Taking up statistical thermodynamics: Equilibrium fluctuations and irreversibility

The microscopic explanation of the physical phenomena represented by a macroscopic theory is often cast in terms of the reduction of the latter to a more fundamental theory, which represents the same phenomena at the microscopic level, albeit in an idealized way. In particular, the reduction of thermodynamics to statistical mechanics is a much discussed case-study in philosophy of physics. Based on the Generalized Nagel–Schaffner model, the alleged reductive explanation would be accomplished if one finds a corrected version of classical thermodynamics that can be strictly derived from statistical mechanics. That is the sense in which, according to Callender (1999, 2001), one should not take thermodynamics too seriously. Arguably, the sought-after revision is given by statistical thermodynamics, intended as a macroscopic theory equipped with a probabilistic law of equilibrium fluctuations. The present paper aims to evaluate this proposal. The upshot is that, while statistical thermodynamics enables one to re-define equilibrium so as to agree with Boltzmann entropy, it does not provide a definitive solution to the problem of explaining macroscopic irreversibility from a microscopic point of view.     

The Limits of Information

Black holes have their own thermodynamics including notions of entropy and temperature and versions of the three laws. After a light introduction to black hole physics, I recollect how black hole thermodynamics evolved in the 1970s, while at the same time stressing conceptual points which were given little thought at that time, such as why the entropy should be linear in the black hole's surface area. I also review a variety of attempts made over the years to provide a statistical mechanics for black hole thermodynamics. Finally, I discuss the origin of the information bounds for ordinary systems that have arisen as applications of black hole thermodynamics.     

The case for black hole thermodynamics part II: Statistical mechanics

I present in detail the case for regarding black hole thermodynamics as having a statistical-mechanical explanation in exact parallel with the statistical-mechanical explanation believed to underlie the thermodynamics of other systems. (Here I presume that black holes are indeed thermodynamic systems in the fullest sense; I review the evidence for that conclusion in the prequel to this paper.) I focus on three lines of argument: (i) zero-loop and one-loop calculations in quantum general relativity understood as a quantum field theory, using the path-integral formalism; (ii) calculations in string theory of the leading-order terms, higher-derivative corrections, and quantum corrections, in the black hole entropy formula for extremal and near-extremal black holes; (iii) recovery of the qualitative and (in some cases) quantitative structure of black hole statistical mechanics via the AdS/CFT correspondence. In each case I briefly review the content of, and arguments for, the form of quantum gravity being used (effective field theory; string theory; AdS/CFT) at a (relatively) introductory level: the paper is aimed at readers with some familiarity with thermodynamics, quantum mechanics and general relativity but does not presume advanced knowledge of quantum gravity. My conclusion is that the evidence for black hole statistical mechanics is as solid as we could reasonably expect it to be in the absence of a directly-empirically-verified theory of quantum gravity.     

The case for black hole thermodynamics part I: Phenomenological thermodynamics

I give a fairly systematic and thorough presentation of the case for regarding black holes as thermodynamic systems in the fullest sense, aimed at readers with some familiarity with thermodynamics, quantum mechanics and general relativity but not presuming advanced knowledge of quantum gravity. I pay particular attention to (i) the availability in classical black hole thermodynamics of a well-defined notion of adiabatic intervention; (ii) the power of the membrane paradigm to make black hole thermodynamics precise and to extend it to local-equilibrium contexts; (iii) the central role of Hawking radiation in permitting black holes to be in thermal contact with one another; (iv) the wide range of routes by which Hawking radiation can be derived and its back-reaction on the black hole calculated; (v) the interpretation of Hawking radiation close to the black hole as a gravitationally bound thermal atmosphere. In an appendix I discuss recent criticisms of black hole thermodynamics by Dougherty and Callender. This paper confines its attention to the thermodynamics of black holes; a sequel will consider their statistical mechanics.     

Statistical mechanics and thermodynamics: A Maxwellian view

One finds, in Maxwell's writings on thermodynamics and statistical physics, a conception of the nature of these subjects that differs in interesting ways from the way they are usually conceived. In particular, though—in agreement with the currently accepted view—Maxwell maintains that the second law of thermodynamics, as originally conceived, cannot be strictly true, the replacement he proposes is different from the version accepted by most physicists today. The modification of the second law accepted by most physicists is a probabilistic one: although statistical fluctuations will result in occasional spontaneous differences in temperature or pressure, there is no way to predictably and reliably harness these to produce large violations of the original version of the second law. Maxwell advocates a version of the second law that is strictly weaker; the validity of even this probabilistic version is of limited scope, limited to situations in which we are dealing with large numbers of molecules en masse and have no ability to manipulate individual molecules. Connected with this is his conception of the thermodynamic concepts of heat, work, and entropy; on the Maxwellian view, these are concept that must be relativized to the means we have available for gathering information about and manipulating physical systems. The Maxwellian view is one that deserves serious consideration in discussions of the foundation of statistical mechanics. It has relevance for the project of recovering thermodynamics from statistical mechanics because, in such a project, it matters which version of the second law we are trying to recover.     

Quantum decoherence and the approach to equilibrium (II)

In a previous paper [Hemmo, M & Shenker, O (2003). Quantum decoherence and the approach to equilibrium I. Philosophy of Science, 70, 330–358] we discussed a recent proposal by Albert [(2000). Time and chance. Cambridge, MA: Harvard University Press. Chapter 7] to recover thermodynamics on a purely dynamical basis, using the quantum theory of the collapse of the quantum state of [Ghirardi, G, Rimini, A and Weber, T., (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review, D 34, 470–479]. We proposed an alternative way to explain thermodynamics within no collapse interpretations of quantum mechanics. In this paper some difficulties faced by both approaches are discussed and solved: the spin echo experiments, and the problem of extremely light gases. In these contexts, we point out several ways in which the above quantum mechanical approaches as well as some other classical approaches to the foundations of statistical mechanics may be distinguished experimentally.     

Special cases,composition of causes,and the complexity of nature

It is often held by philosophers of science that special, idealized situations are prior to complex cases in several senses: equations for complex cases are derived from those for special cases by “composing” special case equations; behavior in complex cases is explained in terms of behavior in special cases; one learns the true nature of a property in the special case where it is allowed to work in isolation. In this paper, I argue that a strand of non-equilibrium thermodynamics which attempts to go beyond the limitations of classical non-equilibrium thermodynamics adheres to something that is the reverse of this picture. Thus, the legitimacy (or lack thereof) of this picture lies very near to the heart of foundational issues in non-equilibrium thermodynamics.     

Re-conceiving quantum theories in terms of information-theoretic constraints

This paper examines Bub's interpretation of the foundational significance of the theorem of Clifton, Bub, and Halvorson (CBH) which characterizes quantum theories in terms of information-theoretic constraints. Bub argues that quantum theory must be re-conceived of as a principle theory of information where information is a new physical primitive, to the exclusion of hidden variable theories. I will argue, contrary to Bub, that the CBH theorem cannot be used to exclude hidden variables theories. Drawing inspiration from Bub, I sketch an alternative conception of quantum mechanics as a theory of information, but one which embraces all empirically equivalent quantum theories.     

Can We Explain Thermodynamics By Quantum Decoherence?

Can we explain the laws of thermodynamics, in particular the irreversible increase of entropy, from the underlying quantum mechanical dynamics? Attempts based on classical dynamics have all failed. Albert (1994a,b; 2000) proposed a way to recover thermodynamics on a purely dynamical basis, using the quantum theory of the collapse of the wavefunction of Ghirardi, Rimini and Weber (1986). In this paper we propose an alternative way to explain thermodynamics within no-collapse interpretations of quantum mechanics. Our approach relies on the standard quantum mechanical models of environmental decoherence of open systems, e.g. Joos and Zeh (1985) and Zurek and Paz (1994).     

The emergence and interpretation of probability in Bohmian mechanics

A persistent question about the deBroglie–Bohm interpretation of quantum mechanics concerns the understanding of Born's rule in the theory. Where do the quantum mechanical probabilities come from? How are they to be interpreted? These are the problems of emergence and interpretation. In more than 50 years no consensus regarding the answers has been achieved. Indeed, mirroring the foundational disputes in statistical mechanics, the answers to each question are surprisingly diverse. This paper is an opinionated survey of this literature. While acknowledging the pros and cons of various positions, it defends particular answers to how the probabilities emerge from Bohmian mechanics and how they ought to be interpreted.     

Physics from Fisher information

B. R. Frieden uses a single procedure, called extreme physical information, with the aim of deriving ‘most known physics, from statistical mechanics and thermodynamics to quantum mechanics, the Einstein field equations and quantum gravity’. His method, which is based on Fisher information, is given a detailed exposition in this book, and we attempt to assess the extent to which he succeeds in his task.     

Taking Thermodynamics Too Seriously

This paper discusses the mistake of understanding the laws and concepts of thermodynamics too literally in the foundations of statistical mechanics. Arguing that this error is still made in subtle ways, the article explores its occurrence in three examples: the Second Law, the concept of equilibrium and the definition of phase transitions.     

The five problems of irreversibility

Thermodynamics has a clear arrow of time, characterized by the irreversible approach to equilibrium. This stands in contrast to the laws of microscopic theories, which are invariant under time-reversal. Foundational discussions of this “problem of irreversibility” often focus on historical considerations, and do therefore not take results of modern physical research on this topic into account. In this article, I will close this gap by studying the implications of dynamical density functional theory (DDFT), a central method of modern nonequilibrium statistical mechanics not previously considered in philosophy of physics, for this debate. For this purpose, the philosophical discussion of irreversibility is structured into five problems, concerned with the source of irreversibility in thermodynamics, the definition of equilibrium and entropy, the justification of coarse-graining, the approach to equilibrium and the arrow of time. For each of these problems, it is shown that DDFT provides novel insights that are of importance for both physicists and philosophers of physics.     

A probabilistic foundation of elementary particle statistics. Part I

The long history of ergodic and quasi-ergodic hypotheses provides the best example of the attempt to supply non-probabilistic justifications for the use of statistical mechanics in describing mechanical systems. In this paper we reverse the terms of the problem. We aim to show that accepting a probabilistic foundation of elementary particle statistics dispenses with the need to resort to ambiguous non-probabilistic notions like that of (in)distinguishability. In the quantum case, starting from suitable probability conditions, it is possible to deduce elementary particle statistics in a unified way. Following our approach Maxwell-Boltzmann statistics can also be deduced, and this deduction clarifies its status.Thus our primary aim in this paper is to give a mathematically rigorous deduction of the probability of a state with given energy for a perfect gas in statistical equilibrium; that is, a deduction of the equilibrium distribution for a perfect gas. A crucial step in this deduction is the statement of a unified statistical theory based on clearly formulated probability conditions from which the particle statistics follows. We believe that such a deduction represents an important improvement in elementary particle statistics, and a step towards a probabilistic foundation of statistical mechanics.In this Part I we first present some history: we recall some results of Boltzmann and Brillouin that go in the direction we will follow. Then we present a number of probability results we shall use in Part II. Finally, we state a notion of entropy referring to probability distributions, and give a natural solution to Gibbs' paradox.     

Determinism and Chance

It is generally thought that objective chances for particular events different from 1 and 0 and determinism are incompatible. However, there are important scientific theories whose laws are deterministic but which also assign non-trivial probabilities to events. The most important of these is statistical mechanics whose probabilities are essential to the explanations of thermodynamic phenomena. These probabilities are often construed as ‘ignorance’ probabilities representing our lack of knowledge concerning the microstate. I argue that this construal is incompatible with the role of probability in explanation and laws. This is the ‘paradox of deterministic probabilities’. After surveying the usual list of accounts of objective chance and finding them inadequate I argue that an account of chance sketched by David Lewis can be modified to solve the paradox of deterministic probabilities and provide an adequate account of the probabilities in deterministic theories like statistical mechanics.     

What could be objective about probabilities?

The basic notion of an objective probability is that of a probability determined by the physical structure of the world. On this understanding, there are subjective credences that do not correspond to objective probabilities, such as credences concerning rival physical theories. The main question for objective probabilities is how they are determined by the physical structure.In this paper, I survey three ways of understanding objective probability: stochastic dynamics, humean chances, and deterministic chances (typicality). The first is the obvious way to understand the probabilities of quantum mechanics via a collapse theory such as GRW, the last is the way to understand the probabilities in the context of a deterministic theory such as Bohmian mechanics. Humean chances provide a more abstract and general account of chances locutions that are independent of dynamical considerations.     

Quantum dissidents: Research on the foundations of quantum theory circa 1970

This paper makes a collective biographical profile of a sample of physicists who were protagonists in the research on the foundations of quantum physics circa 1970. We study the cases of Zeh, Bell, Clauser, Shimony, Wigner, Rosenfeld, d’Espagnat, Selleri, and DeWitt, analyzing their training and early career, achievements, qualms with quantum mechanics, motivations for such research, professional obstacles, attitude towards the Copenhagen interpretation, and success and failures. Except for Rosenfeld, they were all dissidents, fighting against the dominant attitude among physicists at the time according to which foundational issues had already been solved by the founding fathers of the discipline. Theirs is a story of success as the foundations of quantum mechanics finally entered the physics mainstream despite the fact that their expectations of breaking down quantum mechanics were not fulfilled.     

Stable regularities without governing laws?

Can stable regularities be explained without appealing to governing laws or any other modal notion? In this paper, I consider what I will call a ‘Humean system’—a generic dynamical system without guiding laws—and assess whether it could display stable regularities. First, I present what can be interpreted as an account of the rise of stable regularities, following from Strevens (2003), which has been applied to explain the patterns of complex systems (such as those from meteorology and statistical mechanics). Second, since this account presupposes that the underlying dynamics displays deterministic chaos, I assess whether it can be adapted to cases where the underlying dynamics is not chaotic but truly random—that is, cases where there is no dynamics guiding the time evolution of the system. If this is so, the resulting stable, apparently non-accidental regularities are the fruit of what can be called statistical necessity rather than of a primitive physical necessity.     

Ontological reduction and molecular structure

In this paper I outline how the debate concerning the intertheoretic reduction of chemistry reaches a stalemate. One way forward is to switch discussion to the issue of ontological reduction and emergence, so I present a counternomic criterion of emergence that should be acceptable to both sides of the discussion. I then examine the bearing on this debate of the symmetry problem in molecular quantum mechanics, as presented by Woolley and Sutcliffe (1977). I conclude by addressing some objections to emergentist positions: that they posit miraculous violations of physical laws; that emergence is obscure and of doubtful coherence; that causal theories of property identity render emergence, under the counternomic criterion, metaphysically impossible.     

A remark on Fuchs’ Bayesian interpretation of quantum mechanics

Quantum mechanics is a theory whose foundations spark controversy to this day. Although many attempts to explain the underpinnings of the theory have been made, none has been unanimously accepted as satisfactory. Fuchs has recently claimed that the foundational issues can be resolved by interpreting quantum mechanics in the light of quantum information. The view proposed is that quantum mechanics should be interpreted along the lines of the subjective Bayesian approach to probability theory. The quantum state is not the physical state of a microscopic object. It is an epistemic state of an observer; it represents subjective degrees of belief about outcomes of measurements. The interpretation gives an elegant solution to the infamous measurement problem: measurement is nothing but Bayesian belief updating in a analogy to belief updating in a classical setting. In this paper, we analyze an argument that Fuchs gives in support of this latter claim. We suggest that the argument is not convincing since it rests on an ad hoc construction. We close with some remarks on the options left for Fuchs’ quantum Bayesian project.