Statistical mechanics Totally Explained

Everything about Statistical Mechanics totally explained

Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. Statistical mechanics, sometimes called statistical physics, can be viewed as a subfield of physics and chemistry.
It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics and mechanics (classical and quantum) at the microscopic level. In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.
This ability to make macroscopic predictions based on microscopic properties is the main advantage of statistical mechanics over thermodynamics. Both theories are governed by the second law of thermodynamics through the medium of entropy. However, entropy in thermodynamics can only be known empirically, whereas in statistical mechanics, it's a function of the distribution of the system on its micro-states.

Fundamental postulate

The fundamental postulate in statistical mechanics (also known as the equal a priori probability postulate) is the following:
ť Given an isolated system in equilibrium, it's found with equal probability in each of its accessible microstates.

This postulate is a fundamental assumption in statistical mechanics - it states that a system in equilibrium doesn't have any preference for any of its available microstates. Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1/Ω.
   This postulate is necessary because it allows one to conclude that for a system at equilibrium, the thermodynamic state (macrostate) which could result from the largest number of microstates is also the most probable macrostate of the system.
   The postulate is justified in part, for classical systems, by Liouville's theorem (Hamiltonian), which shows that if the distribution of system points through accessible phase space is uniform at some time, it remains so at later times.
   Similar justification for a discrete system is provided by the mechanism of detailed balance.
   This allows for the definition of the information function (in the context of information theory):
ť

I = sum_i 
ho_i ln
ho_i = langle ln 
ho 
angle.

When all rhos are equal, I is minimal, which reflects the fact that we've minimal information about the system. When our information is maximal, for example one rho is equal to one and the rest to zero (we know what state the system is in), the function is maximal.
This "information function" is the same as the reduced entropic function in thermodynamics.

Microcanonical ensemble

Since the second law of thermodynamics applies to isolated systems, the first case investigated will correspond to this case. The Microcanonical ensemble describes an isolated system.
   The entropy of such a system can only increase, so that the maximum of its entropy corresponds to an equilibrium state for the system.
   Because an isolated system keeps a constant energy, the total energy of the system doesn't fluctuate. Thus, the system can access only those of its micro-states that correspond to a given value E of the energy. The internal energy of the system is then strictly equal to its energy.
   Let us call

Omega(E)

the number of micro-states corresponding to this value of the system's energy. The macroscopic state of maximal entropy for the system is the one in which all micro-states are equally likely to occur during the system's fluctuations. ť :

S=k_Bln left(Omega (E) 
ight)

ť where

S

is the system entropy, ť

k_B

is Boltzmann's constant

Canonical ensemble

Invoking the concept of the canonical ensemble, it's possible to derive the probability

P_i

that a macroscopic system in thermal equilibrium with its environment, will be in a given microstate with energy

E_i

according to the Boltzmann distribution:
ť :

P_i =

A Hookian spring!
This result is known as the Entropic Spring Result and amounts to saying that upon stretching a polymer chain you're doing work on the system to drag it away from its (preferred) equilibrium state. An example of this is a common elastic band, composed of long chain (rubber) polymers. By stretching the elastic band you're doing work on the system and the band behaves like a conventional spring. What is particularly astonishing about this result however, is that the work done in stretching the polymer chain can be related entirely to the change in entropy of the system as a result of the stretching.

Further Information

Get more info on 'Statistical Mechanics'.

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