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The postulate of equal a priori probabilities

Thermodynamics

The postulate of equal a priori probabilities

flickr.com/Yamanaka Tamaki

The postulate of equal a priori probabilities

The postulate of equal a priori probabilities lies at the heart of statistical mechanics. We try to understand the postulate here as best as we can.

Our general aim here is to study the macroscopic properties of macroscopic systems. By macroscopic systems, we mean systems which are large enough to be visible to the human eye, large enough to contain a considerably large number of particles. And our specific aim here is to understand one of the most fundamental postulates of statistical mechanics – the postulate of equal a priori probabilities.

Defining the state

Say, our macroscopic system is something like a gas inside a sealed box. We wish to describe this gas quantitatively. The most straightforward way to do this is to count the number of particles forming the gas and specify the positions and momenta of each particle, at a specific instant of time. This set of values of the positions and momenta of the gas particles, is what we call the state of the system. If we observe the system at a later instant, the positions and momenta of the particles would have changed, in general, and hence we’d say that the state of the system has changed.

How many states?

With the system set up, let us now make our aim more specific: We want to find the probability of finding our system in a specific state, out of the literally infinite possibilities. Before you scoff us off saying that such a probability would essentially be equal to zero – it’d be like finding the probability of catching a specific mosquito out of the billions that exist out there. But you must agree to the fact that the problem becomes much more solvable if we were to find the probability of finding a mosquito belonging to a specific species, rather than hunt a particular individual. Therefore, we impose certain constraints on our system. Our system has a volume V, containing N number of particles and the energy of the system lies anywhere in the margin E+\Delta E. With these impositions, the number of possible states the system can exist in, reduces.

How many systems?

Now, if you look at the problem from a slightly different perspective, you will appreciate that calculating the probability of our system having a specific state (under the given constraints) is equivalent to calculating the probability of picking a single system, out of a large number of systems, in which each system is in a unique state. I.e., instead of imagining a large number of possibilities for a single system, imagine a large number of systems, each having one possibility. With the mosquito analogy, here you are, instead of finding the number of diseases a single mosquito can spread, you are considering a large number of mosquitoes and assuming that each mosquito spreads only one kind of disease. Then the probability of a particular disease being spread from a single bite of your mosquito, will be equivalent to the probability of any one of the mosquitoes of the large group of mosquitoes, biting (assuming that a single bite will certainly inflict a particular disease!).

Now, this large number of systems isn’t something real. They are mere mental copies of the system that we’re trying to study. All these systems are identical in composition but each has a different state (under the constraints). We call such an imaginary cluster of systems, an ensemble. Our system, that we set out to study, is one of these.

A priori probabilities

We’ve set up this extraordinary setting of systems, only to calculate the probability of finding our system in one of the states that obey the constraints. As explained above, this will be equal to calculating the probability of picking a system from the ensemble, that has any of the required states (under constraints). So, we simply calculate the total number of systems in the ensemble that have these required states and divide by the total number of systems present in the ensemble.

One important detail is that as we calculate the probability, our original system is in thermal equilibrium. Meaning, its “macrostate” does not depend on the passage of time (macrostates involve macro properties such as temperature and pressure) .

Finally, the big question – will the probability of picking a system from the ensemble be the same as that of picking any other? In other words, is it equally likely that our system be in a specific state or in any other, obeying the assumed constraints? The answer is a simple yes. How? Try to answer, why is it that when a coin is tossed randomly, it is equally likely that you get a head or a tail? You’d say that it is common knowledge that a head or a tail are equally likely to turn up. We say the same here: it is obvious or using the specific phrase “a priori”, to conclude that the probability of finding a system in a state is equal to the probability of finding it in any other state of the ensemble. There’s no bias; nothing that would lead us to suppose that one of the states is more likely to be found than another. All states have an equal chance of turning up. Hence, the equal a priori probabilities.

Roshan Sawhil

Roshan Sawhil is a Physics postgraduate who rejoices both doing and explaining Physics. He also finds doing Philosophy as a leisure activity quite interesting. You can find and connect with him on Facebook and Twitter.

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