# Notes on Statistical Thermodynamics A few approaches are described in the following subsections. 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. 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.

### Statistical Thermodynamics

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. 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. 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.

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. 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:. In , 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.

In , 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. Statistical mechanics proper was initiated in the s with the work of Boltzmann, much of which was collectively published in his Lectures on Gas Theory.

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 Fundamentals of Statistical Mechanics — Wikipedia book. From Wikipedia, the free encyclopedia.

Physics of large number of particles' statistical behavior. Particle statistics. Thermodynamic ensembles. Debye Einstein Ising Potts. Main articles: Mechanics and Statistical ensemble. Main articles: Microcanonical ensemble , Canonical ensemble , and Grand canonical ensemble.

Main article: Monte Carlo method. See also: Non-equilibrium thermodynamics. 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.

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.

Introduction to Statistical Thermodynamics

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. Fundamentals of Statistical Mechanics — Wikipedia book Thermodynamics : non-equilibrium , chemical Mechanics : classical , quantum Probability , statistical ensemble Numerical methods: Monte Carlo method , molecular dynamics Statistical physics Quantum statistical mechanics List of notable textbooks in statistical mechanics List of important publications in statistical mechanics.

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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. While a quantum ensemble can contain states with quantum superpositions, a single quantum state cannot be used to represent an ensemble.

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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. From: J. Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons. The Principles of Statistical Mechanics. Dover Publications. Equilibrium and Non-Equilibrium Statistical Mechanics. Physical Review. Bibcode : PhRv.. Uffink, " Compendium of the foundations of classical statistical physics. Fundamentals of Statistical and Thermal Physics.

Journal of Statistical Physics. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs. Exactly solved models in statistical mechanics. Academic Press Inc. Bibcode : JPhC Physical Review B. Bibcode : PhRvB.. Part I. Maxwell, J. Part II.

## Physics 3374 / 6351 - Thermodynamics and Statistical Mechanics

Hoboken, NJ: Wiley. Studies in History and Philosophy of Modern Physics. Published by World Scientific Press. Series on Advances in Statistical Mechanics. Bibcode : stst. Reproduced in The Scientific Papers of J. Willard Gibbs, Vol II , pp. The enigma of probability and physics. Statistical mechanics. Principle of maximum entropy ergodic theory. Share your thoughts with other customers. Write a customer review.

## PHYS / - Thermodynamics and Statistical Mechanics

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