The problem is posed of the prescription of the so-called Boltzmann–Grad limit operator ) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator \, the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente that there is “no time-asymmetric ingredient” in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The (...) proof is couched on the “ab initio” axiomatic approach to the classical statistical mechanics recently developed. Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out. (shrink)

Information, entropy, probability: these three terms are closely interconnected in the prevalent understanding of statistical mechanics, both when this field is taught to students at an introductory level and in advanced research into the field’s foundations. This paper examines the interconnection between these three notions in light of recent research in the foundations of statistical mechanics. It disentangles these concepts and highlights their differences, at the same time explaining why they came to be so closely linked in the literature. In (...) the literature the term ‘information’ is often linked to entropy and probability in discussions of Maxwell’s Demon and its attempted exorcism by the Landauer-Bennett thesis, and in analyses of the spin echo experiments. The direction taken in the present paper is a different one. Here we discuss the statistical mechanical underpinning of the notions of probability and entropy, and this constructive approach shows that information plays no fundamental role in these concepts, although it can be conveniently used in a sense that we shall specify. (shrink)

Statistical mechanics is the name of the ongoing attempt to explain and predict certain phenomena, above all those described by thermodynamics on the basis of the fundamental theories of physics, in particular mechanics, together with certain auxiliary assumptions. In another paper in this journal, Foundations of statistical mechanics: Mechanics by itself, I have shown that some of the thermodynamic regularities, including the probabilistic ones, can be described in terms of mechanics by itself. But in order to prove those regularities, in (...) particular the time asymmetric ones, it is necessary to add to mechanics assumptions of three kinds, all of which are under debate in contemporary literature. One kind of assumptions concerns measure and probability, and here, a major debate is around the notion of “typicality.” A second assumption concerns initial conditions, and here, the debate is about the nature and status of the so-called past hypothesis. The third kind of assumptions concerns the dynamics, and here, the possibility and significance of “Maxwell's Demon” is the main topic of discussions. This article describes these assumptions and examines the justification for introducing them, emphasizing the contemporary debates around them. (shrink)

This article highlights the limitations of typicality accounts of thermodynamic behavior so as to promote an alternative line of research: understanding and accounting for the success of the techniques and equations physicists use to model the behavior of systems that begin away from equilibrium. This article also takes steps in this promising direction. It examines a technique commonly used to model the behavior of an important kind of system: a Brownian particle that has been introduced to an isolated fluid at (...) equilibrium. It also accounts for the success of the model, by identifying and grounding the technique’s key assumptions. (shrink)

Can the second law of thermodynamics explain our mental experience of the direction of time? According to an influential approach, the past hypothesis of universal low entropy also explains how the psychological arrow comes about. We argue that although this approach has many attractive features, it cannot explain the psychological arrow after all. In particular, we show that the past hypothesis is neither necessary nor sufficient to explain the psychological arrow on the basis of current physics. We propose two necessary (...) conditions on the workings of the brain that any account of the psychological arrow of time must satisfy. And we propose a new reductive physical account of the psychological arrow of time compatible with time-symmetric physics, according to which these two conditions are also sufficient. Our proposal has some radical implications, for example, that the psychological arrow of time is fundamental, whereas the temporal direction of entropy increase in the second law of thermodynamics and the past hypothesis is derived from it, rather than the other way around. 1Introduction2A Physical Account of the Psychological Arrow of Time3Necessary Conditions for the Psychological Arrow of Time4The Psychological Arrow of Time via the Second Law of Thermodynamics and the Past Hypothesis5Why the Past Hypothesis Is Insufficient for the Psychological Arrow of Time5.1Stage 1, Version 1: From the past hypothesis to the psychological arrow via typicality5.2Stage 1, Version 2: From the past hypothesis to the psychological arrow via dynamical or causal considerations5.3Stage 2: From entropy gradient in the brain to the psychological arrow of time6Why the Past Hypothesis Is Unnecessary for the Psychological Arrow of Time7A New Reductive Account of the Psychological Arrow of Time. (shrink)

Philosophers of physics have long debated whether the Past State of low entropy of our universe calls for explanation. What is meant by “calls for explanation”? In this article we analyze this notion, distinguishing between several possible meanings that may be attached to it. Taking the debate around the Past State as a case study, we show how our analysis of what “calling for explanation” might mean can contribute to clarifying the debate and perhaps to settling it, thus demonstrating the (...) fruitfulness of this analysis. Applying our analysis, we show that two main opponents in this debate, Huw Price and Craig Callender, are, for the most part, talking past each other rather than disagreeing, as they employ different notions of “calling for explanation”. We then proceed to show how answering the different questions that arise out of the different meanings of “calling for explanation” can result in clarifying the problems at hand and thus, hopefully, to solving them. (shrink)

Uffink and Valente claim that there is no time-asymmetric ingredient that, added to the Hamiltonian equations of motion, allows to obtain the Boltzmann equation within the Lanford’s derivation. This paper is a discussion and a reply to that analysis. More specifically, I focus on two mathematical tools used in this derivation, viz. the Boltzmann–Grad limit and the incoming configurations. Although none of them are time-asymmetric ingredients, by themselves, I claim that the use of incoming configurations, as taken within the B–G (...) limit, is such a time-asymmetric ingredient. Accordingly, this leads to reconsider a kind of Stoßzahlansatz within Lanford’s derivation. (shrink)

Can Brownian motion arise from a deterministic system of particles? This paper addresses this question by analysing the derivation of Brownian motion as the limit of a deterministic hard-spheres gas with Lanford’s theorem. In particular, we examine the role of the Boltzmann-Grad limit in the loss of memory of the deterministic system and compare this derivation and the derivation of Brownian motion with the Langevin equation.

This thesis makes the issue of reconciling the existence of thermodynamically irreversible processes with underlying reversible dynamics clear, so as to help explain what philosophers mean when they say that an aim of nonequilibrium statistical mechanics is to underpin aspects of thermodynamics. Many of the leading attempts to reconcile the existence of thermodynamically irreversible processes with underlying reversible dynamics proceed by way of discussions that attempt to underpin the following qualitative facts: (i) that isolated macroscopic systems that begin away from (...) equilibrium spontaneously approach equilibrium, and (ii) that they remain in equilibrium for incredibly long periods of time. These attempts standardly appeal to phase space considerations and notions of typicality. This thesis considers and evaluates leading typicality accounts, and, in particular, highlights their limitations. Importantly, these accounts do not underpin a large and important set of facts. They do not, for example, underpin facts about the rates in which systems approach equilibrium, or facts about the kinds of states they pass through on their way to equilibrium, or facts about fluctuation phenomena. To remedy these and other shortfalls, this thesis promotes an alternative, and arguably more important, line of research: understanding and accounting for the success of the techniques and equations physicists use to model the behaviour of systems that begin away from equilibrium. Accounting for their success would help underpin not just the qualitative facts the literature has focused on, but also many of the important quantitative facts that typicality accounts cannot. This thesis also takes steps in this promising direction. It outlines and examines a technique commonly used to model the behaviour of an interesting and important kind of system: a Brownian particle that's been introduced to an isolated homogeneous fluid at equilibrium. As this thesis highlights, the technique returns a wealth of quantitative and qualitative information. This thesis also attempts to account for the success of the model and technique, by identifying and grounding the technique's key assumptions. (shrink)

It was first suggested by Albert that the existence of real, physical non-unitarity at the quantum level would yield a complete explanation for the increase of entropy over time in macroscopic systems. An alternative understanding of the source of non-unitarity is presented herein, in terms of the Transactional Interpretation. The present model provides a specific physical justification for Boltzmann’s Stosszahlansatz, thereby changing its status from an ad hoc postulate to a theoretically grounded result, without requiring any change to the basic (...) quantum theory. In addition, it is argued that TI provides an elegant way of reconciling, via collapse, the time-reversible Liouville evolution with the time-irreversible evolution inherent in master equations. The present model is contrasted with the GRW ‘spontaneous collapse’ theory previously suggested for this purpose by Albert. (shrink)