Restoring the Fluctuation–Dissipation Theorem in Kardar–Parisi–Zhang Universality Class through a New Emergent Fractal Dimension

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. Notably, these questions were answered exactly only for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> dimensions. In this work, we propose a new FDT valid for the KPZ problem in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>n</mi></msub></semantics></math></inline-formula>. We present relations between the KPZ exponents and two emergent fractal dimensions, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>f</mi></msub></semantics></math></inline-formula>, of the rough interface, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>n</mi></msub></semantics></math></inline-formula>. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, the surface fractal dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>f</mi></msub></semantics></math></inline-formula> and, through our relations, the noise fractal dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>n</mi></msub></semantics></math></inline-formula>. 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.