From a purely mathematical point of view, however, the two sets of functions are not necessarily the same for ε ε > 0 (for ε ε = 0 they are equal by definition). Indeed, proceeding systematically in the hierarchy, one finds that similarly to f ( 0 ) , each f ( n ) also contains arbitrary functions of r and t whose relation to the physical hydrodynamic fields is a priori unknown. One of the key simplifying assumptions of Chapman–Enskog theory is to assume that these otherwise arbitrary functions can be written in terms of the exact hydrodynamic fields and their spatial gradients. In other words, the space and time dependence of f enters only implicitly through the hydrodynamic fields. This statement is physically plausible because small Knudsen numbers correspond to the hydrodynamic regime, in which the state of the gas is determined solely by the hydrodynamic fields. In the case of f ( 0 ) , the functions n ′ ( r , t ) , v 0 ′ ( r , t ) , and T ′ ( r , t ) are assumed exactly equal to the physical hydrodynamic fields.