The Third Law of Thermodyamics
The first law of thermodynamics is usually associated with equilibrium systems expressing the difference between the inexact 1-form of heat, Q, and the inexact 1-form of Work, W, as a perfect differential. The statement is precisely a statement of topologicial cohomology theory.

The First Law: Q - W = dU.

However, the topological constraint of the first law is not limited to equilibrium systems (which have a topological dimension 2).

The Second Law has to do with entropy increase.

The third law is related to a conjecture that Absolute Zero is not achievable. In certain formulations, the idea of the third law states that by use of reversible processes, absolute zero cannot be reached in a finite number of steps. These ideas are not defined precisely by a mathematical formalism.

In this note a precise expression for the the Third Law of thermodynamics is given as another topological statement expressing a cohomological constraint, this time, the cohomology does not describe topological constraints on the 1-forms of Heat and Work, but on their 3-forms, Q^dQ and W^dW. Like the first law, the Third Law is valid for equilibirium and non-equilibrium systems, and for processes that are reversible or irreversiblw.

The Third Law: Q^dQ - W^dW = d(U dW).

If the system is restricted to be an equilibrium system, then Q^dQ = 0 and W^dW = 0 in order to insure that the Pfaff topological dimension is not more than 2.

If a process is reversible, then Q^dQ = 0.
Then third law implies that for reversible processes, W^dW is exact and equal to the exterior differential of the two form, U dW.

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Last update 01/23/2009
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