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Description of KPZ interface growth by stochastic Loewner evolution

Yusuke Kosaka Shibasaki

Abstract

In this study, we investigate the relationship between the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) equation and the stochastic Loewner equation (SLE), which is a one parameter family of the conformal mappings involving stochasticity. The author shows the correspondence between 1D KPZ equation with height function $h(x,t)=(3t^2x+x^3)/6t$ and Loewner equation driven by a nonlinear stochastic process, wherein the 1D dynamics of interface growth is characterized by Loewner entropy $S_{Loew}\simeq-\ln{t/κ}$. These results were numerically verified with discussions in relation to the universality in non-equilibrium statistical physics.

Description of KPZ interface growth by stochastic Loewner evolution

Abstract

In this study, we investigate the relationship between the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) equation and the stochastic Loewner equation (SLE), which is a one parameter family of the conformal mappings involving stochasticity. The author shows the correspondence between 1D KPZ equation with height function and Loewner equation driven by a nonlinear stochastic process, wherein the 1D dynamics of interface growth is characterized by Loewner entropy . These results were numerically verified with discussions in relation to the universality in non-equilibrium statistical physics.

Paper Structure

This paper contains 9 sections, 2 theorems, 29 equations, 2 figures.

Table of Contents

  1. Model
  2. One-dimensional KPZ equation
  3. Modified SLE for interface growth
  4. Results
  5. Derivation of KPZ equation from modified SLE
  6. KPZ universality class in terms of Loewner entropy
  7. Numerical simulation
  8. Discussion
  9. Conclusion

Key Result

Theorem 1

The ensemble of the height function $h(x,t)$ of the one-dimensional KPZ equation expressed by Eq. (1) behaves as the same law as that of $h(x,t)=(3t^2x+x^3)/6t$ of the modified SLE described by Eqs. (4) and (5) under the approximation that we omit the $o(t^4)$ term of $t\in[0,1]$. $\blacktrianglelef

Figures (2)

  • Figure 1: Log-log plot of $t$ and $W(x_n,t)$ obtained by the numerical simulation. The parameter settings are noted in the main text. The black solid lines show the scalings suggested by the theory of the KPZ universality class, i.e., $t^{1/3}$ in the short time region and $t^{3/2}$ in the long time region, respectively.
  • Figure 2: The scaling of the probability distribution of the Loewner driving force. The calculation method is described in the main text. The black solid line show the scaling suggested by Eq. (36), i.e., $\exp{\left(-S_{Loew}\right)}=p(\eta_s)\propto t^{1}$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2