangements. (Even in solids, these arrangements are not truly constant due to vibrations, etc., but the idea still applies.)

Also, diffusion has often been modeled probabilistically using arguments related to random walks, etc. This works because of the Boltzmann relationship between energies and probabilities, and the large number of molecules involved, even when the diffusing species is present in a low concentration in another medium.  For other systems that are rarefied (including gas phase diffusion), other models apply which may be more molecularly based apply. (In those cases, the arguments are based on kinetic theory, and macroscopic properties are obtained by averaging.)I apologize for the long answer... it is a great question on a fun topic!

Thank you for detailed and interesting answers. I will defend my PhD thesis on February 4 and 4 years ago one of the referees asked me this question when I was defending my post graduate thesis. at that moment my answer was not satisfying but I think I have better ideas now!

Good luck with your defense!!

I disagree with some comments above. The diffusion coefficient (D) is not a thermodynamic property. Even if the temperature and molecular structure influe on the diffusion coeefficient. It is a dynamical property. Because you can talk about "Diffusion coefficient D" only if you have a motion (vibration, libration, rotation, translational diffusion, ...).

@Makha

I also think  that the diffusion coefficient is a dynamical quantity, but things are not so clear. However your explanation is not sufficient to say that it is a dynamical quantity. Also the temperature can be defined if exists motion like translation, vibration and so on, like all the thermodynamic quantities.You can define the diffusion coefficient even if there is no diffusion at all when the gradients are null. Moreover the diffusion coefficients are somewhat related to the viral coefficients (I am talking about gases).

I think it is a dynamical quantities because it appears when there is non-equilibrium and because it is determined by collisions.

First of all I should say that your question is a bit confusing. By the term "Diffusion" you mean self-diffusion (tracer diffusion) or Mutual diffusion (intrinsic diffusion)?

The very basic concept about the diffusion is the self-diffusion coefficient. Consider a water container in which every water molecule is under full thermodynamic equilibrium. Still each water molecule moves (thanks to its thermal or kinetic energy). This motions are random and after a sufficient time is almost equal (in the sense of mean square displacement) for all molecules. Essentially there is no preferential path for the molecules, but they still move around. This coefficient is a measure of mobility of the molecules in the system.

Now consider the same container of water in which we add one molecule of ethanol. If you track the motions of the ethanol, you can still define the self diffusion (better to say tracer diffusion) of ethanol in water. Note that ethanol molecule is experiencing an isotropic environment due to water molecules around.

In the last case, consider a container of water/benzene mixture well mixed at the first instant of time. When you let the system evolve, all molecules have a thermal motion and mobility based on their self diffusion. But what you see after a while, is separation of water and benzene. Why is that? this is solely because the random motions of the molecules has been biased with a thermodynamic (molecules are seeking a state with minimum of potential energy) factor. In the formulation of the mutual diffusion coefficient of A in B, both of these factor are considered. In a very simple case in the form of:

Dm=(xA*DsB+xB*DsA)*f

in this form the Ds means self diffusion of each molecule, Dm is the mutual diffusion coefficient and x is mole fraction of each component. The factor f is the derivative of the free energy respect to change of concentration. This is the thermodynamic factor which biases self (random) diffusion of A and B toward an specific direction.