Crossover transition

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A “crossover” transition is a phenomenon that occurs in physics and other natural sciences , in which a sharp phase transition - which appears only slightly rounded or “smeared” - is only simulated as being sharp.

Mathematical description

The transition phenomenon is typically expressed by the fact that a variable Y, when plotted in a double-logarithmic manner over a variable X, has two adjacent long straight line sections with different slopes, e.g. B. Transition from an imaginary elongated straight line segment with slope 0 to another with slope 1, something like this: ____ , which can be rounded off by the following function (for between 0 and a very large value): ${\ displaystyle \ nearrow}$ ${\ displaystyle X}$

{\ displaystyle Y = 1 + X ^ {2}}

For very small Xs only the first term on the right is important, for very large ones only the second. The result is a straight line with a zero gradient through Y = 1. On the other hand, a straight line with a slope of 2 results from a logarithmic plot, or another finite value when the power X ² changes . The transition between straight line segments is not "sharp", but "rounded", according to the full description. ${\ displaystyle X \ ll 1}$ ${\ displaystyle Y \ cong 1}$ ${\ displaystyle X \ gg 1}$

The “crossover” transition is therefore only “approximately” localized, namely approximately at the right end of the left straight line segment or at the left end of the right segment.

True and fake interpretation

The transition area between the different straight line sections then resembles a rounded (or “smeared”) real phase transition, with the different straight line gradients being incorrectly interpreted as “critical exponents” to the right and left of this simulated real phase transition and the fictitious intersection of the straight lines incorrectly being interpreted as Place of the simulated phase transition.

In reality, there are two differently characteristic areas of one and the same phase.

Application example

As a specific example, consider a magnetic crystal, such as iron. Below the critical temperature T _C , that the magnetization (more precisely, the expected value of the thermal size) applies when approaching from below so-called thereto. Curie temperature , with a characteristic root law increases ( molecular field theory ) . At the same time, the fluctuations of the magnetization grow, namely according to the law with , until this last one with M achieve comparable magnitude. If that is the case, i. H. as T approaches the critical value T _C further , there is - as can be determined experimentally - a transition to a behavior and , with and This transition from the molecular field slopes, e.g. B. from the actual critical values dominated by the fluctuations, such as to , takes place precisely through a "crossover" transition from the molecular field behavior to the actual critical behavior. ${\ displaystyle \ langle M \ rangle _ {T} \ propto (T_ {C} -T) ^ {1/2}}$ ${\ displaystyle \ langle (\ delta M ^ {2}) \ rangle _ {T} \ propto (T_ {C} -T) ^ {- 1}}$ ${\ displaystyle \ langle (\ delta M ^ {2}) \ rangle _ {T}: = \ langle M ^ {2} \ rangle _ {T} - \ langle M \ rangle _ {T} \, ^ {2 }}$ ${\ displaystyle \ langle M \ rangle _ {T} \ propto (T_ {C} -T) ^ {\ beta}}$ ${\ displaystyle \ langle (\ delta M ^ {2}) \ rangle _ {T} \ propto (T_ {C} -T) ^ {- \ gamma}}$ ${\ displaystyle \ beta \ cong 1/3}$ ${\ displaystyle \ gamma \ cong 4/3 \ ,.}$ ${\ displaystyle \ beta _ {Mol.F} = 1/2}$ ${\ displaystyle \ beta \ cong 1/3}$

This “crossover” phenomenon takes place at a “roughly” defined “crossover” temperature close to T _C , about a few percent below it. It can include considerable differences, such as the difference between the molecular field exponent β = 1/2 and the actual, not exactly known, critical value β ≈ 1/3. In any case, the actual “sharp” phase transition occurs at T _C itself.

Why double logarithmic plot?

The logarithmic plotting of the variable X is essential because this is the only way to create the required long segments, which should generally include many powers of ten. The logarithmic plot of the variable Y for the mathematical conversion of power laws into straight line slopes is also important. In addition, the phenomenon must be additive as a whole, which is due to and fulfilled. ${\ displaystyle \ ln {(X_ {1} \ cdot X_ {2})} = \ ln X_ {1} + \ ln X_ {2}}$ ${\ displaystyle \ ln {(Y_ {1} \ cdot Y_ {2})} = \ ln Y_ {1} + \ ln Y_ {2}}$

About the meaning of the term

It is an important task of theoretical physics to differentiate between such only “simulated” transitions from “smeared real phase transitions”, for example compared to experimental physics . The difference becomes visible in the behavior in the so-called thermodynamic borderline case , where in "real" phase transitions the smearing or rounding must disappear. ${\ displaystyle N \ to \ infty}$

literature

W. Gebhardt, U. Krey: Phase transitions and critical phenomena. Vieweg, Braunschweig 1980, ISBN 3-528-08422-7