Critical exponent

Critical exponents are used in the theory of continuous phase transitions to describe the behavior of a physical system in the vicinity of the critical point and to classify the phase transition into universality classes. ${\displaystyle k}$

In the case of continuous phase transitions, the order parameter tends to continuously approach zero as the critical temperature approaches from below , and some higher derivatives of the associated thermodynamic potential show a non-analyte (a jump or a divergence). The higher derivatives can e.g. B. the response functions such as the specific heat , the compressibility or the susceptibility . ${\displaystyle \Psi }$

One observes that the behavior of the order parameter and some of these higher derivatives only depends on the reduced temperature , which indicates the scaled distance to the critical temperature of the phase transition. More precisely, these quantities approximately follow  a power law with an exponent : ${\displaystyle \tau =T/T_{C}-1}$ ${\displaystyle T_{C}}$ ${\displaystyle F}$ ${\displaystyle k}$

${\displaystyle F(\tau )\approx \tau ^{k}=\left({\frac {T-T_{C}}{T_{C}}}\right)^{k}}$

It was observed experimentally and theoretically calculated that the value of the exponent only depends on some basic properties of the system. Systems with the same basic properties show the same power behavior with identical exponents at the phase transition in a finite number of quantities. One speaks therefore of universal behavior and critical exponents. Systems with the same critical exponent belong to the same universality class; their phase transition is fully characterized by the specification of the university class.

The critical exponents of a universality class are not independent of each other, but rather connected by laws of scale .

Mathematical definition

In the vicinity of the critical temperature of a continuous phase transition, the behavior of a physical quantity can be specified as a function of the reduced temperature:

${\displaystyle F(\tau )=A\cdot \tau ^{\lambda }\left(1+b\cdot \tau ^{\lambda _{1}}+\cdots \right)}$

This can be described in the vicinity of the critical temperature ( ) in good approximation with a simple power law: ${\displaystyle \tau \approx 0}$

${\displaystyle {\begin{alignedat}{2}F(\tau )&\propto &&\tau ^{-k}&&\quad {\text{für}}\quad \tau >0\\F(\tau )&\propto (-&&\tau )^{-k}&&\quad {\text{für}}\quad \tau <0\end{alignedat}}}$

The definition of the critical exponent depends on the direction from which the critical temperature is approached:

• from above, d. H. from the disordered phase: ${\displaystyle \tau >0\Leftrightarrow T>T_{C}\quad \Rightarrow k\,{\stackrel {\text{def}}{=}}\,\lim _{\tau \to 0,\tau >0}{\log |F(\tau )| \over \log(\tau )}}$

• from below, d. H. from the ordered phase: ${\displaystyle \tau <0\Leftrightarrow T<T_{C}\quad \Rightarrow k'\,{\stackrel {\text{def}}{=}}\,\lim _{\tau \to 0,\tau <0}{\frac {\log |F(\tau )|}{\log(-\tau )}}}$

There is only one single critical exponent (actually ) for the order parameter , since this can only be determined by approximation from the ordered phase to the critical temperature (in the disordered phase the order parameter is by definition zero). ${\displaystyle \beta }$ ${\displaystyle \beta '}$

universality

The critical exponents are (almost) universal, i.e. that is, they do not depend on the details, but only on some basic properties of the physical system under consideration. According to Griffiths' universality hypothesis, which has now been very well confirmed experimentally and numerically, these basic properties are :

• the dimensionality ${\displaystyle D}$
• the internal or spin dimensionality ${\displaystyle d}$
• the range of the interaction.

To determine the range of the interaction, a distinction is only made between short / medium and long range. Universal behavior occurs only with short- and long-range interactions. In the case of medium-range interactions, the exponents can then still depend on the range.

There are also systems that have non- universal critical exponents at the phase transition , e.g. B. frustrated systems .

Relationship with the physical quantities

The most important critical exponents and the associated physical quantities are listed in the following table. The signs of the exponents differ depending on the physical quantity, since the order parameter converges when the temperature approaches the critical temperature, while the specific heat, susceptibility and correlation length diverge.

Critical exponent	Physical size
${\displaystyle \beta }$	Order parameters ${\displaystyle \Psi \approx (-\tau )^{\beta },{\text{da }}\tau <0}$
${\displaystyle \alpha ,\alpha '}$	Specific heat ${\displaystyle C\approx {\begin{cases}\tau ^{-\alpha },&{\text{wenn }}\tau >0,\\(-\tau )^{-\alpha '},&{\text{wenn }}\tau <0.\end{cases}}}$
${\displaystyle \gamma ,\gamma '}$	Susceptibility ${\displaystyle \chi \approx {\begin{cases}\tau ^{-\gamma },&{\text{wenn }}\tau >0,\\(-\tau )^{-\gamma '},&{\text{wenn }}\tau <0.\end{cases}}}$
${\displaystyle \nu ,\nu '}$	Correlation length ${\displaystyle \xi \approx {\begin{cases}\tau ^{-\nu },&{\text{wenn }}\tau >0,\\(-\tau )^{-\nu '},&{\text{wenn }}\tau <0.\end{cases}}}$
${\displaystyle \eta }$	Correlation function ${\displaystyle \left\langle \psi ({\vec {r_{i}}})\psi ({\vec {r_{j}}})\right\rangle \approx \left(|{\vec {r_{i}}}-{\vec {r_{j}}}|^{(D-2+\eta )}\right)^{-1}{\text{ bei }}T=T_{C}}$
${\displaystyle \delta }$	critical isotherm ${\displaystyle {\text{ bei }}T=T_{C}}$

values

The following table lists the critical exponents from experiments and theoretical calculations. In the experiments, two values ​​are given for the coefficients , the upper number representing the measurement for and the lower number representing the measurement for . The abbreviation 'log' stands for a logarithmic singularity . ${\displaystyle \alpha ,\gamma ,\nu }$ ${\displaystyle \tau >0}$ ${\displaystyle \tau <0}$

Critical exponent	${\displaystyle \alpha }$	${\displaystyle \beta }$	${\displaystyle \gamma }$	${\displaystyle \delta }$	${\displaystyle \nu }$	${\displaystyle \eta }$
Experiment: real gas	log log	0.35	1.37 (± 0.2) 1.0 (± 0.3)	4.4 (± 0.4)	0.64 0.64	0
Experiment: magnet	log log	0.34	1.33 (± 0.03) 1.33 (± 0.03)	≥ 4.2	0.65 (± 0.03) 0.65 (± 0.03)	0
Landau theory	0 (jump)	0.5	1	3	0.5	0
Theory: Ising model (D = 2, d = 1, short-range)	log	0.125	1.75	15th	1	0.25
Theory: Ising model (D = 3, d = 1, short-range)	0.11	0.325	1.24	≈ 4.82	0.63	≈ 0.33
Theory: Heisenberg model (D = 3, d = 3, short-range)	?	0.365	1.39	4.80	0.705	≈ 0.034

(Source: Nolting Volume 6, Statistische Physik, Springer Verlag)
The theoretical values ​​for the Ising model (D = 2, d = 1, short-range) can still be precisely determined; for all other theoretical values, approximation methods such as renormalization group calculations must be used.

The most accurately measured value is for the phase transition of superfluid helium (the so-called lambda transition ). This value was determined in a satellite in order to minimize pressure differences in the liquid. The measurement result exactly matches the theoretical prediction that was obtained with the help of the variational perturbation theory. ${\displaystyle \alpha =-0{,}0127}$

Laws of scale

See also : Van der Waals equation

The idea for the scale laws go back to L. P. Kadanoff , who showed them specifically for the Ising model . They were then confirmed quantitatively by renormalization group calculations. The laws of scale are only secured if the free enthalpy and the correlation functions are generalized homogeneous functions .

First of all, it follows from the laws of scale that the direction from which the critical exponent is determined is not decisive:

${\displaystyle \alpha =\alpha ',~\gamma =\gamma ',~\nu =\nu '}$

Further scaling laws now connect the various critical exponents with one another:

${\displaystyle \alpha +2\beta +\gamma =2}$

${\displaystyle \alpha +\beta \cdot (1+\delta )=2}$

${\displaystyle \beta ={\frac {\gamma }{\delta -1}}}$

${\displaystyle \nu ={\frac {\gamma }{2-\eta }}}$.

If the scaling laws are valid, it is sufficient to determine only two exponents in order to use the above. Formulas to calculate the remaining four exponents.

literature

• Phase Transitions and Critical Phenomena , Vol. 1-20, (Academic Press), Eds .: C. Domb, MS Green and JL Lebowitz
• JM Yeomans, Statistical Mechanics of Phase Transitions (Oxford Science Publications, 1992) ISBN 0198517300
• Hagen Kleinert , Critical Properties of Theories ${\displaystyle \varphi ^{4}}$ , World Scientific (Singapore, 2001) ; Paperback ISBN 981-02-4658-7 (also available online here )
• Wolfgang Nolting, Basic Course Theoretical Physics, Volume 6 - Statistical Physics , Springer Verlag

swell

1. ^ RB Griffiths, Phys. Rev. Lett. 24, 1479 (1970)
2. ↑ Gebhardt, Wolfgang / Krey, Uwe: phase transitions and critical phenomena , Vieweg 1980