The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth
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The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation–dissipation theorem (FDT) for spatial dimension
. Notably, these questions were answered exactly only for
dimensions. In this work, we propose a new FDT valid for the KPZ problem in
dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension
. We present relations between the KPZ exponents and two emergent fractal dimensions, namely
, of the rough interface, and
. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent
, the surface fractal dimension
and, through our relations, the noise fractal dimension
. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.
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