Critical exponents

Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on

These properties of critical exponents are supported by experimental data. The experimental results can be theoretically achieved in mean field theory for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires the renormalization group. Phase transitions and critical exponents appear also in percolation systems. However, here the critical dimension above which mean field exponents are valid is 6 and higher dimensions. Mean field critical exponents are also valid for random graphs, such as Erdős–Rényi graphs, which can be regarded as infinite dimensional systems.

Phase transitions occur at a certain temperature, called the critical temperature ${\displaystyle T_{c}}$. We want to describe the behavior of a physical quantity ${\displaystyle f}$ in terms of a power law around the critical temperature. So we introduce the reduced temperature ${\displaystyle \tau :=(T-T_{c})/T_{c}}$, which is zero at the phase transition, and define the critical exponent ${\displaystyle k}$.

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