Extended Interpretation of the Boltzmann-Maxwell Equation (A Theory of Nonstationary Random Processes)

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Abstract
Usual theories of statistical mechanics of nonuniform state stand on the assumption of approximately stationary random processes. Generally speaking, this assumption is not reasonable when the state concerned deviates considerably from thermal equilibrium. For a system in such a state, it is difficult not only to define but also to observe a complete set of gross variables. In this case the deviations of the actual future values of the gross variables from their expected future values cannot be neglected. A new method of coarse‐graining is proposed as available to such a state. As a concrete example of system we take a gas. We get a coarse‐grained function 〈f〉D by averaging the fine‐grained probability function of one molecule over a series of observations made on the system on different occasions where at each observation the system is initially in a common macroscopic state as determined by our incomplete set of initial conditions. It is shown that 〈f〉D satisfies the Boltzmann‐Maxwell equation. The equation is valid even when the state concerned deviates considerably from thermal equilibrium and the random processes are not stationary. In this situation, however, the interpretation of the equation is different from the usual one; i.e., the equation predicts not the result of an individual observation of the gas concerned but the result obtained by averaging over a series of observations made repeatedly on different occasions on the same gas under a common incomplete set of macroscopic initial conditions where, with the lapse of time from the start, the result of each observation deviates from the others even in the macroscopic sense. For the time being, the theory is given only in the sense of classical mechanics.