There are seemingly two independent formulations of the third law of thermodynamics both originally were stated by Walther Nernst. The first formulation is known as the Nernst heat theorem, and can be phrased as:
*The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.
The second formulation is  dynamical, known as the unattainability principle:
*It is impossible by any procedure, no matter how idealized, to reduce any assembly to absolute zero temperature in a finite number of operations.
At steady state the second law of thermodynamics implies that the total entropy production is non-negative.
When the cold bath approaches the absolute zero temperature,
it is necessary to eliminate the entropy production divergence at the cold side
when , therefore
For  the fulfillment of the second law depends on the entropy production of the other baths, 
which should compensate for the negative entropy production of the cold bath. 
The first formulation of the third law modifies this restriction. 
Instead of  the third law imposes , 
guaranteeing that at  absolute zero the entropy production at the cold bath is zero: . 
This requirement leads to the scaling condition of the heat current .
The second formulation, known as the unattainability principle can be rephrased as;
*No refrigerator can cool a system to absolute zero temperature at finite time.
The dynamics of the cooling process is governed by the equation
where  is the heat capacity of the bath. Taking  and  with , we can quantify this formulation by evaluating the characteristic exponent  of the cooling process,
This equation introduce the relation between the characteristic exponents   and . When  then the bath is cooled to zero temperature in a finite time, which implies a violation of the third law. It is apparent from the last equation, that the unattainability  principle is more restrictive than the Nernst heat theorem.