I'm confused to when a system in equilibrium is to be found in any one of its acessible states with equal probability (accoriding to the postulate of equal a priori probabilities). Reif in his book Funndamentals of Statistical and Thermal Physics states this principle and soon after gives examples of ensembles of systems he calles in "equilbrium" but withdifferent energy states occupied by greater numbers of systems in the ensemble than others. How does this reconcile with the posulate?
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$\begingroup$ The formulation of you question is unclear. But you may be referring to different statistical ensembles. Equal probability refers to the microcanonical ensemble, a system which is CLOSED (think of a gas in a box that does not exchange anything witht the exterior). You can also imagine a gas in a box, but this time, the box has wall with a given temperature, in this case the system is not closed but energy exchange (between the wall and the gas) happens. We would say that this system is described by a canonical ensemble where every microstate has a different weigth. But closed or not, both .. $\endgroup$– SyroccoCommented Jul 13 at 23:03
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$\begingroup$ systems are in equilibrium, in the sense that their macroscopic quantities (energy, volume, pressure ....) do not evolve anymore. It is just that the equilibrium state is described by different probability distributions according to the constraints (is the system closed, open, ...) $\endgroup$– SyroccoCommented Jul 13 at 23:05
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The point is that the microstates have equal probabilities, while the macrostates may have different probabilities. If the macrostate is defined by some energy property, then, for every energy, there is some number of microstates that lead to that energy. That number is called the multiplicity of the macrostate. In equilibrium, the macrostate with the largest multiplicity is the one most likely to feature, just from the postulate of equal a priori probabilities.
The standard examples are things like two-state paramagnets. There, the microstates are given by a bit string corresponding to whether each little spin is pointing up or down. Then, the total number of spins pointing in each direction is the macroscopic property related to energy, and there are different multiplicities for each. For example, there are many more arrangements for 10 spins to have 5 pointing up and 5 pointing down and for all 10 to point down. If all of the arrangements are equally probable, then it is more likely to find a macrostate with 5 up and 5 down.
In other contexts, total energy is fixed but you can list the macrostates by specifying the fraction of the total energy that each subsystem takes. There may be more arrangements when systems A and B each take half of the energy than when system A takes more energy than system B; then, by the law of equal probability for each microstate, one energetic configuration for the global system will be more likely than another.
In short: one must distinguish micro and macrostates Law of equal probability applies to micro. This implies that macro with largest multiplicity is most likely to be seen at equilibrium.
answered Jul 14 at 18:25