# Statistical Mechanics

The statistical mechanics was originally a field of application of mechanics . Nowadays the term is often used synonymously with statistical physics and statistical thermodynamics and thus stands for the (theoretical and experimental) analysis of numerous, fundamental properties of systems of many particles ( atoms , molecules , etc.).

I.a. statistical mechanics provides a microscopic foundation of thermodynamics . It is therefore of great importance for chemistry , especially for physical chemistry , in which one speaks of statistical thermodynamics. In addition, it describes a large number of other thermal equilibrium and non-equilibrium properties that are investigated with the help of modern measurement methods (e.g. scattering experiments ).

In (original) statistical mechanics the state of a physical system is not determined by the trajectories , i.e. H. characterized by the temporal course of locations and impulses of the individual particles or their quantum mechanical states , but rather by the probability of finding such microscopic states .

The statistical mechanics arose primarily through the work of James Clerk Maxwell , Ludwig Boltzmann and Josiah Willard Gibbs , the latter coining the term.

## Central terms

In the following, some terms from statistical physics will be explained, which play an important role in the analysis of properties of thermal equilibrium .

Historically of central importance is the Boltzmann entropy formula (which is also engraved on Ludwig Boltzmann's tombstone ):

${\ displaystyle S = k _ {\ mathrm {B}} \ ln \ Omega.}$ Marked here

• S the (statistical) entropy of a closed system , d. H. of a micro-canonical ensemble .
• ${\ displaystyle k _ {\ mathrm {B}}}$ the Boltzmann constant , like entropy, with the unit joule per Kelvin
• ${\ displaystyle \ Omega}$ the number of microstates (e.g. locations and impulses of all particles in a gas) that are compatible with the thermodynamic state variables of energy, volume and number of particles (Boltzmann called this variable the “complexion number” equal to the statistical weight , sometimes also given as W. , the macroscopic state).

It is thus taken into account that it is not a single microscopic state, but rather all possible states that determine the macroscopic behavior of a physical system.

Statistical ensembles play a crucial role in statistical physics ; a distinction is made between the micro- canonical , the canonical and the grand-canonical ensemble.

A classic and simple example of the application of statistical mechanics is the derivation of the equations of state of the ideal gas and the van der Waals gas .

Are quantum properties ( indistinguishability of the particles) essential, e.g. B. At low temperatures, special phenomena can occur and can be predicted by statistical physics. For example, for systems with integer spin ( bosons ), the Bose-Einstein statistics apply . Below a critical temperature and if the interactions between the particles are sufficiently weak, a special effect occurs in which a large number of particles assume the state of lowest energy, there is a Bose condensation . In contrast, systems with half-integer spin ( fermions ) obey the Fermi-Dirac statistics . Because of the Pauli principle , states of higher energy are also assumed. There is a characteristic upper "energy edge", the Fermi energy . You determined u. a. numerous thermal properties of metals and semiconductors .

The concepts of statistical mechanics can be applied not only to the position and momentum of the particles, but also to others, e.g. B. apply magnetic properties. Here, the modeling of great importance; z. B. attention is drawn to the thoroughly examined Ising model .

## literature

Basics

Textbooks

• Arieh Ben-Naim: Statistical Thermodynamics Based on Information: A Farewell to Entropy. 2008, ISBN 978-981-270-707-9 .
• D. Chandler: Introduction to Modern Statistical Mechanics. 1st ed., Oxford University Press, 1987, ISBN 0-19-504277-8 .
• Torsten Fließbach: Textbook on Theoretical Physics: Statistical Physics. , 2006, ISBN 978-3-8274-1684-1 .
• R. Hentschke: Statistical Mechanics. 1st edition, Wiley-VCH, 2004, ISBN 3-527-40450-3 .
• Wolfgang Nolting, Basic Course Theoretical Physics 6: Statistical Physics. 2005, ISBN 3-540-20505-5 .
• Franz Schwabl: Statistical Mechanics , 2006, ISBN 978-3-54031-095-2 .

Popular scientific literature

Introductions to philosophical topics

• L. Sklar: Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge: CUP 1993.
• D. Albert: Time and Chance. Cambridge, MA: Harvard University Press 2000.
• P. Ehrenfest , T. Ehrenfest : The Conceptual Foundations of the Statistical Approach in Mechanics. Cornell University Press, Ithaca, NY 1959.