Statistical mechanics
Statistical mechanics |
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In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in the fields of physics, biology,[1] chemistry, neuroscience,[2] computer science,[3][4] information theory[5] and sociology.[6] Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.[7][8]
Statistical mechanics arose out of the development of
While classical thermodynamics is primarily concerned with
History
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.[9]
The founding of the field of statistical mechanics is generally credited to three physicists:
- Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates
- James Clerk Maxwell, who developed models of probability distribution of such states
- Josiah Willard Gibbs, who coined the name of the field in 1884
In 1859, after reading a paper on the diffusion of molecules by
Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory.
The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884.[14] 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:
"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."
— J. Clerk Maxwell[15]
"Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched.[16] 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.[17] 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.[18]
Principles: mechanics and ensembles
In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:
- The complete state of the mechanical system at a given time, mathematically encoded as a quantum state vector(quantum mechanics).
- An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the 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 disconnect 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
As is usual for probabilities, the ensemble can be interpreted in different ways:[17]
- 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.
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
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. (By contrast, mechanical equilibrium is a state with a balance of forces that has ceased to evolve.) 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
The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the
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
A
A common approach found in many textbooks is to take the equal a priori probability postulate.[18] 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.
- information entropy).[19]
Other fundamental postulates for statistical mechanics have also been proposed.[9][20][21] For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate.[20][21] One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates:[20]
- The probability density function is proportional to some function of the ensemble parameters and random variables.
- Thermodynamic state functions are described by ensemble averages of random variables.
- The entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.
where the third postulate can be replaced by the following:[21]
- At infinite temperature, all the microstates have the same probability.
Three thermodynamic ensembles
There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.[17] 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 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.[22] The Gibbs theorem about equivalence of ensembles[23] was developed into the theory of concentration of measure phenomenon,[24] which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.[25]
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.[18]
Microcanonical | Canonical | Grand canonical | |
---|---|---|---|
Fixed variables | |||
Microscopic features | Number of microstates | Canonical partition function
|
Grand partition function
|
Macroscopic function | Boltzmann entropy
|
Helmholtz free energy | Grand potential |
Calculation methods
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
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.[18]
- A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few square-lattice Ising model in zero field, hard hexagon model.
Monte Carlo
Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a
The Monte Carlo method 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
- 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.[30]
- For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.[30]
- 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
Many physical phenomena 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. These rates are important in 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
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
One approach to non-equilibrium statistical mechanics is to incorporate
- 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 is often used in electronic quantum transport calculations.
- Stochastic Liouville equation.
Near-equilibrium methods
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
This provides an indirect avenue for obtaining numbers such as
A few of the theoretical tools used to make this connection include:
- Fluctuation–dissipation theorem
- Onsager reciprocal relations
- Green–Kubo relations
- Landauer–Büttiker formalism
- Mori–Zwanzig formalism
Hybrid methods
An advanced approach uses a combination of stochastic methods and
Applications
The ensemble formalism 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,[17]
- regression analysis of gravitational orbits,
- ensemble forecasting of weather,
- dynamics of neural networks,
- bounded-rational potential games in game theory and economics.
Statistical physics explains and quantitatively describes
Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep
Quantum statistical mechanics
See also
- Thermodynamics: non-equilibrium, chemical
- Mechanics: classical, quantum
- statistical ensemble
- Numerical methods: Monte Carlo method, molecular dynamics
- Quantum statistical mechanics
- List of notable textbooks in statistical mechanics
- List of important publications in statistical mechanics
- List of statistical mechanics articles
- Laplace transform
Notes
References
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- ^ Berger, Adam L.; Pietra, Vincent J. Della; Pietra, Stephen A. Della (March 1996). "A maximum entropy approach to natural language processing" (PDF). Computational Linguistics. 22 (1): 39–71. INIST 3283782.
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- ^ a b Uffink, Jos (March 2006). Compendium of the foundations of classical statistical physics (Preprint).
- ^ 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.
- OCLC 52358254.
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- ^ James Clerk Maxwell ,Theory of Heat (London, England: Longmans, Green, and Co., 1871), p. 309
- ISBN 978-90-277-1674-3.
- ^ a b c d e f g Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
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Further reading
- Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Waveland Press. ISBN 978-1-4786-1005-2.
- Müller-Kirsten, Harald J W. (2013). Basics of Statistical Physics (PDF). ISBN 978-981-4449-53-3.
- Kadanoff, Leo P. "Statistical Physics and other resources". Archived from the original on August 12, 2021. Retrieved June 18, 2023.
- Kadanoff, Leo P. (2000). Statistical Physics: Statics, Dynamics and Renormalization. World Scientific. ISBN 978-981-02-3764-6.
- Flamm, Dieter (1998). "History and outlook of statistical physics". arXiv:physics/9803005.
External links
- 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.
- Thermodynamics and Statistical Mechanics by Richard Fitzpatrick
- Cohen, Doron (2011). "Lecture Notes in Statistical Mechanics and Mesoscopics". arXiv:1107.0568.
- 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.