Confusing as hell. Sorry, but it is. The problem is, you state this is a 3-d random walk, apparently subject to constraints that the points lie within a unit sphere. No problem with that, at least in theory. But you never state what steps are allowed. And then you have a crazy scheme to generate the steps. Sorry, but you do, and I'm not positive where you came up with that scheme, partially because you never state sufficient information about the random walk to know if what you are doing has even a shred of mathematical justification behind it. I don't think it does. But then, only you know what kind of walk this is, and that has been kept a secret.
So all I can really do is look at your code. Looking...
You start out with 400 particles, all at the center of the sphere. Then you move one particle, chosen at random. So essentially, you are running 400 parallel random walks. No real problem with that, except that it is inefficient as hell to move only one point at a time, but your computer, your big cup of coffee.
Next, for the chosen particle to be CONSIDERED for a move, you pick a location somewhere on the surface of the unit sphere. It is not a uniformly distributed point on that sphere. But then why should that be important? ;-) We don't know why you are doing things this way yet. So where is the point chosen? If we assume theta is interpreted as a longitudonal angle, then you picked some point randomly on the interval [0,2*pi]. So anywhere around the sphere. But at what lattitude? That part is interesting.
The second coordinate in a spherical coordinate system would logically be distributed to lie in the interval [-pi/2,pi/2]. Not uniformly so, because there should be more points near the equator. I suppose one might be able to derive the necessary distribution. I might even do so along the way. But lets see what you got.
phi=1./(cos(2*rand(1,10000)-1));
hist(phi,100)
Hmm. Not even remotely what I might expect. So this is your first problem. But not your only problem.
Next, given that wrongly chosen point on the surface of the sphere, you now form the sum of that coordinate with the (x,y,z) coordinates of some point inside the sphere. If the result of that sum is still inside the sphere, you accept the move, otherwise, you reject it. Ok, that does keep you always inside the sphere, so good in that respect.
Sigh. WHAT? WHY?
So, anyway, It is obvious as to why your points tend to drift to the north pole. You have an insane way of choosing the step in your random walk, that is not remotely uniformly distrbuted in direction. Sorry for calling it insane. But there is some element of that in your choice. ;-) Perhaps you just misinterpreted some scheme you found to generate points on the surface of a sphere. (There are WAY easier ways to generate uniformly distributed points on the surface of a sphere. But I'm not even positive why you are writing your random walk in this way.)
Next, you have some scheme where this random walk seems to involve time. That too is not at all clear what you are doing, or why you did it that way. Sorry. But all I have to go on is your code, and the somewhat confusing comments in the code, as I try to reconcile the code you wrote with the limited description you gave. Hey, I love that you did use fairly frequent comments in the code. Otherwise I'd be totally in the dark. Oh, wait, I still feel like the lights are pretty dim here.
So before I go too far in this quest, I need to understand what the goal is here. What kind of random walk is this? For example, I would point out that depending where one of your particles lies, the next step is not uniformly distributed in either direction or in step length. Is that your goal? You need to provide some help in these things, that is, if you want help, at least from me.
