statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

, configuration entropy is the portion of a system's entropy that is related to discrete representative positions of its constituent particles. For example, it may refer to the number of ways that atoms or molecules pack together in a mixture, alloy or glass, the number of conformations of a molecule, or the number of spin configurations in a magnet. The name might suggest that it relates to all possible configurations or particle positions of a system, excluding the entropy of their velocity or

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

, but that usage rarely occurs.

Calculation

If the configurations all have the same weighting, or energy, the configurational entropy is given by Boltzmann's entropy formula :

S = k_B \, \ln W,

where ''k''_''B'' 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 ''P''_''n'', the configurational entropy of the system is given by :

S = - k_B \, \sum_^W P_n \ln P_n,

which in the perfect disorder limit (all ''P''_''n'' = 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 The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy ...

and is analogous to that of Shannon's information entropy. The mathematical field of

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

, and in particular the

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

of combinations and

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s 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, usually denoted as a configurational integral.

Notes

References

* Statistical mechanics Thermodynamic entropy Philosophy of thermal and statistical physics Entropy {{thermodynamics-stub

Calculation

See also

Notes

References