Download Statistical Mechanics of Phase Transition

Statistical Mechanics of Phase Transition
Phase transition is a large scale phenomenon involving many atoms and molecules. This
is a consequence of intermolecular interactions (with the sole exception of Bose-Einstein
condensation).
There are few questions one would like to ask about phase transition :
1. What are the characteristics of the phenomenon ?
2. Why does it arise ?
3. How does it happen ?
The most important characteristic of phase transition is that the system exhibits
macroscopic change in response to a microscopic or infinitesimal change in a control
parameter. Let us consider a few examples :
1. Gas-liquid condensation/evaporation
2. Liquid-solid melting/freezing
3. Paramagnetic-ferromagnetic phase transition in solids
4. Order-disorder transition in alloys
5. Phase separation in binary mixtures
6. Superfluid transition in liquid He from normal to superfluid state
7. Superconducting transition
8. Metal-insulator transition
9. Sol-gel transition in polymers
10. Helix-coil transition in DNA
11. Protein folding
Classification of phase transition:
Ehrenfest Classification: The first attempt at classifying phase transitions was the
Ehrenfest classification which grouped phase transitions based on the degree of nonanalyticity involved.
Under this scheme phase transitions are labeled by the lowest derivative of the free
energy that is discontinuous at the transition.
First order phase transition exhibits discontinuity in the 1st derivative of the free
energy with a thermodynamic variable.
Ex.- solid/liquid/gas transitions.
Second order phase transitions have a discontinuity in a 2nd derivative of the free
energy.
Ex.- ferromagnetic phase transition.
Under Ehrenfest classification scheme there could in principle be third, fourth, and
higher order phase transitions.
Modern Classification: The Ehrenfest scheme is an inaccurate method of classifying
phase transitions, for it does not take into account the case where a derivative of free
energy diverges. For instance , in the ferromagnetic transition, the heat capacity diverges
to infinity.
In modern classification scheme, phase transitions are divided into two broad
categories.
1st Order Phase Transition:- Phase transition which involves a latent heat.
Ex.- solid/liquid/gas transitions and Bose-Einstein condensation
Continuous Phase Transition / 2nd Order Phase Transition:- These have no associated
latent heat.
Ex.- ferromagnetic transition and superfluid transition.
Some Other Characteristics of 1st Order Phase Transition:1. Meta-stability:- In 1st order phase transition little bit away from coexistence (the point
at which two phases are in equilibrium ) there exists a state inside the two phase region of
phase diagram where system can stay, in a phase having free energy minimum although
there is some other global minimum, for a finite time. This state is called metastable state
and the phenomenon is known as metastability.
A metastable system with metastable state (1),
unstable state (2) and stable state (3)
2. Hysterisis:- This is another characteristic of 1st order phase transition. Hysterisis
represents history dependence of a physical system. In magnetic transition under applied
magnetic field (1st order) if some amount of field is applied to a miagnetic system having
zero magnetization and then turn it off then the system does not come back to initial state,
some amount of magnetization remains in the system. Further if we apply the magnetic
field in opposite direction then the magnetization comes back to the zero value at a
certain value of the magnetic field . In this way if we keep varying the applied magnetic
field then we will see that the system never comes back to its initial state ( zero
magnetization at zero applied field) and we will see a loop. This phenomenon is known
as hysteresis and the loop is called hysteresis loop.
M
H
Magnetization versus applied magnetic field
Free Energy Surfaces:
T > Tc

T = Tc

T < Tc

H<0


T > Tc



T = Tc
T < Tc

H=0
T > Tc
T = Tc





T < Tc

H>0



The Landau free energy density for various values of T and H. The  indicates the
value of  at which  achieves its global minimum. The right-most column of graphs
depicts the 1st order transition, which occurs for T < Tc as H is varied from a
negative to a positive value. The central row depicts the continuous transition, which
occurs for H = 0 as T is varied from above Tc to below Tc.