Critical exponent

Critical exponent used in the theory of continuous phase transitions for describing the behavior of a physical system in the vicinity of the critical point and the classification of the phase transition in universality classes.

For continuous phase transitions the order parameter Ψ is when approaching the critical temperature continuously to zero and some higher derivatives of the corresponding thermodynamic potential show a non - Analyzität ( a jump or a divergence). These higher order derivatives may be, for example, the response functions such as the specific heat, or compressibility of the susceptibility. In this case, it is observed that the behavior of the order parameter and some of these higher derivatives is dependent only on the reduced temperature, which indicates the scaled distance from the critical temperature of phase transition. More specifically, these variables F follow approximately a power law with an exponent:

It has been observed to both experimentally and theoretically calculated that the value of the exponent depends only on a few fundamental properties of the system. Systems with the same basic characteristics thus show the phase transition in a finite number of variables, the same behavior with identical power exponent. One speaks therefore of universal behavior and critical exponents. Systems with the same critical exponents belong to the same universality class, and their phase transition is completely characterized by specifying the Universitalitätsklasse. The critical exponent of a universality class are not independent but connected by scaling laws.

Mathematical definition

Near the critical temperature of a continuous phase transition can specify the behavior of a physical quantity as a function of reduced temperature:

In good approximation can describe this near the critical temperature very well with a simple power law:

The definition of the critical exponent is dependent on which direction one approaches the critical temperature:

If one speaks of the disordered phase, is so you will be in the ordered phase. Since the order parameter is in the disordered phase by definition zero, there is only one critical exponents for the order parameter (actually ). One can determine the exponent of the order parameter from one side only.

The universality of the critical exponents

The critical exponents are ( almost) universal, ie they do not depend on the details, but only of some basic properties of the considered physical system. So says the - now very well experimentally and numerically confirmed - universality hypothesis of Griffiths that the critical exponents only:

• The dimensionality,
• The range of the interaction,
• The internal or Spindimensionalität the system

Depend.

To determine the range of the interaction, a distinction only between kurz-/mittel- and long range. Only for short and long-range interactions raises a universal behavior. At medium range interactions, the exponent can then depend on the range. However, there are more systems to demonstrate the non-universal critical exponent at the phase transition, such as frustierte systems.

Relation between the critical exponents and the physical quantities

In the following Table the most important critical exponent and its associated physical parameters are tabulated.

Values ​​of the critical exponent

In the following table the critical exponent from experiments and theoretical calculations are listed. In the experiments, two values ​​for the coefficients are given, the top number of the measurement, and the lower figure represents the measurement of. The abbreviation 'log' stands for a logarithmic singularity.

(Source: Nolting Volume 6, Statistical Physics, Springer Verlag ) The theoretical values ​​for the Ising model (D = 2, d = 1, short range ) are yet be precisely determined, for all other theoretical values ​​must approximation methods such as renormalization group can be used. The most accurately measured value of is -0.0127 for the phase transition of superfluid helium ( the so-called lambda- transition). The value was determined in the satellite in order to minimize differences in pressure in the liquid. The measurement result agrees exactly with the theoretical prediction, which was won with the help of variational perturbation theory.

Scaling laws

The idea for the scaling laws go on L.P. Kadanoff back, which showed it specifically for the Ising model. Quantitatively confirmed by renormalization group you were then. Secured the scaling laws are only if the free energy and the correlation functions are generalized homogeneous functions. First, it follows from the scaling laws that the direction from which the critical exponent is determined not critical:

Other scaling laws now connect the various critical exponents together.

Are the scaling laws in effect extends the determination of two exponents.