**Harvey Brown**

**University of Oxford, Wolfson**

*Remarks on the arrow of time and the behaviour of entropy in (order-theoretic) axiomatisations of thermodynamics*

A notable feature of Carathéodory’s 1909 axiomatic version of thermodynamics, based on the binary relation (preorder) between states associated with adiabatic accessibility, is that his axioms lead to an account of irreversible processes in which entropy can either always decrease or always increase. In this talk I try to make sense of Carathéodory’s claim that resort to experiment is supposed to resolve this issue. I also address the question as to whether an analogous ambiguity in the behaviour of entropy lurks in the more recent (1999) axiomatisation of thermodynamics due to Lieb and Yngvason.

**Leah Henderson**

**Carnegie Mellon University**

*Can the second law be compatible with time reversal invariant dynamics?*

It is commonly thought that there is some tension between the second law of thermodynamics and the time reversal invariance of the microdynamics. This idea has been challenged by Jos Uffink, who argues that the relationship between the second law and time reversal invariance depends on the formulation of the second law. Uffink claims that a recent version of the second law due to Elliott Lieb and Jakob Yngvason allows irreversible processes, yet is time reversal invariant.

In this talk, I will first attempt to clarify the intuitive argument for incompatibility between thermodynamic irreversibility and time reversal invariant dynamics. I will then argue, contrary to Uffink, that Lieb and Yngvason’s version of the second law is not time reversal invariant.

**Jos Uffink**

**University of Minnesota**

*Entropy, Entanglement, Utility*

This talk explores a formal analogy between the study of entanglement in quantum theory, entropy in classical thermodynamics, and utility in decision theory. Roughly speaking, I will argue that in all three cases, the mathematical problem arises of finding and characterizing those functions that respect a given pre-ordering relation, subject to certain auxilliary conditions. Moreover, theorems have been obtained in these three separate areas that might be applied to them in common. It is my main purpose to draw attention to these analogies, and argue how they might be useful in thermodynamics and quantum theory.

**Jakob Yngvason**

**University of Vienna**

*The entropy concept for non-equilibrium states*

In earlier work, we presented a foundation for the second law of classical thermodynamics in terms of the entropy principle. More precisely, we provided an empirically accessible axiomatic derivation of an entropy function defined on all equilibrium states of all systems that has the appropriate additivity and scaling properties, and whose increase is a necessary and sufficient condition for an adiabatic process between two states to be possible. Here, after a brief review of this approach, we address the question of defining entropy for non-equilibrium states. Our conclusion is that it is generally not possible to find a unique entropy that has all relevant physical properties. We do show, however, that one can define two entropy functions, called S− and S+, which, taken together, delimit the range of adiabatic processes that can occur between non-equilibrium states. The concept of comparability of states with respect to adiabatic changes plays an important role in our reasoning. Reference: dx.doi.org/10.1098/rspa.2013.0408