For an isolated system (a system that does not exchange energy or mass with its surroundings)

The Maximum Entropy Production Principle: Its Theoretical

Foundations and Applications to the Earth System

Statistical mechanics is the application of probability to physical theories. It explains the macroscopic

properties of systems such as temperature and pressure in terms of the microscopic arrangements of the

elements that comprise the system. Real world systems typically have a very large number of individual

molecules. For an isolated system (a system that does not exchange energy or mass with its surroundings)

at equilibrium, we should expect it to be in the most probable macroscopic state which corresponds to

the greatest number of different ways that the individual molecules can rearrange themselves. This

probabilistic feature becomes effectively a law when we deal with systems that have extremely large

numbers of individual elements.

Figure 1 gives a spatial demonstration of this probabilistic basis of

entropy during the evolution from an initial low entropy to maximum entropy state in a rigid box that

contains a number of gas molecules. It was Bolzmann who showed at thermodynamic equilibrium, one

can compute the entropy of a system with:

S = kB ln (1)

where kB is the Boltzmann constant and translates the microscopic energy of particles to the macroscopic

property of temperature and is the number of different microstates possible for a particular macrostate.

In Figure 1, the two macrostates have all the gas molecules in one side of the box or evenly distributed

throughout the box. As the number of molecules increases, the difference in for these two different

macrostates increases. When we deal with the number of molecules that would be contained within a

litre of air at room temperature and sea level pressure, the difference in entropy becomes extremely large

and the probability of all molecules being in one side of the box is so small as to be safely ignored. It

is the fact that many real world systems of interest are composed of a very large number of individual

elements that leads to the power of statistical mechanics.