# Statistical mechanics

**Statistical mechanics**is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of Mechanics, which is concerned with the motion of particles or objects when subjected to a force. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in every day life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum). In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.

Table of contents |

2 Connection with thermodynamics 3 Variable particle number 4 Further development |

## Microscopic Entropy, the Boltzmann Factor and the Partition Function

At the heart of statistical mechanics lies Boltzmann's definition of entropy of a physical system:

- The entropy of a macroscopic state is proportional to the logarithm of the number of microscopic states corresponding to it.

*k*. See microcanonical ensemble.

From this definition it is possible to deduce the fact that, if a system is in contact with a heat bath, the probability of a microstate of energy *E* is proportional to

*T*arises from the fact that the system is in equilibrium with the heat bath (see canonical ensemble). This quantity is called the

**Boltzmann factor**. The probabilities of the various microstates must add to one, and the normalization factor is the partition function:

*i*th microstate of the system. The partition function is a measure of the number of states accessible to the system at a given temperature. See Derivation of the partition function for a proof of Boltzmann's factor and the form of the partition function from first principles.

To sum up, the probability of finding a system at temperature *T* in a particular state with energy *E*_{i} is

## Connection with thermodynamics

The partition function can be used to find the expected (average) value of any microscopic property of the system, which can then be related to macroscopic variables. For instance, the expected value of the microscopic energy *E* is *interpreted* as the microscopic definition of the thermodynamic variable internal energy (*U*)., and can be obtained by taking the derivative of the partition function with respect to the temperature. Indeed,

*E*> as

*U*, the following microscopic definition of internal energy:

- -
*kT*ln*Z*=*U*-*TS*=*F*

Having microscopic expressions for the basic thermodynamic potentials *U* (internal energy), *S* (entropy) and *F* (free energy) is sufficient to derive expressions for other thermodynamic quantities. The basic strategy is as follows. There may be an intensive or extensive quantity that enters explicitly in the expression for the microscopic energy *E*_{i}, for instance magnetic field (intensive) or volume (extensive). Then, the conjugate thermodynamic variables are derivatives of the internal energy. For instance, the macroscopic magnetization (extensive) is the derivative of *U* with respect to the (intensive) magnetic field, and the pressure (intensive) is the derivative of *U* with respect to volume (extensive).

## Variable particle number

This is the version for systems which don't allow an exchange of matter. Otherwise, we would have to introduce chemical potentials, μ_{j}, j=1,...,n and replace the partition function with

_{ij}is the number of j

^{th}species particles in the i

^{th}configuration. Sometimes, we also have other variables to add to the partition function, one corresponding to each conserved quantity. Most of them, however, can be safely interpreted as chemical potentials. For the rest of this article, we will ignore this complication and pretend chemical potentials don't matter. See grand canonical ensemble.

## Further development

where is the average value of property . This equation can be applied to the internal energy, , and pressure, :

Helmholtz free energy: | |

Internal energy: | |

Pressure: | |

Entropy: | |

Gibbs free energy: | |

Enthalpy: | |

Constant Volume Heat Capacity: | |

Constant Pressure Heat Capacity: | |

Chemical potential: |

It is often useful to consider the energy of a given molecule to be distributed among a number of modes. For example, translational energy refers to that portion of energy associated with the motion of the center of mass of the molecule. Configurational energy refers to that portion of energy associated with the various attractive and repulsive forces between molecules in a system. The other modes are all considered to be internal to each molecule. They include rotational, vibrational, electronic and nuclear modes. If we assume that each mode is independent (a very questionable assumption!!!!!) the total energy can be expressed as the sum of each of the components:

Expressions for the various molecular partition functions are shown in the following table.

Nuclear | |

Electronic | |

vibrational | |

rotational (linear) | |

rotational (non-linear) | |

Translational | |

Configurational (ideal gas) |

These equations can be combined with those in the first table to determine the contribution of a particular energy mode to a thermodynamic property. For example the "rotational pressure" could be determined in this manner. The total pressure could be found by summing the pressure contributions from all of the individual modes, ie:

See also:

- Fluctuation Dissipation Theorem
- Mean Field Theory
- Ludwig Boltzmann
- Paul Ehrenfest
- important publiactions in statistical mechanics

General subfields within physics
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Classical mechanics | Condensed matter physics | Continuum mechanics | Electromagnetism | General relativity | Particle physics | Quantum field theory | Quantum mechanics | Solid state physics | Special relativity | Statistical mechanics | Thermodynamics |