r \propto t = (T - T_{c}) / T_{c}

and distinguish the exponents

α, β, γ, δ, ν, η

M \sim (- t)^{β}, χ \sim ∣ t ∣^{- γ}, C_{s} \sim ∣ t ∣^{- α}, M (t = 0) \sim sgn (h) ∣ h ∣^{1/ δ}, ξ \sim ∣ t ∣^{- ν}, G (r; t = 0) \sim r^{- (D - 2 + η)} .

The Landau approximation predicts these powers by minimizing the Landau free energy and differentiating the equation of state. At an ordinary quartic point,

M^{2} = - r / u

χ^{- 1} = r

above and

2∣ r ∣

below, and

h = u M^{3}

on the critical isotherm. The equilibrium singular free energy is proportional to

- r^{2}

below, giving a heat-capacity jump. The quadratic fluctuation denominator is

κ k^{2} + V^{''}

, so

ν = 1/2

and

η = 0

(α, β, γ, δ, ν, η) = (0, \frac{1}{2}, 1, 3, \frac{1}{2}, 0) .

At a tuned tricritical point, the preceding solution gives

M \propto (- r)^{1/4}

V_{m i n} \propto - ∣ r ∣^{3/2}

and

χ^{- 1} = 4∣ r ∣

below. At

r = 0

h = v M^{5}

. The curvature still vanishes linearly, and the Gaussian gradient term still gives zero anomalous dimension. Therefore

(α, β, γ, δ, ν, η) = (\frac{1}{2}, \frac{1}{4}, 1, 5, \frac{1}{2}, 0) .

These mean-field tricritical exponent calculations require approach along the tuned quartic-zero direction. Fluctuations modify the powers below the appropriate upper critical dimension, and marginal effects can add logarithms at it.