text: A definition of thermodynamic entropy can be based entirely on certain properties of the relation  of adiabatic accessibility that are taken as axioms in the Lieb-Yngvason approach. In the following list of properties of the  operator, a system is represented by a capital letter, e.g. X, Y or Z. A system X whose extensive parameters are multiplied by   is written . (e.g. for a simple gas, this would mean twice the amount of gas in twice the volume, at the same pressure.)  A system consisting of two subsystems X and Y is written (X,Y). If  and  are both true, then each system can access the other and the transformation taking one into the other is reversible. This is an equivalence relationship written . Otherwise, it is irreversible. Adiabatic accessibility has the following properties:
  *Reflexivity: 
  *Transitivity: If  and  then 
  *Consistency: if  and  then 
  *Scaling Invariance: if  and  then 
  *Splitting and Recombination:  for all 
  *Stability: if  then 
  The entropy has the property that  if and only if  and  if and only if  in accord with the Second Law. If we choose two states  and  such that  and assign entropies 0 and 1 respectively to them, then the entropy of a state X where  is defined as: