This chapter describes the canonical ensemble, which is an ensemble of systems that are all prepared in the same way with the same number of particles, volume, shape, magnetic field, and electric field. However, the energy of each system fluctuates. The chapter details experiments that involve condensed matter wherein a system is placed in contact with a heat bath that eventually reaches thermal equilibrium. It also clarifies how the system being studied together with the heat bath or combined system is thermally isolated so that its total energy is constant. The chapter illustrates a Boltzmann probability distribution, wherein probabilities are drawn along the horizontal axis and the energies along the vertical. It highlights the use of the Boltzmann probability distribution on systems that are in contact with a heat bath, granted the number of particles in the system is constant.

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## The canonical ensemble

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## Continuous phase transitions

This chapter explores the occurrence of a continuous phase transition when the minimum in the thermodynamic potential evolves smoothly into two or more equal minima. It refers to an example of the liquid–gas phase transition held at constant pressure wherein the system is on the liquid–gas coexistence line and the thermodynamic potential is the generalized Gibbs free energy. It also analyses the temperatures above the critical temperature wherein there is no evidence of two phases and the system has a uniform density which results in a single phase called the fluid phase. The chapter points out that bubbles of gas and drops of liquid in the system are intermingled on all length-scales from the macroscopic to the atomic. It elaborates that the bubbles represent fluctuations of the system away from the average density.

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## Entropy and the second law of thermodynamics

This chapter focuses on entropy, which is a function of state for an ideal gas that uses an integrating factor on the inverse of the temperature according to the ideal gas scale. It refers to the use of the second law of thermodynamics to prove that the most efficient engine operating between two heat baths is a Carnot engine. It also discusses the temperature scale that can be defined from the ratio of heat flows in one cycle of the Carnot engine. The chapter defines a thermal reservoir of heat bath as a body that has a huge heat capacity wherein temperature does not change, such as a block of copper that can act as a thermal reservoir for small additions of heat energy. It looks at the simplest expressions of the second law of thermodynamics that relate to a cycle of processes in which heat flows between hot and cold reservoirs that produce work.

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## Fermi and Bose particles

This chapter reviews the techniques of the grand canonical ensemble to give an exact treatment for both Bose and Fermi particles. It looks at the results of Maxwell-Boltzmann statistics for the average number of particles in sufficiently high temperatures and in a single-particle state for both fermions and bosons. It also examines the exclusion principle in Fermi particles, which means that there cannot be two particles in the same single-particle state, while there is no such restriction for Bose particles. The chapter clarifies that the resulting formulae for the average number of particles in a single-particle state are different at low temperatures. It describes Fermi particles at low temperatures that have a large kinetic energy, an energy which can be much larger than the interaction between particles.

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## The first law of thermodynamics

This chapter introduces classical thermodynamics, which is based on empirical principles resulting from experimental observation. One such principle is that heat and work are both forms of energy and that energy is always conserved, which provides the basis for the first law of thermodynamics. The second law of thermodynamics is based on the principle that heat flows from a hotter to a colder body. The principles of thermodynamics only use macroscopic concepts, such as temperature and pressure, and are justified by their success in explaining experimental observations. The chapter outlines two objectives in the statistical approach: deriving the laws of thermodynamics and deriving the formulae for properties of macroscopic systems. It provides a brief summary of the foundations and essential ideas of the first law of thermodynamics.

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## Ginzburg–Landau theory

This chapter discusses the extension of the theory to allow spatial variation in the order parameter and determining the shape of the interface and the surface contribution to the free energy. It describes the size of fluctuations of the order parameter, determining whether they are sufficiently small to be ignored or whether they dominate the properties of the system and require careful treatment. It discusses Vitaly Ginzburg and Lev Landau's sophisticated model which allows the order parameter to vary in space. The chapter outlines the benefits of Ginzburg and Landau's model, which includes the description of the interface between phases and estimation of the surface free energy. It elaborates how the model predicts the existence of topological singularities in the order parameter, such as quantified vortices in superfluid systems.

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## The ideas of statistical mechanics

This chapter discusses the microscopic basis of entropy, how to calculate the equation of state for a system, and the physical basis of the second law that ensures that entropy increases as a function of time. It cites Ludwig Boltzmann's hypothesis, which states that the entropy of a system is related to the probability of its state and its basis is statistical. The chapter also focuses on statistical mechanics, wherein thermal properties are calculated by using statistical ideas together with a knowledge of the laws of mechanics obeyed by the particles that make up a system. The chapter highlights that the power of statistical mechanics is grounded in the fact that it uses simple ideas and involves general techniques. It demonstrates the notion of statistical probability, which is obtained by considering a series of identical measurements made over a period of time on a single system.

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## Identical particles

This chapter introduces the partition function that involves a sum over independent quantum states of the N-particle system. It outlines the problem of distinguishability, that is, how to distinguish one electron from another when they are replicas of each other and are all the same. It also emphasizes that electrons have no feature or label based on which they can be told apart, and they repeat the same attributes endlessly. The chapter explains that identical particles lead to complications in the quantum mechanical description of the microscopic state of the system as there are no observable changes in the system when any two particles are exchanged. It analyses the uncertainty principle, wherein the position of a particle cannot be accurately specified at each stage, nor can the precise trajectory of a particle be followed.

### Book

### Roger Bowley and Mariana Sánchez

Introductory Statistical Mechanics explains the ideas and techniques of statistical mechanics—the theory of condensed matter—in a simplean accessible and progressive way. The text starts with the laws of thermodynamics and basic simple ideas of quantum mechanics. The conceptual ideas underlying the subject are explained, and; the mathematical ideas are developed in parallel to give a coherent overall view. The text is illustrated with examples not just from solid-state physics, but also from recent theories of radiation from black holes and recent data on the background radiation from the cosmic background explorer.

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## Maxwell distribution of molecular speeds

This chapter starts with ideas from James Clerk Maxwell's publication the distribution of the speeds of particles in a gas, which he obtained by analysing collision processes between the particles. It recounts how Maxwell showed that there could be a distribution of speeds and calculated the distribution, contradicting people's speculation that the particles of the gas moved with the same speed. It also mentions Ludwig Boltzmann, who treated the kinetic theory of gases from a microscopic point of view using the notion of probability. The chapter details how Boltzmann derived Maxwell's distribution of speeds and related it to the temperature of the particles. It examines particles using quantum mechanics and derives the Maxwell distribution of speeds.

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## Phase transitions

This chapter talks about the cooling of a system that results in the appearance of new states of matter, such as gas that forms a liquid or a liquid that can form a solid when it is cooled enough. It describes the transitions between gas, liquid, solid, and superfluid in different phases of the ecosystem. These are called phase transitions. It also explains that the term 'phase' is a state of matter that is uniform throughout in both chemical composition and in physical state. The chapter highlights the importance of uniformity as physical properties, such as density or conductivity, and chemical composition must be the same throughout the system. It clarifies that the notion of uniformity means that any representative sample of the solid of macroscopic size appears to be the same as any other representative sample.

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## Planck's distribution

This chapter considers the distribution of the energy U among N oscillators of frequency ν, noting that an infinite number of distributions are possible if U is viewed as a divisible without limit. It discusses the theory of black-body radiation, which was developed by Max Planck in 1900 and gave birth to quantum physics. It defines a black body as one that absorbs all the radiation incident upon it. No radiation is reflected so that it looks black when it is cold. The chapter explains that objects emit electromagnetic radiation at a rate which varies in relation to their temperature. This radiation cannot be seen because its wavelength is the wrong frequency for the human eye. The chapter states that a body in thermal equilibrium with its surroundings emits and absorbs radiation at exactly the same rate.

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## Probability and statistics

This chapter explores two notions of probability used in science: classical probability and statistical probability. It clarifies that the classical notion of probability assigns, a priori, equal probabilities to all possible outcomes of an event while statistical probability measures the relative frequency of an event. It also explains that simple events, which are events that cannot be broken up into simpler parts, can build up compound events, that is, aggregates of simple events. The chapter also introduces the acid test for determining whether two events are independent of each other. It reviews the use of classical probability and the importance of counting the number of ways in which various events can happen, leading to the concepts of arrangement, permutations, and combinations.

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## Systems with variable numbers of particles

This chapter explains that a system can have a variable number of particles when particles can enter or leave or because a reaction takes place inside the system. It cites a sealed vessel containing a pure substance with both liquid and gas phases present as an example that demonstrates the presence of a meniscus separating the gas from the liquid. This meniscus allows for determining which atoms to exist in the gas phase and in the liquid phase. The chapter also discusses how gas atoms can condense into liquid or how liquid can evaporate into gas. The chapter highlights the application of mathematical language to reactions of elementary particles or reactions that occurred in the early stages of the universe. It outlines the subject of reactions that occur frequently in science, such as solid-state physics, chemistry, and phase equilibrium.