Randomness and Integral Forcing

Consider a system described by a multi-dimensional state vector x. The evolution of x is governed by a set of equations in the form of dx/dt. x is a component of x. F(x(t)), the differential forcing of x, is a deterministic function of x. The solution of such a system often exhibits randomness, where the solution at one time is independent of the solution at another more distant time. This study investigates the mechanism responsible for such randomness. We do so by exploring the integral forcing of x, G_T(t) = ∫_t^t+T F(x(t′))dt′, which links the solution at two distant times, t and t+T.

We show that, for a system in equilibrium, G_T(t) can be expressed as G_T(t) = c_T+d_T x(t) +f_T(t), which consists of (apart from the constant c_T) a dissipating component d_Tx(t) with a negative d_T and a fluctuating component f_T(t). This expression aligns with the idea of the fluctuation-dissipation theorem that for a system in equilibrium, anything that generates fluctuations must also damp the fluctuations. We show further that for a sufficiently large value of T, G_T(t) emerges as a unified forcing. This forcing has a dissipating component characterized by d_T = –1 and a fluctuating component that resembles a white noise. The evolution of x from time t to time t+T, which is described by x(t+T)=x(t)+G_T(t) nominally, is then described by x(t+T) = c_T+f_T(t). This evolution is random, since x(t+T) is independent of x(t). This evolution is also irreversible, since the dissipating component of G_T(t) cancels with x(t) little by little and eventually completely by the time when G_T(t) emerges and generates x(t+T). The unified forcing results from interactions of x(t) with other components of x that are completed during the forward integration over the time span [t,t+T). It represents a forcing that cannot be included in the differential forcing F. In general, randomness and irreversibility are inherent features of a multi-dimensional physical system in equilibrium.