A Method for Restoring the Uniqueness of Temperature and Its Application to the Malthus-Verhulst Equation with a Stochastic Term

摘要

A version of the renormalization group (renormgroup) method is developed, which is called the method for restoring the uniqueness of temperature. This method is applied to the Malthus-Verhulst equation with a stochastic term. This equation from mathematical biology is reduced to a quantum field problem for the one-dimensional case. To establish the dependence of the temperature of the stochastic term on the scale of the block-spin variables, the problem is renormalized using the quantum field renormgroup method (the Wilson technique and the minimal subtraction scheme). As a result of renormalization, the dependence of the temperature of the stochastic term on the scale of the block-spin variables turns out to be the same but ambiguous in both cases. To resolve this difficulty, a special procedure for restoring the uniqueness of the temperature dependence is developed; this procedure makes it possible to determine the dependence of the stochastic term temperature on the scale of the block-spin variables and calculate the correlation length.