Dissipative structures and the solidification

 

One of the problems which emerged primarily in the thermodynamics of irreversible processes is the classification of the dissipative structures. It is closely related to the question if there exists a general criterion for the development of the dissipative structure save the somewhat weak fluctuations based treatment. In short, some fluctuations will be due to the nonequilibrium damped, the other amplified and the fluctuation spectrum will be no more "white". As a consequence the amplified fluctuations - with the periods in the narrow time and space intervals will grow to the macroscopic dimensions and they are in the fact the emerging dissipative structure.

This is a very general view and it may be very cumbersome way to work out any particular dissipative structure save with the numerical simulation methods.

We shall try here also an other way for the criteria and for the estimation of the dissipative structure evolution.

 

Fluctuations

 

Here we show on a very classical way the connection of the fluctuating quantities with the transport equations. We derive it just to have the red line to the next chapter concerning the transport processes.

 

Consider the set of the macroscopic variables Aj, which fluctuate around the mean value A0j. It is assumed that they are dependent on a greater number of the macroscopic variables. they can be then treated as the random variables. So another random variables can be set as

Their distribution is assumed to correspond to the Gaussian distribution

where is the term

the deviation of the entropy from the equilibrium and k Boltzmann constant. The thermodynamic force Xi, conjugated to the variable 

is

Assumed is also the linear relation

Then is the entropy production

We have

and it follows

We assume also

and we get

then

and

 

where from

 

The last equation is nothing else as the equation analogous to the heat conduction equation. We assume that this relation will be also valid for any thermodynamic quantity, especially for the derivatives of the thermodynamic potentials. It is also obvious that the coupling coefficient Lij is a kinetic coefficient as the heat conduction, diffusion coefficient etc.

 

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