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STATISTICAL MECHANICS

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STATISTICAL MECHANICS

Statistical mechanics is a branch of physics that applies probability theory, which contains mathematical tools for dealing with large populations, to the study of the thermodynamic behavior of systems composed of a large number of particles.

Statistical mechanics provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic bulk properties of materials that can be observed in everyday life.

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FUNDAMENTAL POSTULATES OF STATISTICAL MECHANICS

1. Any gas may be considered to be composed of

molecules that are in motion and behave like

very small elastic spheres.

2. All cells in the phase space are of equal size.

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FUNDAMENTAL POSTULATES OF STATISTICAL MECHANICS

3. All accessible microstates corresponding to possible macrostates are equally probable. This s called the Postulate of equal a priori probability.

4. The equilibrium state of a gas corresponds to the macrostate of maximum probability.

5. The total number of molecules is constant.

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ENSEMBLES

A system s defined as a collection of a no.of particles.

An ensemble s defined as a collection of large no.of macroscopically identical, but essentially independent systems

Macroscopically identical-Each of the system constituting an ensemble satisfies the sane macroscopic conditions, eg- Volume, energy, pressure, total no.of particles etc.

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ENSEMBLES

Independent systems- The systems constituting an ensemble r mutually non-interacting. In an ensemble system play the same role as the non-interacting molecules do in gas.

In simpler form, A system consisting of a large number of microscopic paricles is called an ENSEMBLE.

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Classical statistics

In classical mechanics all the particles (fundamental and composite particles, atoms, molecules, electrons, etc.) in the system are considered distinguishable.

This means that one can label and track each individual particle in a system.

As a consequence changing the position of any two particles in the system leads to a completely different configuration of the entire system.

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Classical statistics

There is no restriction on placing more than one particle in any given state accessible to the system.

Classical statistics is called Maxwell-Boltzmann statistics (or M-B statistics).

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The Maxwell-Boltzmann Distribution

•At thermal equilibrium, the distribution of particles among the available energy states will take the most probable distribution consistent with the total available energy and total number of particles.

•Every specific state of the system has equal probability.

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The Maxwell-Boltzmann DistributionThe Maxwell-Boltzmann distribution is the classical distribution function for distribution of an amount of energy between identical but distinguishable particles.

There is no restriction on the number of particles which can occupy a given state.

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Quantum statistics

The fundamental feature of quantum mechanics that distinguishes it from classical mechanics is that particles of a particular type are indistinguishable from one another.

This means that in an assembly consisting of similar particles, interchanging any two particles does not lead to a new configuration of the system.

quantum statistics is further divided into the following two classes on the basis of symmetry of the system.

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QUANTUM STATISTICS

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The Fermi-Dirac Distribution

The Fermi-Dirac distribution applies to fermions, particles with half-integer spin which must obey the Pauli exclusion principle.

Each type of distribution function has a normalization term multiplying the exponential in the denominator which may be temperature dependent.

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The Bose-Einstein Distribution

The Bose-Einstein distribution describes the statistical behavior of integer spinparticles (bosons).

At low temperatures, bosons can behave very differently than fermions because an unlimited number of them can collect into the same energy state, a phenomenon called "condensation".

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The Bose-Einstein Distribution

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Comparison between Distributions

Boltzmann FermiDirac

Bose Einstein

indistinguishableZ=(Z1)N/N!

nK<<1

spin doesn’t matter

localized particles don’t overlap

gas moleculesat low densities

“unlimited” number ofparticles per state

nK<<1

indistinguishableinteger spin 0,1,2 …

bosons

wavefunctions overlaptotal symmetric

photons 4He atoms

unlimited number ofparticles per state

indistinguishablehalf-integer spin 1/2,3/2,5/2 …

fermions

wavefunctions overlaptotal anti-symmetric

free electrons in metalselectrons in white dwarfs

never more than 1particle per state

1exp

1

Tk

n

B

k 1exp

1

Tk

n

B

k

Tk

n

B

k exp

1

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APPLIED STATISTICS

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REFERENCE

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disbex.html

http://ocw.mit.edu/courses/physics/8-044-statistical-physics-i-spring-2008/

Theoritical physics by K.Murugesan.

Statistical Mechanics by

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