Ambiguities in order-theoretic formulations of thermodynamics
  Since the 1909 work of Carath\'eodory, formulations of thermodynamics have gained ground which highlight the role of the the binary relation of adiabatic accessibility between equilibrium states. A feature of Carath\'eodory's system is that the version therein of the second law contains an ambiguity about the nature of irreversible adiabatic processes, making it weaker than the traditional Kelvin-Planck statement of the law. This paper attempts first to clarify the nature of this ambiguity, by defining the arrow of time in thermodynamics by way of the Equilibrium Principle ("Minus First Law"). It then argues that the ambiguity reappears in the important 1999 axiomatisation due to Lieb and Yngvason. 
Robert Marsland III, Harvey R. Brown and Giovanni Valente
physics.class-ph