# Configuration entropy

In statistical mechanics, configuration entropy is the portion of a system's entropy that is related to the position of its constituent particles rather than to their velocity or momentum. It is physically related to the number of ways of arranging all the particles of the system while maintaining some overall set of specified system properties, such as energy.

It can be shown[1] that the variation of configuration entropy of thermodynamic systems (e.g., ideal gas, and other systems with a vast number of internal degrees of freedom) in thermodynamic processes is equivalent to the variation of the macroscopic entropy defined as dS = δQ/T, where δQ is the heat exchanged between the system and the surrounding media, and T is temperature. Therefore, configuration entropy is the same as macroscopic entropy.

## Calculation

The configurational entropy in the microcanonical ensemble is related to the number of possible configurations W at a given energy E by Boltzmann's entropy formula

${\displaystyle S=k_{B}\,\ln W,}$

where kB is the Boltzmann constant and W is the number of possible configurations. In a more general formulation, if a system can be in states n with probabilities Pn, the configurational entropy of the system is given by

${\displaystyle S=-k_{B}\,\sum _{n=1}^{W}P_{n}\ln P_{n},}$

which in the perfect disorder limit (all Pn = 1/W) leads to Boltzmann's formula, while in the opposite limit (one configuration with probability 1), the entropy vanishes. This formulation is called the Gibbs entropy formula and is analogous to that of Shannon's information entropy.

The mathematical field of combinatorics, and in particular the mathematics of combinations and permutations is highly important in the calculation of configurational entropy. In particular, this field of mathematics offers formalized approaches for calculating the number of ways of choosing or arranging discrete objects; in this case, atoms or molecules. However, it is important to note that the positions of molecules are not strictly speaking discrete above the quantum level. Thus a variety of approximations may be used in discretizing a system to allow for a purely combinatorial approach. Alternatively, integral methods may be used in some cases to work directly with continuous position functions.

A second approach used (most often in computer simulations, but also analytically) to determine the configurational entropy is the Widom insertion method.