KPZ dynamics from a variational perspective: potential landscape, time behavior, and other issues

Original · EN

The deterministic KPZ equation has been recently formulated as a gradient flow, in a nonequilibrium potential (NEP) Φ[h(x,t)]=[ν2(∇ h)²-λ2∫ₕ₀₍ₓ,₀₎ʰ⁽ˣ,ᵗ⁾dψ(∇ψ)²]. This NEP---which provides at time t the landscape where the stochastic dynamics of h(x,t) takes place---is however unbounded, and its exact evaluation involves all the detailed histories leading to h(x,t) from some initial configuration h₀(x,0). After pinpointing some consequences of these facts, we study the time behavior of the NEP's first few moments and analyze its signatures when an external driving force F is included. We finally show that the asymptotic form of the NEP's time derivative Φ[h] turns out to be valid for any substrate dimensionality d, thus providing a valuable tool for studies in d>1.