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Kelvin played a big part in the development of statistical mechanics, both for equilibrium and non-equilibrium. This chapter reviews these developments, taking a particular interest in Kelvin's own contributions. Topics covered include Kelvin and thermoelectricity, gas modeled as a collection of molecules, the reversibility paradox, mathematical probability models, and Boltzmann's equations.

This chapter discusses equilibrium statistical mechanics for systems more complicated than monatomic gases, as well as the problem of the trend towards equilibrium of these systems. Ludwig Boltzmann is credited for having begun this branch of statistical mechanics with a basic paper written in 1884, in which he formulated the hypothesis that some among the possible steady distributions can be interpreted as macroscopic equilibrium states. This fundamental work by Boltzmann was taken up again, widened, and expounded in a classical treatise by Josiah Willard Gibbs. In his paper, Boltzmann described statistical families of steady distributions, which he called orthodes. Boltzmann showed that there are at least two ensembles of this kind, the ergode (Gibbs's microcanonical ensemble) and the holode (Gibbs's canonical ensemble). This article also explains why statistical mechanics is usually attributed to Gibbs and not to Boltzmann, the problem of trend to equilibrium and ergodic theory, and Max Planck's work on statistical mechanics.

This chapter unfolds a central philosophical problem of statistical mechanics. This problem lies in a clash between the Static Probabilities offered by statistical mechanics and the Dynamic Probabilities provided by classical or quantum mechanics. The chapter looks at the Boltzmann and Gibbs approaches in statistical mechanics and construes some of the great controversies in the field — for instance the Reversibility Paradox — as instances of this conflict. It furthermore argues that a response to this conflict is a critical choice that shapes one's understanding of statistical mechanics itself, namely, whether it is to be conceived as a special or fundamental science. The chapter details some of the pitfalls of the latter ‘globalist’ position and seeks defensible ground for a kind of ‘localist’ alternative.

quantum-mechanical branching is a time-directed process, while the underlying equations of quantum theory admit no preferred direction of time. The chapter demonstrates that this apparent incompatibility is an aspect of the better-known apparent incompatibility between the directedness in time of statistical mechanics and the underlying time-symmeric microphysics. The chapter sketches a unified way of resolving this apparent incompatibility, related to but distinct from the currently-popular idea of a ‘past hypothesis’.

This chapter examines the subject of polymers experiencing external potential fields that vary in space. The statistical mechanical problem is reduced to the solution of Fokker-Planck equations, and operators for polymer segment density and stress are introduced. Analytical approximation schemes for solving the Fokker-Planck equations and evaluating operators are presented. Effective numerical methods are also described, with an emphasis on spectral collocation techniques.

Computational modelling of materials behaviour is becoming a reliable tool of scientific investigation, complementary to traditional theory and experimentation. The Multiscale Materials Modelling (MMM) approach reflects the realization that continuum and atomistic analysis methods are complementary. This chapter describes the most popular numerical techniques of each component that make up the MMM paradigm for modelling nano- and micro-systems: Quantum Mechanics (QM), Molecular Dynamics (MD), Monte Carlo (MC), Dislocation Dynamics (DD), Statistical Mechanics (SM), and Continuum Mechanics (CM).

This chapter presents a self-contained introduction to some recent developments in nonequilibrium quantum statistical mechanics. In the elementary framework of finite dimensional quantum systems it introduces the concept of entropy production and discusses various generating functionals which encode the statistical properties of its fluctuations. It explores the physical interpretations of these functionals and their mathematical properties. In particular, it investigates their relations to linear response theory, full counting statistics, and quantum hypothesis testing. The chapter discusses very briefly more technical issues linked to the thermodynamic limit as well as to the large time limit. These two limits turn the elementary, finite time fluctuation theory into a powerful machinery, which yields important information on the asymptotic behaviour of infinitely extended quantum systems, e.g., open systems coupled to infinite reservoirs. An essential mathematical tool to deal with such a system is Tomita–Takesaki's modular theory of von Neumann algebras. The power of this theory is somewhat shadowed by its technical aspects. Finite dimensional quantum systems are special since all the structures and results of this machinery can be described by elementary tools.

The idea that science has discovered that our universe is governed by laws of nature is arguably an important part of the modern scientific world‐view. The distinction between laws and other truths discovered by science is essential in at least some scientific reasoning, for example in statistical mechanics. The idea that laws of nature really govern the universe—and that they are not merely regularities that form an elegant system, as they are according to proponents of Humeanism (e.g. David Lewis)—turns out to play a crucial role in much scientific reasoning having to do with so‐called fine‐tuning. This chapter offers a formulation of the law‐governed world‐picture and outlines an argument for a new account of lawhood that vindicates that picture.

There is increasing interest in applying the methods of statistical mechanics and of transport theory to systems which are neither atomistic, nor in equilibrium, but which still fulfill a remaining tenet of statistical physics which is that systems can be completely defined by a very small number of parameters and can be constructed in a reproducible way. Powders fall into this category. This chapter sets up a framework for describing the state of the powder, basing the development on anologies with statistical mechanics. It is argued that powders have an entropy, but it will be the volume which plays the role of the energy in normal statistical mechanics: the energy corresponding to the powder's thermal temperature of 300 K is negligible. In statistical mechanics, the most fundamental entry is via the microcanonical ensemble. This is a closed system, contained in a volume V, which is assumed to take up all configurations subject to the Hamiltonian function taking the value E with equal probability.

This chapter discusses three fundamental questions which the majority of the physics community believes are not worthy of attention and a minority believes are crucial and in urgent need of attention. The first concerns the “anthropic principle”: to what extent is it an “explanation” of basic physical data, such as the dimensionality of space-time, the values of the fundamental constants, etc., to observe that were they appreciably different, human life and consciousness could not have evolved to the point of asking the question? The second has to do with the “arrow of time”: how is the everyday sense of the “flow” of time from past to future consistent with the invariance of the laws of physics under time reversal? The third problem is how to incorporate the occurrence of definite outcomes within the framework of quantum mechanics (the “quantum measurement problem”).

This chapter defends and refines a specific objectivist interpretation of probabilities in statistical mechanics. For ergodic systems, probabilities are defined as time-averages. For other systems, ergodic decomposition is applied, and stochastic nomological machines are used to assign probabilities over the members of the decomposition. The relevance of this analysis to the Boltzmann and Gibbs approaches to statistical mechanics is discussed. The chapter shows that the proposed definition of probabilities matches a Boltzmann-like approach particularly well if the sharp distinction between equilibrium and non-equilibrium is given up and if more emphasis is laid upon the global time profile of entropy. The chapter furthermore argues that the alleged weaknesses of the time-average definition of probability are avoided.

The statistical mechanics of cross-linked macromolecules requires simultaneous treatment of random polymer configurations, excluded-volume interactions, and the quenched disorder of the cross-links, as well as the topological constraints imposed by impenetrable chains. Such a description was pioneered by Deam and Edwards. This work is reviewed and a discussion presented of subsequent efforts to understand the unique elastic properties of networks as well as the critical phenomena of the vulcanization transition.

This chapter discusses elementary statistical mechanical models of ideal chains, i.e., polymers in isolation and not experiencing interactions with other polymers. Freely jointed and bead-spring discrete chain models are introduced along with continuous Gaussian chain and wormlike chain models. Methods of analysis are based on the solution of Fokker-Planck equations.

The microscopic laws of motion offer no sense of time, but in the world at large, time marches on. Macroscopic systems progress irreversibly from one state of thermodynamic equilibrium to another, driven forward by the statistical imperatives of large numbers and overwhelming odds. Energy becomes increasingly dispersed and harder to extract, even as the total amount remains constant. The entropy of the universe grows and grows.

Albert Einstein is the only scientist who's genius was comparable to that of Newton's. Their personalities and lifestyles were completely different, but they were both consumed by the desire to know. In 1905, the miracle year, Einstein gave quantum mechanics its true beginning by working out the theory of photoelectricity, created a new statistical mechanics by studying Brownian motion, gave a fully formed theory of special relativity, and derived his famous mass-energy equation. This monumental accomplishment was matched only by Newton's work during the plague years. A decade later, Einstein single handedly developed general relativity, the deepest and most beautiful of all scientific theories.