Transport coefficients are mathematical constants that arise in differential equations in times and places that describe physical phenomena in the area of non-equilibrium statistical mechanics. These transport coefficients describe the behavior of the physical system when there is a deviation from equilibrium state. This deviation of the system from equilibrium can be caused due to an external force such as in the case of viscosity and mobility or it can be caused due to a gradient in density flow during a particular system change such as occurs in diffusion or in heat conduction.
In this article, I will discuss how physical phenomena that are characterized by coefficient transports are related to each other by a mathematical link such as the relationship between the diffusion coefficient and the mobility coefficient. The phenomena that is discussed here is the diffusion phenomena. All transport coefficients in thermodynamics include constants from the kinetic theory of gases, the most important of which is called the free mean path which has the symbol l. Its physical meaning is the mean path between collisions of gas particles.
The diffusion phenomena can be described using a mathematical differential equation which involves a change in the density of the gas according to the following equation:
J = - D(dn/dx)
This equation means that the number of particles that pass through a surface unit and in time unit is proportional to the first derivative of the particles number according to the displacement x.
Where D is called the diffusion coefficient and is related to constants from the kinetic theory of gases by the following equation:
Where v is the mean velocity of the gas particles and l is the mean free path, or the path between collisions of two gas particles.
The diffusion equation satisfies also the equation of continuity which states that:
dn/dt = (dn/dx)*2
This is so due to the conservation of particles that is obeyed by the particles of the gas. This equation is a classical equation that does not involve quantum mechanical effects. It is interesting that this equation is similar to the quantum mechanical differential equation that is known as the Schroedinger equation. It is first order derivative in time and second order derivative in displacement. This is exactly occurs in the Schroedinger equation.
One would postulate in this case to transform this classical equation into quantum mechanical operators equation that has momentum and energy operators instead of these undefined mathematical operators by multiplying the equation by certain constants. One then would investigate if there are physical systems that obey this quantum mechanical equation.
It is interesting that this quantum mechanical equation is followed also by the heat conduction phenomenon which has also similar differential form with different coefficients than those appearing in the diffusion equation. The heat conduction phenomenon also obeys the equation of continuity due to the conservation of heat energy.