Christof Wetterich derived an exact evolution equation for the scale dependence of an effective action, which allows for the computation of the effective potential. This equation is particularly useful for theories with massless modes in less than four dimensions, such as those relevant for high-temperature phase transitions in particle physics and critical exponents in statistical mechanics. The average action $ \Gamma_{k} $ is an effective action for field averages, with the average taken over a volume $ \sim k^{-d} $, effectively integrating out high-momentum modes. The equation for the effective potential $ U_{k} $ includes terms involving the wave function renormalization $ Z_{k} $ and the inverse average propagator $ P(q) $, which has a physical interpretation in terms of $ \rho $-dependent masses of small fluctuations. The evolution equation for $ \Gamma_{k} $ is derived using the exact propagator and is expressed as a partial differential equation in terms of the variables $ t $ and $ \rho $. This equation has been shown to produce accurate results for various physical systems, including phase transitions and critical exponents. The equation is related to the renormalization group improved one-loop equation and has been used to describe the Kosterlitz-Thouless phase transition and high-temperature phase transitions in four-dimensional theories. The evolution equation is derived by considering the generating functional $ W_{k}[J] $ and its relation to the effective action $ \Gamma_{k} $ through a Legendre transformation. The equation is shown to be exact and can be truncated for practical use, allowing for the computation of the effective potential. The scale $ k $ acts as an infrared cutoff in the propagator, ensuring that large momentum contributions are exponentially suppressed. The equation is particularly useful for theories with massless modes and has been applied to various systems, including scalar field theories and phase transitions. The derivation and application of this equation have provided valuable insights into the behavior of quantum field theories and their phase transitions.Christof Wetterich derived an exact evolution equation for the scale dependence of an effective action, which allows for the computation of the effective potential. This equation is particularly useful for theories with massless modes in less than four dimensions, such as those relevant for high-temperature phase transitions in particle physics and critical exponents in statistical mechanics. The average action $ \Gamma_{k} $ is an effective action for field averages, with the average taken over a volume $ \sim k^{-d} $, effectively integrating out high-momentum modes. The equation for the effective potential $ U_{k} $ includes terms involving the wave function renormalization $ Z_{k} $ and the inverse average propagator $ P(q) $, which has a physical interpretation in terms of $ \rho $-dependent masses of small fluctuations. The evolution equation for $ \Gamma_{k} $ is derived using the exact propagator and is expressed as a partial differential equation in terms of the variables $ t $ and $ \rho $. This equation has been shown to produce accurate results for various physical systems, including phase transitions and critical exponents. The equation is related to the renormalization group improved one-loop equation and has been used to describe the Kosterlitz-Thouless phase transition and high-temperature phase transitions in four-dimensional theories. The evolution equation is derived by considering the generating functional $ W_{k}[J] $ and its relation to the effective action $ \Gamma_{k} $ through a Legendre transformation. The equation is shown to be exact and can be truncated for practical use, allowing for the computation of the effective potential. The scale $ k $ acts as an infrared cutoff in the propagator, ensuring that large momentum contributions are exponentially suppressed. The equation is particularly useful for theories with massless modes and has been applied to various systems, including scalar field theories and phase transitions. The derivation and application of this equation have provided valuable insights into the behavior of quantum field theories and their phase transitions.