Statistical Mechanics

Statistical mechanics makes predictions about the macroscopic properties of a system, such as pressure, temperature, internal energy, heat capacity using information only about the microscopic (or molecular) nature of the system. This is mainly based on the use of statistics since the system under consideration posses a large number of molecules. In order to calculate the thermodynamic, transport and chemical properties it requires (a) the potential energy of interactions between atoms and molecules as a function of spatial configuration and (b) molecular geometry resulting from interactions For e.g let us consider the effect of the pressure of gas on its container. The so called pressure is due to the collisions of gas molecules with the container walls. Although each force exerted on the wall is considered as one collision, however it is almost infinitesimal; since there are 1024 molecule-wall collisions per second for each square centimeter of surface for a gas at standard conditions. The resultant infinite number of collisions of so called forces is the finite force or pressure. So it is obvious that the pressure we obtain, is an average over infinite molecular collisions. It can also be termed as a measured pressure or a longtime average (on a molecular scale) of many molecular collisions. Likewise other macroscopic properties of a system can be related to longtime averages of the corresponding microscopic processes.

The process of identifying the development of a macroscopic theory is to start with a statistical or probabilistic description. It means that we will be no longer be interested in the position and speed of the individual molecules in the container but will keep a information on the probability distribution of the velocities of all molecules. For such a procedure we have to determine the average values of properties and the probability distributions with the available constraints.  The constraints here refer to the states of the overall system such as (a)  fixed temperature, volume, and number of molecules; (b) fixed pressure, volume, and number of molecules. Here we define the ensemble of states as the collection of possible states consistent with the given constraints. Depending on these constraints, certain names of the ensembles are given for e.g the canonical ensemble refers to all states having fixed temperature, volume, and number of molecules. The microcanonical ensemble indicates all states consistent with fixed total energy, volume, and number of molecules. The constraints for the grand canonical ensemble are fixed volume, temperature, and chemical potential  or the partial molar Gibbs energy.

Macroscopic and Microscopic state

A macroscopic state of a gas molecule can be specified by giving the values of temperature, volume, and number of molecules or moles. Similarly the microscopic state requires the position vector and velocity vector of each particle. So for the microstate to be same with the macrostate the following condition must be met:

(a) the number of molecules in the microstate must be the same as the number of
molecules in the macroscopic state;
(b) all the position vectors must reside within the fixed volume
(c) the energy of the microscopic and macroscopic state must be equal

It can be shown that a very large number of microstates are equal to any macroscopic state of the system. Let us consider the distribution of k velocities for a particle. Thus for example, consider the two microstates ( v1,v2,v3…..vk) and ( v1,v3,v2…..vk) for a system of identical particles. The only difference between these two states is that in the second state particle 3 has the velocity that particle 2 had in the first state, and vice versa. Thus there are k! ways this set of velocities could be assigned to the k identical particles. Thus the counting of microstates with the conditions above would over count the number of microstates since we are differentiating between microstates that are indistinguishable.