# How important is statistical mechanics

The Statistical Mechanics was originally an application area of ​​mechanics. Nowadays the term is often used synonymously Statistical Physics used and thus stands for the (theoretical and experimental) analysis of numerous, fundamental properties of systems of many particles (atoms, molecules, etc.). Among other things, statistical mechanics provides a microscopic foundation for 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 no longer determined by the exact "trajectories", i.e. H. characterized by the exact temporal course of position and momentum, of the individual particles or their pure quantum mechanical state, but rather by the probability of finding such microscopic states.

Statistical mechanics was mainly created through the work of James Clerk Maxwell, Ludwig Boltzmann and Josiah Willard Gibbs, the latter coining the term. In the following, some terms from statistical physics will be explained that play an important role in the analysis of properties of thermal equilibrium.

Historically of central importance is Boltzmann's entropy formula (which is also engraved on Ludwig Boltzmann's tombstone): Marked here S. the (statistical) entropy of a closed system, i.e. one micro-canonical ensembles. The quantity Ω indicates the number of microstates (e.g. locations and impulses of all particles in a gas) that are compatible with the thermodynamic state quantities of energy, volume and number of particles (Boltzmann called this quantity the "complexion number" = statistical weight (sometimes also W) of the macroscopic state). The constant kB. is called the Boltzmann constant and has the unit of entropy [Joule / Kelvin].

So it is taken into account that not a single microscopic condition, but rather all possible states determine the macroscopic behavior of a physical system. Statistical ensembles play a decisive role in statistical physics; a distinction is made between the micro-canonical, the canonical and the grand-canonical ensemble.

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

Are quantum properties (indistinguishability of the particles) essential, e.g. B. at low temperatures, special phenomena can occur and are predicted by statistical physics. For systems with integer spin (bosons), the Bose-Einstein statistics apply. Below a critical temperature and when 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: Bose condensation occurs.

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. It determines, among other things. 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. Modeling is of great importance here; z. B. attention is drawn to the thoroughly examined Ising model.

### Special works

• D. Chandler: Introduction to Modern Statistical Mechanics, 1st ed., Oxford University Press, 1987, ISBN 0195042778.
• R. Hentschke: Statistical Mechanics, 1st ed., Wiley-VCH, 2004, ISBN 3527404503.

### 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, Ithaca, NY: Cornell University Press 1959
• Gibbs, J .: Elementary Principles in Statistical Mechanics, New York: Dover 1960