Since this is something that has been on my find for awhile, I wanted to get the thought-ball rolling. I have a couple of theories, but I want to see what other people think. Also, this is going to be a crazy in-depth description of "reincarnation" and why I think it's possible for the most part (so please bear with me).
So basic biology here: Humans are made of tissues (skin, muscle, bone etc) which are made up of cells (skin cells, hair cells, blood cells etc) which are made up of molecules (lipids, amino acids, proteins etc) which are also made up of even smaller particles known as atoms (electrons, protons, neutrons) which are also made up of even much smaller particles known as subatomic particles (quarks and it get’s even smaller in theory.) So this is the order you need to keep in mind. From largest to smallest.
The smallest known particles (atoms) are floating around in clouds of energy. They are never really touching one another. So apply this same principle to the larger scale. One would assume that since nothing’s really touching anything you could simply walk through a wall, but we know that’s impossible. There’s a barrier that separates all things from intersecting one another and turning into one hot mess of atoms. There’s barriers all around us at all times. It’s what keeps the atoms that make up the cells of our arms from colliding and merging with the atoms that make up the structure of the wall. Other wise we’d all just be transient beings, never being able to bind together. Think of a cloud of smoke that just wisps away and never stays in that big mass it once was. It dissipates into nothingness.
So what keeps us together? What prevents those from attracting and merging with some other atom? Sometimes nothing. The term free radical, refers to a lost particle that lost it’s connection with another particle and now it’s wandering about trying to find something to attract to. However, when it does find something to attract to, it often pushes something else out of the way.
This free radical causes a chain-reaction of breakdowns (basically chaos: the unbalance to balance, the disorder to order). This entropy (equalization/breakdown) doesn’t necessarily mean the destruction, but rather the rebuilding of something else, we'll say balance. That free radical is just going to keep going around bumping things out of order until it starts to form something else.
The first law of conservation of energy is that energy cannot be create or destroyed, but rather transformed. So when a person dies, what is really happening? Since our bodies are made of energy (see very first thing I explained) something has to come along (say a free-radical) that has to bounce everything out of order, in order to create something new. The energy itself is not destroyed but rather reformed. As a person dies, we see a deterioration of body tissues, wrinkly skin, loss of bodily function etc etc. Until we see a person die. But what is death? Does it involve a soul? What is a soul? Where will all those subatomic free-radical particles end up?
Look at all the little pictures to make one big picture. All the atoms that make up the molecules, that make up cells, that make up tissues, that make up a person. This large scale being is being de-constructed back into it’s small-scale parts. Everything is there, that was there before, but what happens to all those atoms?
So I pose the question: is it possible that reincarnation is actually the process of one person's (or objects) sub-atomic particles binding together to form another "physical" object and that we are all made from the same energy?
[Note: If you see an incorrect statement, or something inaccurate, please tell me! Apologizing now if I rambled. I'd like to hear what other Atheists have to say about this topic...]
Let's not talk about "energy" as though it = soul, the way some addle-brained New Agers do. The fact is that when a being (plant or animal) dies, it simply comes apart. The molecules are ingested by scavengers and bacteria. The bones will eventually either be eaten by the animals who can digest them or they will simply be worn down by time.
There's nothing mysterious about it at all, and nothing about the process that hints and entering a new body as the same being. That is impossible to the degree of being almost silly.