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Fluctuations
Equations (2.4) and (2.5) imply that the probability that a given microstate μ occurs is P = exp{ [ F - H(μ))]/kBT} = exp{-S/kB}.
Since the number of different microstates is so huge, we are not only interested in probabilities of individual microstates but also in probabilities of macroscopic variables, such as internal energy U.
Η Η
allstates allstates
U U H ( ) ( ) e / e
We first form the moments (where b = 1/ kBT; the average energy is denoted < U> and U is a fluctuating quantity ),
( 2. 4 )^ F = -kBT lnZ^ (2.5)
( ) /
Z
e
P
kB T
H
Fluctuations
Similar fluctuation relations exist for many other quantities, for example the isothermal susceptibility χ = (∂/∂H)T is related to fluctuations of the magnetization M = ∑σ i as
Phase Transitions: Order Parameter
- The distinguishing feature of most phase transitions is the appearance of a non-zero value of an 'order parameter', i.e. of some property of the system which is non-zero in the ordered phase but identically zero in the disordered phase.
- The order parameter is defined differently in different kinds of physical systems.
- In a ferro-magnet it is simply the spontaneous magnetization.
- In a liquid-gas system it will be the difference in the density between the liquid and gas phases at the transition;
- for liquid crystals the degree of orientational order is telling.
Phase Transitions: Order Parameter
- An order parameter may be a scalar quantity or may be a multicomponent (or even complex) quantity.
- Depending on the physical system, an order parameter may be measured by a variety of experimental methods
- such as neutron scattering, where Bragg peaks of superstructures in antiferromagnets allow the estimation of the order parameter from the integrated intensity,
- oscillating magnetometer measurement directly determines the spontaneous magnetization of a ferromagnet,
- while NMR is suitable for the measurement of local orientational order.
Phase Transition: First Order
- These remarks will concentrate on systems which are in thermal equilibrium and
- which undergo a phase transition between a disordered state and
- one which shows order that is which can be described by an appropriately defined order parameter.
- If the first derivatives of the free energy are discontinuous at the transition temperature Tc the transition is termed first order.
- The magnitude of the discontinuity is unimportant in terms of the classification of the phase transition, but there are diverse systems with either very large or rather small 'jumps'.
Phase Transition: Second Order
- For second order phase transitions first derivatives are continuous;
- These transitions are at some temperature Tc and 'field' H.
- These are characterized by singularities in the second derivatives of the free energy;
- and properties of rather disparate systems can be related by considering not the absolute temperature but rather the reduced distance from the transition ε= abs( 1 – T/Tc );
- Note that in the 1960s and early 1970s the symbol ε was used to denote the reduced distance from the critical point.
Second order Phase Transition
In contrast, at a second order transition the two free energy curves meet tangentially.
Phase Diagrams
- Phase transitions occur as one of several different thermodynamic fields is varied.
- Thus, the loci of all points at which phase transitions occur form phase boundaries in a multidimensional space of thermodynamic fields.
- The classic example of a phase diagram is that of water, shown in pressure temperature space in Fig. 2.4, in which lines of first order transitions separate ice-water, water-steam, and ice-steam.
- The three first order transitions join at a 'triple point', and the water-steam phase line ends at a 'critical point' where a second order phase transition occurs.
Phase Diagrams for Ferromagnets
A much simpler phase diagram than for water occurs for the two-dimensional Ising ferromagnet with Hamiltonian :
where σi = ±1 represents a 'spin' at lattice site i which interacts with nearest neighbors on the lattice with interaction constant J.
In many respects this model has served as a 'fruit fly' system for studies in statistical mechanics.
Phase Diagrams for Ferromagnets
When J >0, At low temperatures a first order transition occurs as H is
swept through zero, and the phase boundary terminates at the
critical temperature Tc as shown in Fig. 2.4.
- In this model it is easy to see, by invoking the symmetry involving reversal of all the spins and the sign of H, that the phase boundary must occur at H = 0
- so that the only remaining 'interesting' question is the location of the critical point.
- Of course, many physical systems do not possess this symmetry.
