Biological and biologically-relevant Physics

Thus white noise is uniform in Fourier space, at least for frequencies below the inverse correlation time. Another way of looking at this is that the distribution of white noise is smooth in real space. Therefore it is also uniform with the same distribution under any linear transformation whose Jacobian is 1. The Jacobian of a symmetric Fourier transform (using the prefactor 1/ √ 2π) must have modulus 1 since the product of four Fourier transforms is the identity (two are the identity plus space-inversion). Thus the amplitude is the same in Fourier space as in real space. The equation Fext+Fth = ma is called a Langevin equation. In an isotropic and uniform space Fext = Fext(v) which equals zero at zero velocity. Expanding F about v = 0 gives Fext ≈ −ζv, where the proportionality is negative to prevent v from blowing up as t → ∞. ζ is the viscous friction coefficient and B = 1/ζ is called the mobility; the time constant for velocity decay is the product mB. The damping and forcing terms in Langevin’s equation are not independent; their balance is constrained so that each component of velocity corresponds on average to (1/2)kBT of energy. For given F(t) we can solve the equation of motion via a Laplace transform.