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Considering the action for the theory $\lambda\phi^{4}$ for a massive scalarbosonic field as an entropy functional on the space of coupling constants andon the space of fields, we determine the gradient flows for the scalar field,the mass, and the self-interaction parameter. When the flow parameter isidentified with the energy scale, we show that there exist phase transitionsbetween unbroken exact symmetry scenarios and spontaneous symmetry breakingscenarios at increasingly high energies. Since a non-linear heat equationdrives the scalar field through a {\it reaction-diffusion} process, in generalthe flows are not reversible, mimicking the renormalization group flows of the$c$-theorem; the deformation of the field at increasingly high energies can bedescribed as non-linear traveling waves, or solitons associated to self-similarsolutions

Deforming the theory lambda-phi-4 along the parameters and fields gradient flows