Foundations of Statistical Mechanics

This paper presents a list of one million digits having the property that all shustrings over this list are of uniform length, at the minimum value of six.  The similar sequence from RAND Corporation is not nearly so well disordered as is this digit sequence given in this paper.  Indeed, we claim that no sequence of one million digits represents greater disorder than is represented by the sequence of one million purely random digits that is given in this paper.

DOI: 10.13140/2.1.3232.5444

According to the orthodox interpretation of bounce cosmologies, the universe was born from an entropy reducing phase in a previous universe. To defend the thesis that the whole of physical reality was caused to exist a finite time ago, William Lane Craig and co-author James Sinclair have argued the low entropy interface between universes should instead be understood as the beginning of two universes. Here, I present Craig and Sinclair with a dilemma. On the one hand, if the direction of time is reducible, as friends of the Mentaculus-e.g., David Albert, Barry Loewer, and David Papineau-maintain, then there is reason to think that the direction of time and the entropic arrow of time align. But on that account, efficient causation is likely reducible to non-causal phenomena. In consequence, contrary to Craig and Sinclair's theological aims, things can begin to exist without causes. On the other hand, if the direction of time is not reducible, Craig and Sinclair's interpretation of bounce cosmologies is unjustified. Lastly, a reply to a potential objection motivates a discussion of how to interpret bounce cosmologies on the tensed theory of absolute time favored by Craig and Sinclair. I offer two interpretations of bounce cosmologies that, given a tensed theory of absolute time, are preferable to those Craig and Sinclair offer, yet inconsistent with their project in natural theology; on one interpretation, the universe does not require a supernatural cause and, on the other, bounce cosmologies represent the universe as never having begun to exist.

Statistical mechanics is a strange theory. Its aims are debated, its methods are contested, its main claims have never been fully proven, and their very truth is challenged, yet at the same time, it enjoys huge empirical success and gives us the feeling that we understand important phenomena. What is this weird theory, exactly? Statistical mechanics is the name of the ongoing attempt to apply mechanics (classical, as discussed in this paper, or quantum), together with some auxiliary hypotheses, to explain and predict certain phenomena, above all those described by thermodynamics. This paper shows what parts of this objective can be achieved with mechanics by itself. It thus clarifies what roles remain for the auxiliary assumptions that are needed to achieve the rest of the desiderata. Those auxiliary hypotheses are described in another paper in this journal, Foundations of statistical mechanics: The auxiliary hypotheses.

In a recent paper Stud. Hist. Philos. Mod. Phys. 36, 355 2005 it is argued that to properly understand the thermodynamics of Landauer’s principle it is necessary to extend the concept of logical operations to include indeterministic operations. Here we examine the thermodynamics of such operations in more detail, extending the work of Landauer to include indeterministic operations and to include logical states with variable entropies, temperatures, and mean energies. We derive the most general statement of Landauer’s principle and prove its universality, extending considerably the validity of previous proofs. This confirms conjectures made that all logical operations may, in principle, be performed in a thermodynamically reversible fashion, although logically irreversible operations would require special, practically rather difficult, conditions to do so. We demonstrate a physical process that can perform any computation without work requirements or heat exchange with
the environment. Many widespread statements of Landauer’s principle are shown to be special cases of our generalized principle.

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 an 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.

In this thesis we study the informational underpinnings of thermodynamics and statistical mechanics. To this purpose, we use an abstract framework—general probabilistic theories—, capable of describing arbitrary physical theories, which allows one to abstract the informational content of a theory from the concrete details of its formalism. In this framework, we extend the treatment of microcanonical thermodynamics, namely the thermodynamics of systems with a well-defined energy, beyond the known cases of classical and quantum theory. We formulate two requirements a theory should satisfy to have a well-defined microcanonical thermodynamics. We adopt the recent approach of resource theories, where one studies the transitions between states that can be accomplished with a restricted set of physical operations. We formulate three different resource theories, differing in the choice of the restricted set of physical operations.

To bridge the gap between the objective dynamics of particles and the subjective world of probabilities, one of the core issues in the foundations of statistical mechanics, we propose four information-theoretic axioms. They are satisfied by quantum theory and more exotic alternatives, including a suitable extension of classical theory where classical systems interact with each other creating entangled states. The axioms identify a class of theories where every mixed state can be modelled as the reduced state of a pure entangled state. In these theories it is possible to introduce well-behaved notions of majorisation, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. The three resource theories define the same notion of resource if and only if, on top of the four axioms, the dynamics of the underlying theory satisfy a condition called “unrestricted reversibility”. Under this condition we derive a duality between microcanonical thermodynamics and pure bipartite entanglement.

Certain results, most famously in classical statistical mechanics and complex systems, but also in quantum mechanics and high-energy physics, yield a coarse-grained stable statistical pattern in the long run. The explanation of these results shares a common structure: the results hold for a 'typical' dynamics, that is, for most of the underlying dynamics. In this paper I argue that the structure of the explanation of these results might shed some light --a different light-- on philosophical debates on the laws of nature. In the explanation of such patterns, the specific form of the underlying dynamics is almost irrelevant. The conditions required, given a free state-space evolution, suffice to account for the coarse-grained lawful behaviour. An analysis of such conditions might thus provide a different account of how regular behaviour can occur. This paper focuses on drawing attention to this type of explanation, outlining it in the diverse areas of physics in which it appears, and discussing its limitations and significance in the tractable setting of classical statistical mechanics.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

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 (classical or quantum), 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.

According to the orthodox interpretation of bounce cosmologies, the universe was born from an entropy-reducing phase in a previous universe. To defend the thesis that the whole of physical reality was caused to exist a finite time ago, Craig and Sinclair have argued the low-entropy interface between universes should instead be understood as the beginning of two universes. Here, I present Craig and Sinclair with a dilemma. On the one hand, if the direction of time is reducible, as friends of the Mentaculus—for example, Albert, Loewer, and Papineau—maintain, then there is reason to think that the direction of time and the entropic arrow of time align. But on that account, efficient causation is likely reducible to non-causal phenomena. In consequence, contrary to Craig and Sinclair’s theological aims, things can begin to exist without causes. On the other hand, if the direction of time is not reducible, Craig and Sinclair’s interpretation of bounce cosmologies is unjustified. Lastly, ...

The recent use of typicality in statistical mechanics for foundational purposes has stirred an important debate involving both philosophers and physicists. In this paper I try to approach the problem from the point of view of the explanatory power of typicality. I claim three main points. (1) Typicality-claims, the basic ingredients of the typicality explanations, show a crucial semantic duality. (2) This duality is mirrored by the duality between constitutive and causal explanation. I therefore conclude that typicality is strictly related to the  mechanistic explanation. (3) For historical reasons, only constitution has been analyzed in statistical mechanics, while causation has been neglected. I argue that this is a mistake and, most possibly, the reason of the philosophical discontent toward typicality.

We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms---Causality, Purity Preservation, Pure Sharpness, and Purification---identify a class of theories where every mixed state can be modelled as the marginal of a pure entangled state and where every unsharp measurement can be modelled as a sharp measurement on a composite system. This class of theories, called sharp theories with purification, includes quantum theory both with complex and real amplitudes, as well as a suitable extension of classical probability theory where classical systems can be entangled with other, non-classical systems. Theories satisfying our axioms support well-behaved notions of majorization, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. We conjecture that every theory admitting a sensible thermodynamics must be extendable to a sharp theory with purification.

Schulman (Entropy 7(4):221–233, 2005) has argued that Boltzmann’s intuition,
that the psychological arrow of time is necessarily aligned with the thermodynamic
arrow, is correct. Schulman gives an explicit physical mechanism for this
connection, based on the brain being representable as a computer, together with certain
thermodynamic properties of computational processes. Hawking (Physical Origins
of Time Asymmetry, Cambridge University Press, Cambridge, 1994) presents
similar, if briefer, arguments. The purpose of this paper is to critically examine the
support for the link between thermodynamics and an arrow of time for computers.
The principal arguments put forward by Schulman and Hawking will be shown to
fail. It will be shown that any computational process that can take place in an entropy
increasing universe, can equally take place in an entropy decreasing universe. This
conclusion does not automatically imply a psychological arrow can run counter to
the thermodynamic arrow. Some alternative possible explanations for the alignment
of the two arrows will be briefly discussed

In this review article we hope to give a self-contained, though certainly
not exhaustive, description of the Ising model and of some selected rigorous results illustrating properties which throw some light on the general nature of the phenomenon of the phase transitions far from the
critical point and which, hopefully, should not be a peculiarity
of the simplicity of the model.

Microcanonical thermodynamics studies the operations that can be performed on systems with well-defined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, ii) noisy operations, generated by the interaction with ancillas in the microcanonical state, and iii) unital operations, defined as the operations that preserve the microcanonical state. We focus our attention on a class of physical theories, called sharp theories with purification, where these three sets of operations exhibit remarkable properties. Firstly, each set is contained into the next. Secondly, the convertibility of states by unital operations is completely characterised by a majorisation criterion. Thirdly, the three sets are equivalent in terms of state convertibility if and only if the dynamics allowed by theory satisfy a suitable condition, which we call unrestricted reversibility. Under this condition, we derive a duality between the resource theories of microcanonical thermodynamics and the resource theory of pure bipartite entanglement.

Are principles of information processing necessary to demonstrate the consistency of statistical mechanics? Does the physical implementation of a computational operation have a fundamental thermodynamic cost, purely by virtue of its logical properties? These two questions lie at the centre of a large body of literature concerned with the Szilard engine (a variant of the Maxwell's demon thought experiment), Landauer's principle (supposed to embody the fundamental principle of the thermodynamics of computation) and possible connections between the two. A variety of attempts to answer these questions have illustrated many open questions in the foundations of statistical mechanics.

We consider both the dynamics within and towards the supercycle attractors along the
period-doubling route to chaos to analyze the development of a statistical-mechanical structure.
In this structure the partition function consists of the sum of the attractor position distances
known as supercycle diameters and the associated thermodynamic potential measures the rate of
approach of trajectories to the attractor. The configurational weights for finite 2N, and infinite
N →∞, periods can be expressed as power laws or deformed exponentials. For a finite period the
structure is undeveloped in the sense that there is no true configurational degeneracy, but in the
limit N →∞ this is realized together with the analog property of a Legendre transform linking
entropies of two ensembles. We also study the partition functions for all N and the action of the
central limit theorem via a binomial approximation.