statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

, configuration entropy is the portion of a system's

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

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 Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

momentum In Newtonian mechanics, momentum (: momenta or momentums; 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. ...

, 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 The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...

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 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 as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

, and in particular the

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

of combinations and

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

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,

atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...

s or

molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...

s. 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 Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

{{thermodynamics-stub

Calculation

See also

Notes

References