Evolution criterion and renormalisation group for DLA

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Abstract
A renormalisation group method is presented to derive the fractal dimension for the diffusion-limited aggregation (DLA). In a non-equilibrium steady state the dissipation energy and the conductivity of the surface layer play the roles of the free energy and the coupling constant in an equilibrium state. The renormalisation group transformation of the conductivity of the surface layer is obtained in the replacement of the coupling constant. The renormalisation group equation has a non-trivial solution where the derivative at the fixed point has a positive value of less than one. This is consistent with the theorem of minimum entropy production. The fractal dimension is expressed in terms of the derivative of the renormalisation group equation at the fixed point. The fractal dimension df=1.628 is found by using the 2*2 small cell renormalisation.