This paper presents a study of collective motion in self-propelled particle systems, inspired by biological phenomena such as flocking, schooling, and pedestrian movement. The authors investigate how collective motion emerges from local interactions in a noisy environment, using a model of self-propelled particles (SPPs) that mimic the behavior of organisms. The model is based on particles aligning their direction of motion with the average direction of their neighbors, with some randomness introduced to simulate environmental noise. The study shows that even in one-dimensional systems, an ordered phase can exist for finite noise levels, a finding that contrasts with equilibrium systems where order is destroyed at finite temperatures. The model is compared to equilibrium systems like ferromagnets, where the analogy lies in the alignment of directions, but the dynamics are different due to the non-equilibrium nature of the SPP system. The paper discusses the phase transitions in two and three dimensions, showing that the SPP system exhibits different behavior from equilibrium systems, with critical exponents and scaling laws that differ from mean-field predictions. In one dimension, the model is analyzed in detail, showing that the system can exhibit ordered motion even with noise, and that the critical line in the density-noise plane follows a power-law relationship. The authors also develop a continuum theory to describe the SPP system, which includes equations for the velocity and density fields. They analyze the stability of domain wall solutions and show that the system can exhibit ordered motion due to the instability of these walls. The study highlights the similarities and differences between SPP systems and equilibrium systems, showing that SPP systems can exhibit critical phenomena similar to those in equilibrium systems, but with distinct characteristics. The results suggest that SPP systems can be well characterized using the framework of classical critical phenomena, but also show surprising features when compared to analogous equilibrium systems.This paper presents a study of collective motion in self-propelled particle systems, inspired by biological phenomena such as flocking, schooling, and pedestrian movement. The authors investigate how collective motion emerges from local interactions in a noisy environment, using a model of self-propelled particles (SPPs) that mimic the behavior of organisms. The model is based on particles aligning their direction of motion with the average direction of their neighbors, with some randomness introduced to simulate environmental noise. The study shows that even in one-dimensional systems, an ordered phase can exist for finite noise levels, a finding that contrasts with equilibrium systems where order is destroyed at finite temperatures. The model is compared to equilibrium systems like ferromagnets, where the analogy lies in the alignment of directions, but the dynamics are different due to the non-equilibrium nature of the SPP system. The paper discusses the phase transitions in two and three dimensions, showing that the SPP system exhibits different behavior from equilibrium systems, with critical exponents and scaling laws that differ from mean-field predictions. In one dimension, the model is analyzed in detail, showing that the system can exhibit ordered motion even with noise, and that the critical line in the density-noise plane follows a power-law relationship. The authors also develop a continuum theory to describe the SPP system, which includes equations for the velocity and density fields. They analyze the stability of domain wall solutions and show that the system can exhibit ordered motion due to the instability of these walls. The study highlights the similarities and differences between SPP systems and equilibrium systems, showing that SPP systems can exhibit critical phenomena similar to those in equilibrium systems, but with distinct characteristics. The results suggest that SPP systems can be well characterized using the framework of classical critical phenomena, but also show surprising features when compared to analogous equilibrium systems.