Probabilities in Statistical Mechanics: What are they?

This paper addresses the question of how we should regard the probability distributions introduced into statistical mechanics. It will be argued that it is problematic to take them either as purely ontic, or purely epistemic. I will propose a third alternative: they are almost objective probabilities, or epistemic chances. The definition of such probabilities involves an interweaving of epistemic and physical considerations, and thus they cannot be classified as either purely epistemic or purely ontic. This conception, it will be argued, resolves some of the puzzles associated with statistical mechanical probabilities: it explains how probabilistic posits introduced on the basis of incomplete knowledge can yield testable predictions, and it also bypasses the problem of disastrous retrodictions, that is, the fact the standard equilibrium measures yield high probability of the system being in equilibrium in the recent past, even when we know otherwise. As the problem does not arise on the conception of probabilities considered here, there is no need to invoke a Past Hypothesis as a special posit to avoid it

Ruin probabilities with dependence on the number of claims within a fixed time window

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival time distributions depending on the claims that arrived within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the Bonus-Malus feature. We discuss first asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process. Via an appropriate change of measure, ruin probabilities could be computed to a closed form formulae. Additionally, we present simulated results via the importance sampling method, which further permit an in-depth analysis of a few concrete cases

Semiclassical transition probabilities by an asymptotic evaluation of the S matrix for elastic and inelastic collisions. Bessel uniform approximation

It has been observed in the past that the usual Airy uniform approximation gives probabilities greater than one, especially for near elastic collisions. By mapping the phase onto −ζ cos y + ky + A rather than (1∕3)y^3 − ζy + A one obtains a uniform approximation involving Bessel functions of the first kind, which approaches unity for the elastic collision. This Bessel uniform approximation is no more complicated than the Airy and also gives good agreement with exact quantum results, even if probabilities are large

Continuity properties of a factor of Markov chains

Starting from a Markov chain with a finite alphabet, we consider the chain obtained when all but one symbol are undistinguishable for the practitioner. We study necessary and sufficient conditions for this chain to have continuous transition probabilities with respect to the past

One-sided versus two-sided stochastic descriptions

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and Disordered Systems

A new algorithm for finding survival coefficients employed in reliability equations

Product reliabilities are predicted from past failure rates and reasonable estimate of future failure rates. Algorithm is used to calculate probability that product will function correctly. Algorithm sums the probabilities of each survival pattern and number of permutations for that pattern, over all possible ways in which product can survive

Quantum Pasts and the Utility of History

From data in the present we can predict the future and retrodict the past. These predictions and retrodictions are for histories -- most simply time sequences of events. Quantum mechanics gives probabilities for individual histories in a decoherent set of alternative histories. This paper discusses several issues connected with the distinction between prediction and retrodiction in quantum cosmology: the difference between classical and quantum retrodiction, the permanence of the past, why we predict the future but remember the past, the nature and utility of reconstructing the past(s), and information theoretic measures of the utility of history. (Talk presented at the Nobel Symposium: Modern Studies of Basic Quantum Concepts and Phenomena, Gimo, Sweden, June 13-17, 1997)Comment: 22pages, uses REVTEX 3.