In mathematics, the Kardar–Parisi–Zhang (KPZ) equation is a non-linear stochastic partial differential equation, introduced by Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang in 1986. It describes the temporal change of a height field h ( x → , t ) {\displaystyle h({\vec {x}},t)} with spatial coordinate x → {\displaystyle {\vec {x}}} and time coordinate t {\displaystyle t} :
∂ h ( x → , t ) ∂ t = ν ∇ 2 h + λ 2 ( ∇ h ) 2 + η ( x → , t ) . {\displaystyle {\frac {\partial h({\vec {x}},t)}{\partial t}}=\nu \nabla ^{2}h+{\frac {\lambda }{2}}\left(\nabla h\right)^{2}+\eta ({\vec {x}},t)\;.}
Here, η ( x → , t ) {\displaystyle \eta ({\vec {x}},t)} is white Gaussian noise with average
⟨ η ( x → , t ) ⟩ = 0 {\displaystyle \langle \eta ({\vec {x}},t)\rangle =0} and second moment
⟨ η ( x → , t ) η ( x → ′ , t ′ ) ⟩ = 2 D δ d ( x → − x → ′ ) δ ( t − t ′ ) , {\displaystyle \langle \eta ({\vec {x}},t)\eta ({\vec {x}}',t')\rangle =2D\delta ^{d}({\vec {x}}-{\vec {x}}')\delta (t-t'),}
ν {\displaystyle \nu } , λ {\displaystyle \lambda } , and D {\displaystyle D} are parameters of the model, and d {\displaystyle d} is the dimension. In one spatial dimension, the KPZ equation corresponds to a stochastic version of Burgers' equation with field u ( x , t ) {\displaystyle u(x,t)} via the substitution u = − λ ∂ h / ∂ x {\displaystyle u=-\lambda \,\partial h/\partial x} . Via the renormalization group, the KPZ equation is conjectured to be the field theory of many surface growth models, such as the Eden model, ballistic deposition, and the weakly asymmetric single step solid on solid process (SOS) model. A rigorous proof has been given by Bertini and Giacomin in the case of the SOS model.
KPZ universality class Many interacting particle systems, such as the totally asymmetric simple exclusion process, lie in the KPZ universality class. This class is characterized by the following critical exponents in one spatial dimension (1 + 1 dimension): the roughness exponent α = 1 2 {\displaystyle \alpha ={\tfrac {1}{2}}} , growth exponent β = 1 3 {\displaystyle \beta ={\tfrac {1}{3}}} , and dynamic exponent z = 3 2 {\displaystyle z={\tfrac {3}{2}}} . In order to check if a growth model is within the KPZ class, one can calculate the width of the surface:
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