Velocity auto correlation function of a confined Brownian particle

Motivated by the simple models of molecular motor obeying a linear force-velocity relation, we have studied stochastic dynamics Brownian particle in presence velocity dependent force, $$f_s(1-\frac{v}{v_0})$$ where $$f_{s}$$ is constant. The position and auto correlation functions different situations are calculated exactly. We observed that function shows an exponentially decaying behaviour with time saturates to constant value asymptotic limit, for fixed $$f_s$$ . It attains saturation faster increase value. When confined harmonic well, spectral density exhibits symmetric corresponding damped oscillatory before zero long limit. With viscous coefficient, non-systematic variation observed. Further, sinusoidal driving velocities increases amplitude transient regime. For shift relative average basically thermal contribution total correlation. Moreover, athermal regime, entirely due force.