Einstein Nonlocality, Space-Time Structure, and Thermodynamics

Abstract

The problem of a possible connection between the different arrows of time observed in physical phenomena has recently gained interest,^1-3 but has still to be considered as open. Moreover, the arrow most important for our everyday life—the thermodynamic one—is plagued by ambiguities in the statistical definition of entropy, which apparently has to be based on some observer-related concept of “coarse-graining” or “relevance”.^3,4 A feature common to all useful relevance concepts (formally expressed by idempotent operators in the space of phase-space densities or density matrices) is some kind of locality required if entropy is to be approximately additive,

$$ S\; = \;\int {\;{{\rm{d}}_3}{\rm{rs}}\left( {\rm{r}} \right)}$$

Additivity would be lacking in a pure ensemble definition of entropy

$$ {{\rm{S}}_{\rm{e}}}\; = \; - \int {{\rm{dp}}\;{\rm{dq}}\;\rho \left( {{\rm{p,q}}} \right)} \;\ln \;\rho \left( {{\rm{p,q}}} \right)$$

since ρ(p,q) necessarily contains correlations between different particles even over astronomical distances. They are usually explained as caused by interactions in the past. In particular, the additivity (or locality) of entropy neglects macroscopic correlations which are very relevant though of little numerical contribution.