Entropic forces have recently attracted considerable attention as ways to reformulate, retrodict, and perhaps even "explain" classical Newtonian gravity from a rather specific thermodynamic perspective. In this article I point out that if one wishes to reformulate classical Newtonian gravity in terms of an entropic force, then the fact that Newtonian gravity is described by a conservative force places significant constraints on the form of the entropy and temperature functions. (These constraints also apply to entropic reinterpretations of electromagnetism, and indeed to any conservative force derivable from a potential.) The constraints I will establish are sufficient to present real and significant problems for any reasonable variant of Verlinde's entropic gravity proposal, though for technical reasons the constraints established herein do not directly impact on either Jacobson's or Padmanabhan's versions of entropic gravity. In an attempt to resolve these issues, I will extend the usual notion of entropic force to multiple heat baths with multiple "temperatures" and multiple "entropies".

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

We characterize the variation functions of computable Lipschitz functions. We show that a real z is computably random if and only if every computable Lipschitz function is differentiable at z. Beyond these principal results, we show that a real z is Schnorr random if and only if every Lipschitz function with L1-computable derivative is differentiable at z.

Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically, and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. However, we show that in general one cannot compute conditional probabilities. Specifically, we construct a pair of computable random variables (X, Y) in the unit interval whose conditional distribution P[Y|X] encodes the halting problem.Nevertheless, probabilistic inference has proven remarkably successful in practice, even in infinite-dimensional continuous settings. We prove several results giving general conditions under which conditional distributions are computable. In the discrete or dominated setting, under suitable computability hypotheses, conditional distributions are computable. Likewise, conditioning is a computable operation in the presence of certain additional structure, such as independent absolutely continuous noise.