Statistical mechanics
NVEMicrocanonical NVTCanonical µVTGrand canonical NPHIsoenthalpic–isobaric NPTIsothermal–isobaric

Statistical mechanics is one of the pillars of modern physics. It is necessary for the fundamental study of any physical system that has many degrees of freedom. The approach is based on statistical methods, probability theory and the microscopic physical laws.^{[1][2][3][note 1]}

It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.

Statistical mechanics.with emphasis on the “statistical,” and least-possible reference to the “mechanics.” While the mechanical details change from system to system, the statistical digestion of those details is system-independent, and it is that fact that accounts for the compelling universality of thermodynamics. Table of Contents. Chapter 11: Statistical Mechanics of Interacting Systems: The Method of Quantized Fields 11.1 The formalism of second quantization 11.2 Low-temperature behavior of an imperfect Bose gas 11.3 Low-lying states of an imperfect Bose gas 11.4 Energy spectrum of a Bose liquid 11.5 States with quantized circulation 11.6.

Statistical mechanics shows how the concepts from macroscopic observations (such as temperature and pressure) are related to the description of microscopic state that fluctuates around an average state. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.^[1]

Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

• 2Statistical thermodynamics
  • 2.3Calculation methods
• 3Non-equilibrium statistical mechanics

Principles: mechanics and ensembles[edit]

In physics, there are two types of mechanics usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:

1. The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).
2. An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)

Using these two concepts, the state at any other time, past or future, can in principle be calculated.There is however a disconnection between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.

Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinates. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,^{[note 2]} and can be compactly summarized as a density matrix.

As is usual for probabilities, the ensemble can be interpreted in different ways:^[1]

• an ensemble can be taken to represent the various possible states that a single system could be in (epistemic probability, a form of knowledge), or
• the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (empirical probability), in the limit of an infinite number of trials.

These two meanings are equivalent for many purposes, and will be used interchangeably in this article.

Statistical Physics Text Landau Reichl

However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.^{[note 3]} The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

Statistical thermodynamics[edit]

The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.

Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.

Fundamental postulate[edit]

A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).^[1]There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.^[1] Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.

A common approach found in many textbooks is to take the equal a priori probability postulate.^[2] This postulate states that

For an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.

The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:

• Ergodic hypothesis: An ergodic system is one that evolves over time to explore 'all accessible' states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.
• Principle of indifference: In the absence of any further information, we can only assign equal probabilities to each compatible situation.
• Maximum information entropy: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest Gibbs entropy (information entropy).^[4]

Other fundamental postulates for statistical mechanics have also been proposed.^[5]

Three thermodynamic ensembles[edit]

There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.^[1] These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

Microcanonical ensemble

describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.

Canonical ensemble

describes a system of fixed composition that is in thermal equilibrium^{[note 4]} with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.

Grand canonical ensemble

describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.

For systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.^[6] The Gibbs theorem about equivalence of ensembles^[7] was developed into the theory of concentration of measure phenomenon,^[8] which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.^[9]

Important cases where the thermodynamic ensembles do not give identical results include:

• Microscopic systems.
• Large systems at a phase transition.
• Large systems with long-range interactions.

In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.^[2]

Thermodynamic ensembles[1]
Microcanonical	Canonical	Grand canonical
Fixed variables	N, E, V	N, T, V	μ, T, V
Microscopic features	Number of microstates ${\displaystyle W}$	${\displaystyle Z=\sum _k{e}^{-E_k/k_{B}T}}$	${\displaystyle \mathcal{Z}=\sum _k{e}^{-(E_k-\mu N_k)/k_{B}T}}$
Macroscopic function	${\displaystyle S=k_B\log W}$	${\displaystyle F=-k_{B}T\log Z}$	${\displaystyle \mathrm{\Omega }=-k_{B}T\log \mathcal{Z}}$

Calculation methods[edit]

Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.

Exact[edit]

There are some cases which allow exact solutions.

• For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics).
• Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of Maxwell–Boltzmann statistics, Fermi–Dirac statistics, and Bose–Einstein statistics.^[2]
• A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few toy models.^[10] Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model.

Monte Carlo[edit]

One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

• The Metropolis–Hastings algorithm is a classic Monte Carlo method which was initially used to sample the canonical ensemble.
• Path integral Monte Carlo, also used to sample the canonical ensemble.

Other[edit]

• For rarefied non-ideal gases, approaches such as the cluster expansion use perturbation theory to include the effect of weak interactions, leading to a virial expansion.^[3]
• For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.^[3]
• Molecular dynamics computer simulations can be used to calculate microcanonical ensemble averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.
• Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.

Non-equilibrium statistical mechanics[edit]

There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:

• heat transport by the internal motions in a material, driven by a temperature imbalance,
• electric currents carried by the motion of charges in a conductor, driven by a voltage imbalance,
• spontaneous chemical reactions driven by a decrease in free energy,
• friction, dissipation, quantum decoherence,
• systems being pumped by external forces (optical pumping, etc.),
• and irreversible processes in general.

All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)

In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.

Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.

Stochastic methods[edit]

One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

• Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term 'statistical mechanics' had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).

  The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the 'interesting' information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.

  A quantum technique related in theme is the random phase approximation.
• BBGKY hierarchy:In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.
• Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions):A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach often used in electronic quantum transport calculations.

Near-equilibrium methods[edit]

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or 'know' how it came to be away from equilibrium.^[3]:664

This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation-dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.

A few of the theoretical tools used to make this connection include:

Hybrid methods[edit]

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.^[11][12]

Applications outside thermodynamics[edit]

The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:

• propagation of uncertainty over time,^[1]
• regression analysis of gravitational orbits,
• ensemble forecasting of weather,
• dynamics of neural networks,
• bounded-rational potential games in game theory and economics.

History[edit]

In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.^[5]

In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.^[13] This was the first-ever statistical law in physics.^[14] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.^[15] Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.

Statistical mechanics proper was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory.^[16] Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.

The term 'statistical mechanics' was coined by the American mathematical physicist J. Willard Gibbs in 1884.^{[17][note 5]} 'Probabilistic mechanics' might today seem a more appropriate term, but 'statistical mechanics' is firmly entrenched.^[18] Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous.^[1] Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.^[2]

See also[edit]

Fundamentals of Statistical Mechanics – Wikipedia book

• Thermodynamics: non-equilibrium, chemical
• Mechanics: classical, quantum
• Probability, statistical ensemble
• Numerical methods: Monte Carlo method, molecular dynamics

Notes[edit]

1. ^The term statistical mechanics is sometimes used to refer to only statistical thermodynamics. This article takes the broader view. By some definitions, statistical physics is an even broader term which statistically studies any type of physical system, but is often taken to be synonymous with statistical mechanics.
2. ^The probabilities in quantum statistical mechanics should not be confused with quantum superposition. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.
3. ^Statistical equilibrium should not be confused with mechanical equilibrium. The latter occurs when a mechanical system has completely ceased to evolve even on a microscopic scale, due to being in a state with a perfect balancing of forces. Statistical equilibrium generally involves states that are very far from mechanical equilibrium.
4. ^The transitive thermal equilibrium (as in, 'X is thermal equilibrium with Y') used here means that the ensemble for the first system is not perturbed when the system is allowed to weakly interact with the second system.
5. ^According to Gibbs, the term 'statistical', in the context of mechanics, i.e. statistical mechanics, was first used by the Scottish physicist James Clerk Maxwell in 1871. From: J. Clerk Maxwell, Theory of Heat (London, England: Longmans, Green, and Co., 1871), p. 309: 'In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus.'

References[edit]

1. ^ ^abcdefghiGibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
2. ^ ^abcdeTolman, R. C. (1938). The Principles of Statistical Mechanics. Dover Publications. ISBN9780486638966.
3. ^ ^abcdBalescu, Radu (1975). Equilibrium and Non-Equilibrium Statistical Mechanics. John Wiley & Sons. ISBN9780471046004.
4. ^Jaynes, E. (1957). 'Information Theory and Statistical Mechanics'. Physical Review. 106 (4): 620–630. Bibcode:1957PhRv..106..620J. doi:10.1103/PhysRev.106.620.
5. ^ ^abJ. Uffink, 'Compendium of the foundations of classical statistical physics.' (2006)
6. ^Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw–Hill. p. 227. ISBN9780070518001.
7. ^Touchette, Hugo (2015). 'Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels'. Journal of Statistical Physics. 159 (5): 987–1016. arXiv:1403.6608. doi:10.1007/s10955-015-1212-2.
8. ^Ledoux, Michel (2005). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs. 89. doi:10.1090/surv/089. ISBN9780821837924..
9. ^Gorban, A. N.; Tyukin, I. Y. (2018). 'Blessing of dimensionality: Mathematical foundations of the statistical physics of data'. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 376 (2118): 20170237. doi:10.1098/rsta.2017.0237. PMC5869543.
10. ^Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press Inc. ISBN9780120831807.
11. ^Altshuler, B. L.; Aronov, A. G.; Khmelnitsky, D. E. (1982). 'Effects of electron-electron collisions with small energy transfers on quantum localisation'. Journal of Physics C: Solid State Physics. 15 (36): 7367. Bibcode:1982JPhC...15.7367A. doi:10.1088/0022-3719/15/36/018.
12. ^Aleiner, I.; Blanter, Y. (2002). 'Inelastic scattering time for conductance fluctuations'. Physical Review B. 65 (11): 115317. arXiv:cond-mat/0105436. Bibcode:2002PhRvB..65k5317A. doi:10.1103/PhysRevB.65.115317.
13. ^See:
    • Maxwell, J.C. (1860) 'Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,'Philosophical Magazine, 4th series, 19 : 19–32.
    • Maxwell, J.C. (1860) 'Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another,'Philosophical Magazine, 4th series, 20 : 21–37.
14. ^Mahon, Basil (2003). The Man Who Changed Everything – the Life of James Clerk Maxwell. Hoboken, NJ: Wiley. ISBN978-0-470-86171-4. OCLC52358254.
15. ^Gyenis, Balazs (2017). 'Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium'. Studies in History and Philosophy of Modern Physics. 57: 53–65. arXiv:1702.01411. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001.
16. ^Ebeling, Werner; Sokolov, Igor M. (2005). Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems. Edited by Ebeling Werner & Sokolov Igor M. Published by World Scientific Press. Series on Advances in Statistical Mechanics. 8. pp. 3–12. Bibcode:2005stst.book.....E. doi:10.1142/2012. ISBN978-90-277-1674-3. (section 1.2)
17. ^J. W. Gibbs, 'On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics.' Proceedings of the American Association for the Advancement of Science, 33, 57-58 (1884). Reproduced in The Scientific Papers of J. Willard Gibbs, Vol II (1906), pp. 16.
18. ^Mayants, Lazar (1984). The enigma of probability and physics. Springer. p. 174. ISBN978-90-277-1674-3.

External links[edit]

Wikimedia Commons has media related to Statistical mechanics.

• Philosophy of Statistical Mechanics article by Lawrence Sklar for the Stanford Encyclopedia of Philosophy.
• Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials. SklogWiki is particularly orientated towards liquids and soft condensed matter.
• Statistical Thermodynamics - Historical Timeline
• Thermodynamics and Statistical Mechanics by Richard Fitzpatrick
• Lecture Notes in Statistical Mechanics and Mesoscopics by Doron Cohen
• Videos of lecture series in statistical mechanics on YouTube taught by Leonard Susskind.
• Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28.

Description

Statistical Mechanics explores the physical properties of matter based on the dynamic behavior of its microscopic constituents. After a historical introduction, this book presents chapters about thermodynamics, ensemble theory, simple gases theory, Ideal Bose and Fermi systems, statistical mechanics of interacting systems, phase transitions, and computer simulations.

This edition includes new topics such as BoseEinstein condensation and degenerate Fermi gas behavior in ultracold atomic gases and chemical equilibrium. It also explains the correlation functions and scattering; fluctuationdissipation theorem and the dynamical structure factor; phase equilibrium and the Clausius-Clapeyron equation; and exact solutions of one-dimensional fluid models and two-dimensional Ising model on a finite lattice. New topics can be found in the appendices, including finite-size scaling behavior of Bose-Einstein condensates, a summary of thermodynamic assemblies and associated statistical ensembles, and pseudorandom number generators. Other chapters are dedicated to two new topics, the thermodynamics of the early universe and the Monte Carlo and molecular dynamics simulations.

This book is invaluable to students and practitioners interested in statistical mechanics and physics.

• Bose-Einstein condensation in atomic gases
• Thermodynamics of the early universe
• Computer simulations: Monte Carlo and molecular dynamics
• Correlation functions and scattering
• Fluctuation-dissipation theorem and the dynamical structure factor
• Chemical equilibrium
• Exact solution of the two-dimensional Ising model for finite systems
• Degenerate atomic Fermi gases
• Exact solutions of one-dimensional fluid models
• Interactions in ultracold Bose and Fermi gases
• Brownian motion of anisotropic particles and harmonic oscillators

Readership

Graduate and Advanced Undergraduate Students in Physics. Researchers in the field of Statisical Physics

Chapter 1: The Statistical Basis of Thermodynamics

1.1 The macroscopic and the microscopic states

1.2 Contact between statistics and thermodynamics: physical significance of the number Ω(N, V, E)

1.3 Further contact between statistics and thermodynamics

1.4 The classical ideal gas

1.5 The entropy of mixing and the Gibbs paradox

1.6 The “correct” enumeration of the microstates

Problems

Chapter 2: Elements of Ensemble Theory

2.1 Phase space of a classical system

2.2 Liouville’s theorem and its consequences

2.3 The microcanonical ensemble

2.4 Examples

2.5 Quantum states and the phase space

Problems

Chapter 3: The Canonical Ensemble

3.1 Equilibrium between a system and a heat reservoir

3.2 A system in the canonical ensemble

3.3 Physical significance of the various statistical quantities in the canonical ensemble

3.4 Alternative expressions for the partition function

3.5 The classical systems

3.6 Energy fluctuations in the canonical ensemble: correspondence with the microcanonical ensemble

3.7 Two theorems — the “equipartition” and the “virial”

3.8 A system of harmonic oscillators

3.9 The statistics of paramagnetism

3.10 Thermodynamics of magnetic systems: negative temperatures

Problems

Chapter 4: The Grand Canonical Ensemble

4.1 Equilibrium between a system and a particle-energy reservoir

4.2 A system in the grand canonical ensemble

4.3 Physical significance of the various statistical quantities

4.4 Examples

4.5 Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles

4.6 Thermodynamic phase diagrams

4.7 Phase equilibrium and the Clausius–Clapeyron equation

Problems

Chapter 5: Formulation of Quantum Statistics

5.1 Quantum-mechanical ensemble theory: the density matrix

5.2 Statistics of the various ensembles

5.3 Examples

5.4 Systems composed of indistinguishable particles

5.5 The density matrix and the partition function of a system of free particles

Problems

Chapter 6: The Theory of Simple Gases

6.1 An ideal gas in a quantum-mechanical microcanonical ensemble

6.2 An ideal gas in other quantum-mechanical ensembles

6.3 Statistics of the occupation numbers

6.4 Kinetic considerations

6.5 Gaseous systems composed of molecules with internal motion

6.6 Chemical equilibrium

Problems

Chapter 7: Ideal Bose Systems

7.1 Thermodynamic behavior of an ideal Bose gas

7.2 Bose-Einstein condensation in ultracold atomic gases

7.3 Thermodynamics of the blackbody radiation

7.4 The field of sound waves

7.5 Inertial density of the sound field

7.6 Elementary excitations in liquid helium II

Problems

Chapter 8: Ideal Fermi Systems

8.1 Thermodynamic behavior of an ideal Fermi gas

8.2 Magnetic behavior of an ideal Fermi gas

8.3 The electron gas in metals

8.4 Ultracold atomic Fermi gases

8.5 Statistical equilibrium of white dwarf stars

8.6 Statistical model of the atom

Problems

Chapter 9: Thermodynamics of the Early Universe

9.1 Observational evidence of the Big Bang

9.2 Evolution of the temperature of the universe

9.3 Relativistic electrons, positrons, and neutrinos

9.4 Neutron fraction

9.5 Annihilation of the positrons and electrons

9.6 Neutrino temperature

9.7 Primordial nucleosynthesis

9.8 Recombination

9.9 Epilogue

Problems

Chapter 10: Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions

10.1 Cluster expansion for a classical gas

10.2 Virial expansion of the equation of state

10.3 Evaluation of the virial coefficients

10.4 General remarks on cluster expansions

10.5 Exact treatment of the second virial coefficient

10.6 Cluster expansion for a quantum-mechanical system

10.7 Correlations and scattering

Problems

Chapter 11: Statistical Mechanics of Interacting Systems: The Method of Quantized Fields

11.1 The formalism of second quantization

11.2 Low-temperature behavior of an imperfect Bose gas

11.3 Low-lying states of an imperfect Bose gas

11.4 Energy spectrum of a Bose liquid

11.5 States with quantized circulation

11.6 Quantized vortex rings and the breakdown of superfluidity

11.7 Low-lying states of an imperfect Fermi gas

11.8 Energy spectrum of a Fermi liquid: Landau’s phenomenological theory21

11.9 Condensation in Fermi systems

Problems

Chapter 12: Phase Transitions: Criticality, Universality, and Scaling

12.1 General remarks on the problem of condensation

12.2 Condensation of a van der Waals gas

12.3 A dynamical model of phase transitions

12.4 The lattice gas and the binary alloy

12.5 Ising model in the zeroth approximation

12.6 Ising model in the first approximation

Statistical Mechanics Physics Pdf

12.7 The critical exponents

12.8 Thermodynamic inequalities

12.9 Landau’s phenomenological theory

12.10 Scaling hypothesis for thermodynamic functions

12.11 The role of correlations and fluctuations

12.12 The critical exponents ν and η

12.13 A final look at the mean field theory

Problems

Chapter 13: Phase Transitions: Exact (or Almost Exact) Results for Various Models

13.1 One-dimensional fluid models

13.2 The Ising model in one dimension

13.3 The n-vector models in one dimension

13.4 The Ising model in two dimensions

13.5 The spherical model in arbitrary dimensions

13.6 The ideal Bose gas in arbitrary dimensions

13.7 Other models

Problems

Chapter 14: Phase Transitions: The Renormalization Group Approach

14.1 The conceptual basis of scaling

14.2 Some simple examples of renormalization

14.3 The renormalization group: general formulation

14.4 Applications of the renormalization group

14.5 Finite-size scaling

Problems

Chapter 15: Fluctuations and Nonequilibrium Statistical Mechanics

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15.1 Equilibrium thermodynamic fluctuations

15.2 The Einstein–Smoluchowski theory of the Brownian motion

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15.3 The Langevin theory of the Brownian motion

15.4 Approach to equilibrium: the Fokker–Planck equation

15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem

15.6 The fluctuation–dissipation theorem

15.7 The Onsager relations

Problems

Chapter 16: Computer Simulations

16.1 Introduction and statistics

16.2 Monte Carlo simulations

16.3 Molecular dynamics

16.4 Particle simulations

16.5 Computer simulation caveats

Problems

Details

Reviews

'An excellent graduate-level text. The selection of topics is very complete and gives to the student a wide view of the applications of statistical mechanics. The set problems reinforce the theory exposed in the text, helping the student to master the material' --Francisco Cevantes

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'Making sense out of the world around us in one of the most appealing facets of physics. One may start by putting together seemingly isolated observations and as the different pieces start to fall into place, more complicated arrangements and more fundamental explanations are sought. This is indeed the case for instance when trying to understand the behaviour of a collection of particles. On the one hand, thermo- dynamics provides us with a satisfactory explanation of the macroscopic phenomena observed, however, in order to get to the core of the physical system it becomes necessary to take into account the microscopic constituents of the system as well as the fact that quantum mechanical effects are at play. This is the realm of statistical mechanics and the subject of one of the most widely recognised textbooks around the globe: Pathria’s Statistical Mechanics.…The original style of the book is kept, and the clarity of explanations and derivations is still there. I am convinced that this third edition of Statistical Mechanics will enable a number of new generations of physicists to gain a solid background of statistical physics and that can only be a good thing.' --Contemporary Physics

Statistical Mechanics explores the physical properties of matter based on the dynamic behavior of its microscopic constituents. After a historical introduction, this book presents chapters about thermodynamics, ensemble theory, simple gases theory, Ideal Bose and Fermi systems, statistical mechanics of interacting systems, phase transitions, and computer simulations.

This edition includes new topics such as BoseEinstein condensation and degenerate Fermi gas behavior in ultracold atomic gases and chemical equilibrium. It also explains the correlation functions and scattering; fluctuationdissipation theorem and the dynamical structure factor; phase equilibrium and the Clausius-Clapeyron equation; and exact solutions of one-dimensional fluid models and two-dimensional Ising model on a finite lattice. New topics can be found in the appendices, including finite-size scaling behavior of Bose-Einstein condensates, a summary of thermodynamic assemblies and associated statistical ensembles, and pseudorandom number generators. Other chapters are dedicated to two new topics, the thermodynamics of the early universe and the Monte Carlo and molecular dynamics simulations.

This book is invaluable to students and practitioners interested in statistical mechanics and physics.

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