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Macrostates and Microstates
Published in Jeffrey Olafsen, Sturge’s Statistical and Thermal Physics, 2019
The multiplicity Ω is important for the following reason. The fundamental assumption of statistical mechanics is that a macrosystem in a given macrostate is equally likely to be in any of theΩmicrostates which constitute that macrostate. When we observe a particular macrostate in an experiment, we are averaging over all the microstates that it comprises, giving equal weight (often called the a priori probability) to each. In quantum theory, the assumption that all states of the same energy are equally likely to be occupied is best formulated, and can be justified, in terms of density matrix theory, which is beyond the scope of this book. 2 In classical statistical mechanics, this assumption follows from the ergodic hypothesis. The ergodic hypothesis has gone through a number of metamorphoses, 3 but in essence states that a macrosystem passes rapidly through all the Ω possible microstates, spending an equal amount of time in each. Thus, a time average is the same as an average over the microstates. This hypothesis used to be called the ergodic theorem, until it was shown to be not strictly true. However, in a large system, the ergodic hypothesis is an excellent approximation to the truth and we will assume it. It follows that the statistical weight of a macrostate is its multiplicity Ω; by this we mean that, other things being equal, the probability of observing a particular macrostate is proportional to Ω for that macrostate.
On dynamics of Volterra and non-Volterra cubic stochastic operators
Published in Dynamical Systems, 2022
Uygun U. Jamilov, Akhror. Yu. Khamrayev
The trajectory (orbit) of V for an initial value is defined by In statistical physics, an ergodic hypothesis proposes a connection between dynamics and statistics. In the classical theory, the assumption was made that the average time spent in any region of phase space is proportional to the volume of the region in terms of the invariant measure, or, more generally, that time averages may be replaced by space averages. For nonlinear (quadratic) dynamical system (2), (3) Ulam [33] suggested an analogue of a measure-theoretic ergodicity in the form of the following ergodic hypothesis: a QSO V is said to be ergodic if the limit exists for any .