Statistical weight


In statistical mechanics, the statistical weight is the relative probability of a particular feature of a state. It refers to the configuration entropy of a system. The configuration is the instantaneous arrangement of all the particles of the system throughout the distinct energy levels of the system. The macrostate of the system will reflect those configurations of the particles which are more likely; we assume that the probability of a particle being at each level is equal. It is denoted which can also be referred to as multiplicity and is the of entropy.
where is the total number of particles and the number of particles in any one of the energy levels available. The weight relates the microstate to the number of ways particles can be arranged to match the macrostate it expresses the nanoscopic in the scale of our reality.
If the energy associated with the feature is, the statistical weight is given by the Boltzmann factor, where is the Boltzmann constant and is the temperature in kelvins.
The statistical weight is a convenient shorthand that is often used in transfer matrix solutions of problems in statistical mechanics.
In statistical mechanics, we always seek the number of microstates corresponding to a given macrostate. It is called the thermodynamic probability or the statistical weight of the macrostate and is denoted by. This variable is related to the thermodynamic variable entropy. You will observe that the relation between the entropy and the thermodynamic probability forms the basis of entire statistical analysis.

Derivation of relation between entropy and thermodynamic probability

We now search for a microscopic explanation for S in terms of. We start from the TdS equation
T dS = dU + P dV �dN .....
So,
S =� f .....
Also
= g ......
Note: In adiabatic changes, for example, volume increase with no heat added. dS = 0 i.e. S is constant. Also dnj = 0 so is also constant.
During irreversible changes, the total S in the universe increases. The total of the universe also increases because the equilibrium macrostate must be the maximum.