But the measurement in the experiment always takes some time. Even a short macro time is quite long from a micro point of view. For example, at 0℃ and 1 atmospheric pressure, the gas molecules in the volume of 1 cubic centimeter collide about 10 29 times per second, and even in such a short macro time as 10 (-6) seconds, the collision reaches 10 23 times. Therefore, the physical quantities measured macroscopically are all long-term microscopic average values, which can be considered as. However, this (limit) average value cannot be calculated from micromechanical analysis, because the initial data of phase orbits can not be determined. In order to explain macroscopic physical phenomena by micromechanical analysis, the following basic principles (or basic assumptions) are put forward in statistical mechanics: for a balanced physical system, the average value of physical quantities measured by probability in phase space should be equal to the time average value of this physical quantity along a track.

In order to support the introduction of this basic principle, Boltzmann put forward the so-called ergodic hypothesis, which holds that a phase trajectory can run all over (or fill) the whole energy surface. Later, some people put forward the quasi-ergodic hypothesis that a phase trajectory can arbitrarily approach any point on the energy surface. However, the mathematical research points out that the above ergodic hypothesis cannot be established, and the quasi ergodic hypothesis is not enough to guarantee "phase average = time average". Therefore, the future research on the mathematical basis of statistical mechanics will focus on the condition of "phase average = time average", and the system that meets this condition is called ergodicity or ergodicity. Since 1930s, marked by the work of many mathematicians, such as g d boekhoff, J von Neumann and α я Qin Xin, ergodicity research has become an important branch of mathematics.