What kind of physical processes are well modeled as poisson processes? For instance, it is easy to understand how Brownian motions appear: when an object receives many small shocks that are roughly i.i.d., the resulting path will look like Brownian motions.
What type of physical processes give rise to Poisson processes?
For instance, with one dimension thought of as time: what kind of physically processes would naturally have the "memoryless property" between spikes?

The Poisson distribution tells you the probability of the number of events in a specified time interval if these events are taking place independently (happening of one of them doesn't affect the probability of happening of other potential events).
I think it's better to explain via an example. Suppose you have a shop, and you want to know what is the distribution of the number of customers you get in one hour. Can you model it with a Poisson distribution? Well, does receiving a customer affect the probability of receiving the next one? Most likely yes, because if there are already some customers in your shop, the next person considering entering into your shop might say this is a busy should, I'll go to another one (on the contrary, one might argue that if there are already some costumers in your shop, it will make your shop to look more attractive). The button line is having already some customers has an effect (either positive or negative) on potential future customers. Thus, in this case, you shouldn't model it with a Poisson distribution. On the other hand, if you have a huge shop with many staff it might be reasonable to assume your shop can accommodate many many customers, and thus having some customers doesn't have a bearing on the likelihood of the next person entering your shop. If that is the assumption you want to make, you can go ahead with using Piusson as the distribution of the number of customers per hour.
